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Infinite elliptic hypergeometric series: convergence and difference equations D. I. Krotkova, V. P. Spiridonovba a National Research University Higher School of Economics, Moscow, Russia b Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow Region, Russia
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 12, DOI: https://doi.org/10.4213/sm9874 
Bibliographic databases:
    § 1. IntroductionHypergeometric functions have a very long history [1]. The notion of hypergeometric series was introduced even before Newton proved the binomial theorem for infinite $_1F_0$-series. The systematic study of such functions was launched by Euler, who introduced the gamma function, computed the beta integral in terms of the gamma function, defined the key $_2F_1$-series and considered the second-order differential equation for it. He introduced also the simplest $q$-hypergeometric series and found a number of exact identities for them. A $q$-analogue of the hypergeometric equation was considered by Heine [3] in the middle of the 19th century. The theory of hypergeometric functions was developing further on for a very long time in these two instances, which have found very many applications in various fields of mathematics and theoretical physics. At the turn of the millennium a new type of hypergeometric functions related to elliptic functions was discovered; for a survey, see [11]. It turned out that the properties of elliptic functions explain some of the known beauties of ordinary and $q$-hypergeometric functions. Moreover, at the elliptic level new symmetries appeared and many formulae became much more compact than their degenerate cases known previously. By now, most classical constructions for ordinary hypergeometric functions have been uplifted to the elliptic level both in the univariate and multivariate setting. A major part of applications uses elliptic hypergeometric series, which satisfy by definition a balancing condition imposed on the parameters [8]. Some particular elliptic hypergeometric series arise in applications to the Yang-Baxter equation [2], in the theory of biorthogonal rational functions [10], [11] and in the theory of orthogonal polynomials with spectrum dense on the unit circle [12], [13]. However, some standard constructions are still waiting for proper understanding at the elliptic level. In the present paper we consider one of such problems — a straightforward generalization of higher-order hypergeometric differential equations to finite difference equations of infinite order for theta-hypergeometric series. Additionally, we investigate the convergence of infinite univariate theta hypergeometric and elliptic hypergeometric series for particular restrictions on their parameters. The latter analysis resembles the one performed by Hardy and Littlewood [5] for a particular $q$-hypergeometric series, when $q=\exp(2\pi i\chi)$ for a fixed irrational number $\chi$. A similar convergence problem for infinite series arises at the elliptic level for an arbitrary choice of parameters, in sharp contrast to the $q$-hypergeometric case. We generalize the Hardy-Littlewood criterion to infinite elliptic hypergeometric series and prove the convergence of such series in some particular cases. In particular, we show that for some choice of parameters infinite very-well poised elliptic hypergeometric series ${}_{r+1}V_r$ have a radius of convergence larger than $1$. § 2. Ordinary hypergeometric series and their $q$-analoguesConsider the standard $p$-Pochhammer symbol
$$
\begin{equation}
(a; p)_{\infty}=\prod_{n=0}^\infty (1-ap^n), \qquad (a; p)_s =\frac{(a;p)_\infty}{(ap^s; p)_\infty}, \quad |p|<1,
\end{equation}
\tag{2.1}
$$
and recall the following conventional notation for its $k$-fold product:
$$
\begin{equation*}
(a_1, \dots,a_k; p)_{\infty} :=(a_1; p)_\infty \dotsb (a_k;p)_\infty.
\end{equation*}
\notag
$$
The Jacobi theta function we use is defined by
$$
\begin{equation}
\theta(z; p) :=(z, pz^{-1}; p)_\infty, \qquad z\in\mathbb C^\times,
\end{equation}
\tag{2.2}
$$
and the classical Jacobi triple-product identity gives the following expansion:
$$
\begin{equation}
\theta(z; p)=\frac{1}{(p;p)_\infty}\sum_{n \in \mathbb{Z}}p^{\binom{n}{2}}(-z)^n, \qquad \binom{n}{2}=\frac{n(n-1)}{2}.
\end{equation}
\tag{2.3}
$$
The key reflection and quasiperiodicity properties of this theta function are as follows:
$$
\begin{equation}
\theta(z^{-1}; p)=\theta(pz; p)=-z^{-1}\theta(z; p).
\end{equation}
\tag{2.4}
$$
As a consequence, we have
$$
\begin{equation*}
\theta(p^kz; p)=(-z)^{-k}p^{-\binom{k}{2}}\theta(z; p), \qquad k\in\mathbb Z.
\end{equation*}
\notag
$$
For an integer $n\in \mathbb{Z}_{\geqslant0}$ we define the elliptic shifted factorial, or the elliptic Pochhammer symbol by
$$
\begin{equation}
\theta(t; p; q)_n :=\prod_{m=0}^{n-1}\theta(tq^m; p), \qquad \theta(t; p; q)_0=1,
\end{equation}
\tag{2.5}
$$
and use shorthand notation for the $k$-fold product of elliptic Pochhammer: symbols
$$
\begin{equation*}
\theta(t_1, \dots,t_k; p; q)_n :=\theta(t_1; p; q)_n \dotsb \theta(t_k; p; q)_n.
\end{equation*}
\notag
$$
For $p=0$ we have $\theta(t;0)=1-t$, and the product (2.5) reduces to the standard $q$-Pochhammer symbol In this paper we study the properties of elliptic hypergeometric series. These series can be considered as a natural generalization of classical objects, such as the generalized hypergeometric function [1], which is defined in terms of the Pochhammer symbol $(a)_n\!=\!a(a+1) \dotsb (a+n-1)$ and its $k$-fold product version $(\ell_1, \dots,\ell_k)_n\!=(\ell_1)_n \dotsb(\ell_k)_n$ as follows:
$$
\begin{equation}
_s F_r\biggl(\begin{matrix} a_0, \dots, a_{s-1}\\ b_1, \dots, b_r \end{matrix} \biggm| z\biggr) =\sum_{n=0}^\infty \frac{(a_0, \dots,a_{s-1})_n}{(b_1, \dots,b_r)_n}\frac{z^n}{n!}.
\end{equation}
\tag{2.6}
$$
This function satisfies the following differential equation of finite order in terms of the operator $\delta=z\,{d}/{dz}$
$$
\begin{equation}
\bigl[\delta(\delta+b_1-1) \dotsb (\delta+b_r-1)-z(\delta+a_0) \dotsb (\delta+a_{s-1})\bigr] \cdot _s F_r \biggl(\begin{matrix} a_0, \dots, a_{s-1}\\ b_1, \dots, b_r \end{matrix} \biggm| z\biggr)=0,
\end{equation}
\tag{2.7}
$$
which is easy to verify in a straightforward way. A natural generalization of the $_s F_r$-series, which nowadays is also treated as a classical function, is called the basic, or $q$-hypergeometric series
$$
\begin{equation}
_s \phi_r\biggl(\begin{matrix} t_0, \dots, t_{s-1}\\ w_1, \dots, w_r \end{matrix}; q \biggm| z\biggr)=\sum_{n=0}^\infty \frac{(t_0, \dots,t_{s-1};q)_n}{(w_1, \dots,w_r;q)_n}\, \frac{z^n}{(q;q)_n}.
\end{equation}
\tag{2.8}
$$
This function satisfies the following $q$-difference equation of finite order
$$
\begin{equation}
\begin{aligned} \, \nonumber &\bigl[(1-q^\delta)(1-w_1q^{\delta-1})\dotsb(1-w_rq^{\delta-1}) \\ &\qquad -z(1-t_0q^{\delta})\dotsb(1-t_{s-1}q^{\delta})\bigr]\,{}_s\phi_r \biggl(\begin{matrix} t_0, \dots, t_{s-1}\\ w_1, \dots, w_r \end{matrix}; q \biggm| z\biggr)=0, \end{aligned}
\end{equation}
\tag{2.9}
$$
where the operator $q^\delta$ acts as a $q$-shift operator on arbitrary functions of $z\in\mathbb{C}$: $q^\delta f(z):=f(qz)$. The notation $q^{zd/dz}$ can be justified by the action of this operator on functions admitting a Laurent series expansion:
$$
\begin{equation*}
q^{zd/dz} f(z)=\sum_{k=0}^\infty \frac{(\log q)^k}{k!} \biggl(z\frac{d}{dz}\biggr)^k \sum_{n\in\mathbb{Z}}c_nz^n =\sum_{n\in\mathbb{Z}}c_n\sum_{k=0}^\infty \frac{(\log q)^k}{k!} n^k z^n =f(qz).
\end{equation*}
\notag
$$
For $s=r+1$ our definition of $_{r+1}\phi_r$-series coincides with the one given in [3]. For $s\neq r+1$ it differs by the changes $q, t_j, w_k \to q^{-1}, t_j^{-1}, w_k^{-1}$, and an appropriate scaling of the $z$-variable, respectively. The Pochhammer-Horn type characterization of the ordinary and $q$-hypergeometric series $\sum_{n\in\mathbb{Z}} c_n$ described above consists in the statement that the ratio of consecutive series coefficients $c_{n+1}/c_n$ is a rational function of $n$ and $q^n$, respectively. Such series terminate below in a natural way after a special choice of one of the free parameters resulting from such a definition ($b_0=1$ and $w_0=q$, respectively), which is reflected in the presence of the term $n!$ in (2.6) and the term $(q;q)_n$ in (2.8). The next natural step of the generalization of series of hypergeometric type, which was made relatively recently, is the series characterized by the assumption that $c_{n+1}/c_n$ is either an elliptic function of $n$ with modulus $p=\exp(2\pi i\tau)$ or a meromorphic theta function of $n$ with the same modulus (see [8] and [11]). This class of hypergeometric series degenerates to the basic case for $p=0$ and, thus, to ordinary hypergeometric series for $p=0$ and $q \to 1$. These infinite series formally satisfy finite difference equations of infinite order, as described below. Note that a particular terminating elliptic hypergeometric series, defining per se an elliptic function of special form of the parameters, was considered for the first time by Frenkel and Turaev in [2]. § 3. Theta hypergeometric seriesConsider now the following formal power series:
$$
\begin{equation}
_s E_r\biggl(\begin{matrix} t_0, \dots, t_{s-1}\\ w_1, \dots, w_r\end{matrix}; q, p \biggm| z\biggr) =\sum_{n=0}^\infty \frac{\theta(t_0, \dots,t_{s-1}; p; q)_n}{\theta(q, w_1, \dots,w_r; p; q)_n} z^n
\end{equation}
\tag{3.1}
$$
(see [8]), with a natural restriction on the values of the parameters $w_k$ which ensures that the coefficients do not have singularities: $\theta(w_k;p;q)_n\neq 0$. For $s=0$ or $r=0$ the parameters $t_j$ or $w_k$, respectively, are absent. Lemma 1. The power series $_s E_r$ satisfies the following formal finite difference equation of infinite order: Proof. Using the properties of the elliptic Pochhammer symbol (2.5) one can easily find the ratio of consecutive coefficients of this series: This is a particular meromorphic theta function of the formal variable $n$, since it satisfies the quasiperiodicity relations which is characteristic for theta functions.
Notice now that $\theta(1; p)=0$ and, as follows from (2.3),
$$
\begin{equation*}
\theta(t q^{\delta}; p) z^n=\frac{1}{(p;p)_\infty}\sum_{k \in \mathbb{Z}} p^{k(k-1)/2}(-t q^{\delta})^k z^n =\theta(t q^n; p) z^n.
\end{equation*}
\notag
$$
Hence, in terms of theta functions of the $q$-shifting operator $q^\delta$ we derive the following formal equality
$$
\begin{equation}
\begin{aligned} \, \nonumber &\bigl[\theta(q^\delta, w_1 q^{\delta-1}, \dots, w_r q^{\delta-1}; p)- z \theta(t_0 q^{\delta}, \dots, t_{s-1}q^{\delta}; p)\bigr]\,{}_s E_r \biggl(\begin{matrix} t_0, \dots, t_{s-1}\\ w_1, \dots, w_r \end{matrix}; q, p \biggm| z\biggr) \\ &\qquad =\sum_{n=1}^\infty \frac{\theta(t_0, \dots,t_{s-1}; p; q)_n z^n}{\theta(q, w_1,\dots, w_r;p;q)_{n-1}} - z\sum_{n=0}^\infty \frac{\theta(t_0, \dots,t_{s-1}; p; q)_{n+1} z^{n}} {\theta(q, w_1,\dots, w_r;p;q)_n}=0. \end{aligned}
\end{equation}
\tag{3.3}
$$
Expressing the theta functions of $q^\delta$ in square brackets as power series of $q^\delta$ by the use of Jacobi’s triple product identity (2.3) we obtain the required result. The lemma is proved. Equation (3.2) for the simplest series $_0E_0(q, p\mid z)$ was derived in [9]. For $p=0$ the difference equation of infinite order (3.3) reduces to the finite-order $q$-difference equation for $q$-hypergeometric series (2.9). Now we consider more attentively the coefficients arising in the infinite bilateral sums in (3.2). For some complex parameters $s_j$, $j=1,\dots, r$, one has
$$
\begin{equation}
\begin{aligned} \, \nonumber &\prod_{j=1}^r \theta(s_j q^{\delta}; p) f(z)=\frac{1}{(p; p)_\infty^r} \sum_{m_1, \dots,m_r \in \mathbb{Z}}\prod_{j=1}^r(-s_j)^{m_j} p^{\sum_{j=1}^r \binom{m_j}{2}} f\bigl(zq^{\sum_{j=1}^r m_j}\bigr) \\ &\qquad =\frac{1}{(p; p)_\infty^r} \sum_{n\in\mathbb Z} \Phi_n(s_1, \dots,s_r;p) f(zq^n), \end{aligned}
\end{equation}
\tag{3.4}
$$
where
$$
\begin{equation}
\Phi_n(s_1, \dots,s_r;p) :=\sum_{\substack{m_1, \dots,m_r \in \mathbb{Z} \\m_1+\dots+m_r=n}} (-s_1)^{m_1} \dotsb (-s_r)^{m_r} p^{\binom{m_1}{2}+\dots+\binom{m_r}{2}}.
\end{equation}
\tag{3.5}
$$
As one can notice, $\Phi_n(s_1, \dots,s_r;p)$ is a certain theta function, which depends on $n$ in some way, and $\Phi_0(s_1, \dots,s_r;p)$ is a theta function on the root system $A_{r-1}$. Lemma 2. The following three identities hold true: Proof. For the first relation notice the transformation
$$
\begin{equation*}
\begin{aligned} \, &\Phi_n(s_1, \dots,s_r;p) =\sum_{\substack{m_1, \dots,m_r \in \mathbb{Z} \\m_1+\dots+m_r=n}} (-s_1)^{m_1} \dotsb (-s_r)^{m_r} p^{\binom{m_1}{2}+\dots+\binom{m_r}{2}} \\ &\qquad =\sum_{\substack{m_1, \dots,m_r \in \mathbb{Z} \\m_1+\dots+m_r=0}} (-s_1)^{m_1} \dotsb (-s_j)^{m_j+n} \dotsb (-s_r)^{m_r} p^{\binom{m_1}{2}+\dots+\binom{m_j+n}{2}+\dots+\binom{m_r}{2}} \\ &\qquad =p^{\binom{n}{2}}(-s_j)^n \Phi_0(s_1, \dots,s_j p^n, \dots,s_r;p). \end{aligned}
\end{equation*}
\notag
$$
A similar computation also yields equality (3.7):
$$
\begin{equation*}
\begin{aligned} \, \Phi_{n+r}(s_1, \dots,s_r;p) &=\sum_{\substack{m_1, \dots,m_r \in \mathbb{Z} \\m_1+\dots+m_r=n+r}} (-s_1)^{m_1} \dotsb (-s_r)^{m_r} p^{\binom{m_1}{2}+\dots+\binom{m_r}{2}} \\ &=\sum_{\substack{m_1, \dots,m_r \in \mathbb{Z} \\m_1+\dots+m_r=n}} (-s_1)^{m_1+1} \dotsb (-s_r)^{m_r+1} p^{\binom{m_1+1}{2}+\dots+\binom{m_r+1}{2}} \\ &=(-1)^rs_1 \dotsb s_r p^n \Phi_n(s_1, \dots,s_r;p). \end{aligned}
\end{equation*}
\notag
$$
Notice that the last identity implies the relation
Now for the last part (3.8), consider the expansion
$$
\begin{equation*}
\theta(s_1 z; p)\dotsb\theta(s_r z; p) =\frac{1}{(p;p)_\infty^r}\sum_{n=-\infty}^{\infty} z^n \Phi_n(s_1, \dots,s_r;p)
\end{equation*}
\notag
$$
similar to (3.4). Using that for primitive $r$th roots of unity we have $\sum_{k=0}^{r-1}\zeta^{kn}=0$ for $n\neq \ell r$, a Fourier-type transformation gives
$$
\begin{equation*}
\begin{aligned} \, & \frac{1}{r}\sum_{m=0}^{r-1} \theta(s_1 z \zeta^m; p)\dotsb\theta(s_r z \zeta^m; p)=\frac{1}{(p;p)_\infty^r}\sum_{n=-\infty}^{\infty} z^{nr}\Phi_{nr}(s_1, \dots,s_r; p) \\ &\qquad =\frac{(p^r;p^r)_\infty}{(p;p)_\infty^r} \theta(-(-z)^r s_1 \dotsb s_r; p^r)\Phi_{0}(s_1, \dots,s_r; p), \end{aligned}
\end{equation*}
\notag
$$
where we have used relation (3.9). An explicit expression for the function $\Phi_0(s_1, \dots,s_r; p)$ obviously follows. The lemma is proved. Finally, we arrive at the following statement. Theorem 3. The theta-hypergeometric series (3.1) is a formal power series solution of the infinite order $q$-difference equation Equation (3.10) is obtained simply after applying the operator relation (3.4) to both operator theta function products in square brackets in the first line of (3.3). For example, for $r=2$ we can find an explicit expression for the function $\Phi_0(s_1, \dots,s_r;p)$ that can be obtained from the representation (3.8) after substituting in $z=s_2^{-1}$:
$$
\begin{equation*}
\Phi_0(s_1,s_2;p)=\sum_{m\in\mathbb Z} \biggl(\frac{s_1}{s_2}\biggr)^m p^{m^2} =(p^2;p^2)_\infty \theta\biggl(-p\frac{s_1}{s_2};p^2\biggr).
\end{equation*}
\notag
$$
This means that the formal theta-hypergeometric series
$$
\begin{equation*}
{}_{2}E_1\biggl(\begin{matrix}a,b\\ c\end{matrix}\biggm|z\biggr)
\end{equation*}
\notag
$$
satisfies the following explicit compact equation
$$
\begin{equation}
\sum_{n\in\mathbb Z} (-1)^n p^{\binom{n}{2}} \biggl( \theta\biggl(-\frac{q}{c}p^{n+1};p^2\biggr) - za^n \theta\biggl(-\frac{a}{b}p^{n+1};p^2\biggr) \biggr) {}_2 E_1 \biggl(\begin{matrix} a,b\\ c\end{matrix}; q, p \biggm| q^n z\biggr)=0,
\end{equation}
\tag{3.11}
$$
which can be considered as a direct theta-functional analogue of the hypergeometric equation for the Euler-Gauss $_2F_1$-function. It can take different forms obtained after choosing different parametrization of the $\Phi_n$-coefficients. In particular, one can interchange the parameters $a$ and $b$. Note that in all cases the $q$-shifted functions $f(q^n z)$ enter the equation only with a coefficient linear in the independent variable $z$. Clearly, any solution of (3.11) can be multiplied by an arbitrary elliptic function $\varphi(z)=\varphi(qz)$ remaining a solution of the same equation. Splitting the sum in (3.11) into even and odd values of the index $n$ and eliminating the $p^n$-factors in the arguments of theta functions we arrive at the equation
$$
\begin{equation}
\begin{aligned} \, \nonumber &\sum_{k\in\mathbb Z}\biggl\{ p^{k(k-1)}\biggl[ \theta\biggl(-\frac{pq}{c};p^2\biggr)\biggl(\frac{c}{q}\biggr)^k -z\theta\biggl(-\frac{pa}{b};p^2\biggr)(ab)^k \biggr] f(q^{2k}z) \\ &\qquad -p^{k^2}\biggl[ \theta\biggl(-\frac{q}{c};p^2\biggr)\biggl(\frac{c}{q}\biggr)^{k+1} -zb\theta\biggl(-\frac{a}{b};p^2\biggr)(ab)^k \biggr] f(q^{2k+1}z)\biggr\}=0. \end{aligned}
\end{equation}
\tag{3.12}
$$
For fixed parameters $a$, $b$, $c$, $d$ and $q$, only the terms with summation index $k=0,1$ give nontrivial contributions to the limit as $p\to 0$, provided that the function $f(z)$ depends on $p$ in a nonsingular way. This yields the $q$-hypergeometric equation (2.9) for the function $_2\varphi_1$. Supposedly, a finite-difference equation of infinite order should have a rich space of solutions with many constants of integration. At the moment we cannot characterize the whole space of solutions of equation (3.12). However, it is possible to find solutions with the isolated singularity at the point $z=0$. Indeed, any such function can be expanded in a Laurent series $f(z)=\sum_{n\in\mathbb{Z}}c_n z^n$ around $z=0$. Substituting this expansion into (3.12) one finds a recurrence relation of the first order for the coefficients $c_n$, which can be solved explicitly in terms of elliptic Pochhammer symbols. For generic values of the parameters the coefficient $d_0$ in the relation $c_{-1}=d_0 c_0$ appears to be equal to zero, $d_0=0$, which means that $c_n=0$ for $n<0$. Therefore, the full series appears to be analytic at the point $z=0$ (provided that it is convergent), and it coincides with the solution
$$
\begin{equation*}
f(z)={}_2 E_1\biggl(\begin{matrix}a,b\\ c\end{matrix}; q, p \biggm| z\biggr)
\end{equation*}
\notag
$$
derived above. Multiplying this solution by a nontrivial elliptic function ${\varphi(qz)=\varphi(z)}$, we obtain a single-valued solution of (3.11), which is not analytic at ${z=0}$. Other solutions can contain components that are not single-valued functions of $z$. Some particular solutions of toy equations of infinite order are described below. Consider eigenfunctions of the simplest difference operator of infinite order $\theta(aq^\delta; p)$. This should give us an idea of the possible structure of solutions of more complicated $q$-difference equations of infinite order. Using the infinite product representation instead of an infinite theta series, we have
$$
\begin{equation*}
\theta(aq^\delta; p) f(z)=\prod_{n=0}^\infty (1-aq^\delta p^n) (1-pa^{-1}q^{-\delta}p^n)f(z)= \lambda f(z).
\end{equation*}
\notag
$$
A single-valued function $f(z)=z^N\theta(bz;q)/\theta(cz;q)$, $N\in\mathbb{Z}$, satisfies the equation $f(qz)=q^Ncb^{-1}f(z)$. Therefore, for an arbitrary set of parameters $b_j, c_j\in \mathbb{C}^\times$ we have an eigenfunction
$$
\begin{equation}
f(z)=z^N\prod_{j=1}^m\frac{\theta(b_jz;q)}{\theta(c_jz;q)}, \quad\text{where } \lambda=\theta\biggl(aq^N\prod_{j=1}^m \frac{c_j}{b_j};p\biggr).
\end{equation}
\tag{3.13}
$$
Whenever the condition $\prod_{j=1}^m(b_j/c_j)=aq^Np^n$, $n\in\mathbb{Z}$, is satisfied, one obtains a kernel function. As a result, any linear combination of such functions with different parameters $N$, $b_j$ and $c_j$ satisfying the same constraint also lies in the kernel of the operator in question. There are also solutions which are not single-valued in $z$ similarly to the general solution of the ordinary hypergeometric equation for the Euler-Gauss hypergeometric $_2F_1$-function. Indeed, consider the function of the form $z^d$, which is not single valued for $d\neq \mathbb{Z}$. One has the formal eigenvalue problem $\theta(aq^\delta; p) z^d= \theta(aq^d; p) z^b$, so that, whenever $aq^d=p^n$ (or $d=-\log ap^n/\log q$), $n\in\mathbb{Z}$, one obtains a kernel function. A linear combination of such functions for different values of $n$, weighted by $q$-elliptic coefficients, yields the following infinite Dirichlet-type series:
$$
\begin{equation}
f(z)=z^{-{\log a}/{\log q}}\sum_{k\in \mathbb{Z}} h_k(z) z^{-k {\log p}/{\log q}}, \qquad h_k(qz)=h_k(z),
\end{equation}
\tag{3.14}
$$
which is a kernel function of the operator $\theta(aq^\delta; p)$, provided that the series is convergent. Also in the single-valued function (3.13) one can replace the factor $z^N$ by $z^d$, impose the constraint $\prod_{j=1}^m(b_j/c_j)=aq^dp^n$, $n\in\mathbb{Z}$, and take linear combinations of such functions for different $b_j$, $c_j$, $d$ and $n$ to obtain even more complicated kernel functions. Suppose that the operator $\theta(aq^\delta; p)$ acts in some Hilbert space of functions and one is interested in inverting it. If at least one of the kernel functions mentioned above belongs to the Hilbert space in question, then the inversion is not possible. Now consider a more complicated operator. Take $w, q, p\in \mathbb{C}^\times$, $K\in\mathbb{Z}$, set $t:=pq^Kw^{-K^2}$ and impose the constraints $|w|,|p|,|t|<1$. Then set
$$
\begin{equation*}
\widehat{\mathcal{O}}=\sum_{n\in \mathbb{Z}} w^{\binom{n}{2}}\theta(w^{nK}z;p) q^{n\delta}.
\end{equation*}
\notag
$$
Expanding $\theta(w^{nK}z;p)$ in a Laurent series in $z$ and changing the order of summations in the emerging double sum we obtain
$$
\begin{equation*}
\widehat{\mathcal{O}}=\frac{(w;w)_\infty (t;t)_\infty}{(p;p)_\infty} \theta\bigl(w^{-\binom{K}{2}}zq^{-K\delta};t\bigr)\theta(-q^\delta;w),
\end{equation*}
\notag
$$
where we have used the relation $z^mq^{-Km\delta}=q^{K\binom{m}{2}}(zq^{-K\delta})^m$. As we see, the operator of the form $\theta(az q^{K\delta}; t)\theta(bq^\delta; w)$ determines the kernel of the $\widehat{\mathcal{O}}$-operator, and the functions described above provide explicit examples of this kernel.
§ 4. Convergence of elliptic hypergeometric seriesNow we return to examining $_s E_r$-series, in particular, we raise the question as to when this formal power series converges. One can readily notice that the substitution $t_i = q^{-N}$ for some fixed $i$ and $N \in \mathbb{Z}_{\geqslant 0}$ leads to the termination of the series
$$
\begin{equation*}
_s E_r\biggl(\begin{matrix} q^{-N}, t_1, \dots, t_{s-1}\\ w_1, \dots, w_r \end{matrix}; q, p \biggm| z\biggr)= \sum_{n=0}^{N} \frac{\theta(q^{-N}, t_1, \dots,t_{s-1}; p; q)_n}{\theta(q, w_1, \dots,w_r; p; q)_n} z^n,
\end{equation*}
\notag
$$
and the resulting sum converges for a trivial reason. The most interesting case corresponds to the elliptic hypergeometric series. Set ${s=r+1}$ and impose the special balancing constraint Then we see that the ratio of consecutive terms of the series
$$
\begin{equation*}
h(n)=\frac{c_{n+1}}{c_n} =z\frac{\theta(q^n t_0, q^n t_1, \dots, q^n t_{r}; p)}{\theta(q^{n+1}, q^n w_1, \dots, q^n w_r; p)}
\end{equation*}
\notag
$$
becomes an elliptic function of the formal variable $n$. Namely, considering $n$ as a complex variable transforms $h(n)$ into a meromorphic function of $n$ with double periodicity
$$
\begin{equation}
h\biggl(n+\frac{2\pi i}{\log q}\biggr)=h\biggl(n+\frac{\log p}{\log q}\biggr)=h(n),
\end{equation}
\tag{4.2}
$$
that is, it is an elliptic function. This gives the name to the series. Equalities (4.2) follow from the fact that $h(n)$ depends only on the function $q^n$, which does not change under the first period shift of the $n$-variable and becomes $pq^n$ after the second period shift. In the latter case the quasiperiodicity of the theta functions (2.4) brings in constant multipliers, which cancel due to the balancing condition (4.1). Now, in order to remove the singularity of $h_n$ at $n=-1$ we fix the special value $t_0=q$ and consider the series
$$
\begin{equation*}
_{r+1} E_r\biggl(\begin{matrix} q, t_1, \dots, t_{r}\\ w_1, \dots, w_r \end{matrix}; q, p \biggm| z\biggr)=\sum_{n=0}^\infty \frac{\theta(t_1, \dots,t_{r}; p; q)_n}{\theta(w_1, \dots,w_r; p; q)_n} z^n, \qquad \prod_{k=1}^{r} t_k=\prod_{k=1}^{r} w_k.
\end{equation*}
\notag
$$
The balancing condition implies that the function
$$
\begin{equation}
H(u):=\frac{\theta(u t_1, \dots, u t_{r}; p)}{\theta(u w_1, \dots, u w_r; p)}
\end{equation}
\tag{4.3}
$$
is a meromorphic function of $u\in\mathbb{C}^\times$ which is invariant under the transformation $u\to pu$ (or a $p$-elliptic function), $H(pu)=H(u)$. Zeros of this function are positioned at the points $u=t_k^{-1} p^\mathbb{Z}$ and its poles sit at $u=w_k^{-1}p^{\mathbb{Z}}$ for $k=1, \dots,r$. In the exponential form we set and then the function $H(\exp(2\pi i x))$ is an elliptic function of $x\in \mathbb{C}$ with fundamental periods $\{1, \tau\}$. Now we choose $q$ carefully. Consider any pair of integers $N, M$, such that the line from $x=0$ to $x=N+M\tau$ does not contain any poles of the function $H(\exp(2\pi i x))$. For such a pair of integers and any real number $\chi \in \mathbb{R}$ we set
$$
\begin{equation}
q=\exp(2\pi i \chi(N+M\tau)), \qquad (N,M)\neq (0,0).
\end{equation}
\tag{4.4}
$$
This implies that for fixed $\chi \in \mathbb{R}$ and integers $N$ and $M$ the parameters satisfy the constraints
$$
\begin{equation}
t_j,w_j\neq p^k \exp(2\pi i\chi (N+M\tau)l), \qquad k\in\mathbb{Z}, \quad l\in \mathbb{Z}_{\leqslant 0}, \qquad j=1,\dots, r.
\end{equation}
\tag{4.5}
$$
Then the series coefficients $c_n/z^n$ neither vanish (that is, the series does not terminate), nor blow up. However, if some of the parameters $w_j$ lie at transcendental points on the straight line from $x=0$ to $x=N+M\tau$, then in the limit as $n\to\infty$ some subsequences of points $q^np^{\mathbb{Z}}$ approach these parameters arbitrarily closely, a singularity arises, and estimating the values of $|H(q^n)|$ becomes a delicate problem. Therefore, in the simplest case one can assume that the constraints (4.5) on the parameters $w_j$ hold for all real values of $\chi$, rather than just for some fixed number. Then the absolute values $|H(q^n)|$ are bounded above by some positive constant $S$. As a result, the series converges for $|z| < S^{-1}$. Suppose that $\chi $ is a rational number, $\chi =a/b$, where $(a,b)=1$, that is, $q^b=p^{Ma}$. Then one has the periodicity $H(q^{n+b})=H(q^n)$, and the series under consideration can be summed to a closed form:
$$
\begin{equation}
\begin{aligned} \, \nonumber &\sum_{n=0}^\infty \frac{\theta(t_1, \dots,t_{r}; p; q)_n}{\theta(w_1, \dots,w_r; p; q)_n} z^n =\sum_{j=0}^\infty\sum_{l=0}^{b-1}\prod_{k=0}^{jb+l-1}H(q^k)z^{jb+l} \\ &\qquad =\sum_{j=0}^\infty R^j z^{jb}\sum_{l=0}^{b-1}\prod_{k=jb}^{jb+l-1}H(q^k)z^{l} =\frac{1+\sum_{l=1}^{b-1}\prod_{k=0}^{l-1} H(q^{k}) z^l}{1-Rz^b}, \end{aligned}
\end{equation}
\tag{4.6}
$$
where we have set $R:=\prod_{k=0}^{b-1} H(q^k)$. Clearly, the radius of convergence in this case is $r_{c}=|R|^{-1/b}$. Since for a nontrivial $R$-function $|R|$ cannot be equal to $1$ for all values of the parameters, when $|R|>1$ we can swap the values of $t_k$ and $w_k$ and obtain $r_c>1$, that is, there always exist series with such a radius of convergence. Note that formula (4.6) defines a meromorphic function of the parameters $t_k$, $w_k$ and $q$ as a rational combination of theta functions, and in general it is neither elliptic, nor even quasiperiodic in these variables. Now consider the case when $\chi $ is a fixed irrational number, with additional constraints on the values of $t_j$, similar to those of $w_j$, so that not only the poles of the function $H(u)$ do not lie on the interval under consideration, but the zeros $t_j^{-1}p^{\mathbb{Z}}$ are also assumed not to lie at transcendental points. In this case the following explicit bound on the radius of convergence can be obtained. By Weyl’s equidistribution theorem, for any irrational $\chi $ the logarithms of the numbers $q^k=\exp(2\pi i k\chi (N+ M\tau))$ are uniformly distributed on the line from $0$ to $N+M\tau$. Therefore, the following limit exists
$$
\begin{equation}
\lim_{n \to \infty} \frac{1}{n} \sum_{k=0}^{n-1} \log |H(q^k)| =\int_{0}^{1} \log \bigl|H\bigl(\exp(2\pi i x (N+M\tau))\bigr)\bigr|\,dx.
\end{equation}
\tag{4.7}
$$
As a result, the leading asymptotics of the coefficients of our series has a simple power form:
$$
\begin{equation*}
|c_n| =\prod_{k=0}^{n-1}|H(q^k)|\,|z|^n =\biggl|z\exp \biggl(\frac{1}{n} \sum_{k=0}^{n-1} \log |H(q^k)|\biggr)\biggr|^n \mathrel{\underset{n\to\infty}{=}}\biggl|\frac{z}{r_c}\biggr|^n,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
r_{c}:=\exp\biggl(-\int_{0}^{1} \log \bigl|H\bigl(\exp(2\pi i x (N+M\tau))\bigr)\bigr|\,dx\biggr).
\end{equation*}
\notag
$$
By the ratio test, we see that our series converges for $|z|<r_c$, that is, $r_c$ is the radius of convergence. The convergence problem for a particular $q$-hypergeometric series when $q=\exp(2\pi i \chi)$ for some irrational $\chi $ was considered for the first time by Hardy and Littlewood [5], where a simpler integral arose, and it was explicitly evaluated, producing the exact value of the radius of convergence. However, in [5] an analogue of $H(u)$ has a singularity and the uniform distribution argument does not work for all irrational $\chi$. For more results of this type for $q$-hypergeometric objects, see [6], [7] and the next section. In particular, Petruska showed in [7] that it is essential for $t_i$ and $w_j$ not to lie in the interval of integration, which we assume. One can pose a natural question — is there any way to compute the integral arising? Consider the following analogue of the above integral, where we assume that $|p|<1$ or $\operatorname{Im}\tau>0$:
$$
\begin{equation}
F_{N,M}(t):=\int_{0}^{1} \log \bigl|\theta\bigl(t\exp(2\pi i x (N+M\tau)); p\bigr)\bigr|\,dx.
\end{equation}
\tag{4.8}
$$
Singularities of the integrand can only be of logarithmic type, hence the integral converges. First we examine its dependence on the parameters $N$ and $M$. For this purpose consider their greatest common divisor $D=\gcd(N,M)$, $D>0$. Then
$$
\begin{equation}
F_{N,M}(t)=F_{N/D,M/D}(t)+\pi \operatorname{Im}\tau \frac{M(D-1)(2M(D+1)-3D)}{6D^2}-\log|t|\frac{M(D-1)}{2D}.
\end{equation}
\tag{4.9}
$$
This equality is a consequence of several elementary transformations. Starting from
$$
\begin{equation*}
\begin{aligned} \, &D\int_{0}^{1} \log \bigl|\theta \bigl(t\exp(2\pi i x (N+M\tau)); p\bigr)\bigr|\,d x \\ &\qquad =\int_{0}^{D} \log\biggl|\theta \biggl(t\exp\biggl(2\pi i x \biggl(\frac{N}{D}+\frac{M}{D}\tau\biggr)\biggr);p\biggr)\biggr|\,dx \\ &\qquad=\sum_{k=0}^{D-1} \int_{0}^{1}\log\biggl|\theta \biggl(t \exp\biggl(2\pi i(x+k)\biggl(\frac{N}{D}+\frac{M}{D}\tau\biggr)\biggr);p\biggr)\biggr|\,dx \\ &\qquad=\sum_{k=0}^{D-1} \int_{0}^{1}\log\biggl|\theta \biggl(p^{Mk/D}t \exp\biggl(2\pi i x \biggl(\frac{N}{D}+\frac{M}{D}\tau\biggr)\biggr);p\biggr)\biggr|\,dx, \end{aligned}
\end{equation*}
\notag
$$
applying the quasiperiodicity relation (2.3) and taking the logarithm of the modulus of the quasiperiodicity multiplier, we obtain
$$
\begin{equation*}
F_{N,M}(t)=F_{N/D,M/D}(t)+\frac{1}{D}\sum_{k=0}^{D-1} \biggl(\frac{M^2k(k+1)}{D^2}\pi\operatorname{Im}\tau-\frac{Mk}{D}\pi \operatorname{Im}\tau-\frac{Mk}{D}\log|t|\biggr).
\end{equation*}
\notag
$$
Computing sums with respect to $k$ we arrive at the result claimed. Next we describe a different representation of $F_{N,M}(t)$ based on the infinite product form of the theta function. We restrict our attention to the case $M>0$ and write
$$
\begin{equation*}
\begin{aligned} \, &\int_{0}^{1} \log \bigl|\theta \bigl(t\exp(2\pi i x (N+M\tau)); p\bigr)\bigr|\,dx \\ &\qquad=\int_{0}^{1} \sum_{k=0}^{M-1} \log \bigl|\theta \bigl(tp^k \exp(2\pi i x (N+M\tau)); p^M\bigr)\bigr|\,dx \\ &\qquad =\sum_{k=0}^{M-1} \int_{0}^{1} \sum_{n=0}^\infty\log \bigl|\bigl(1-tp^k \exp(2\pi i( x+n)(N+M\tau))\bigr) \\ &\qquad\qquad\times\bigl(1-t^{-1}p^{-k} \exp(2\pi i(n+1- x )(N+M\tau))\bigr)\bigr|\,dx \\ &\qquad =\sum_{k=0}^{M-1} \sum_{n=0}^\infty \int_{n}^{n+1}\log \bigl|\bigl(1-tp^k \exp(2\pi i x (N+M\tau))\bigr) \\ &\qquad\qquad\times\bigl(1-t^{-1}p^{-k} \exp(2\pi i x (N+M\tau))\bigr)\bigr|\,dx \\ &\qquad =\sum_{k=0}^{M-1} \int_{0}^\infty \log \bigl|\bigl(1-tp^k \exp(2\pi i x (N+M\tau))\bigr) \\ &\qquad\qquad\times \bigl(1-t^{-1}p^{-k} \exp(2\pi i x (N+M\tau))\bigr)\bigr|\,dx. \end{aligned}
\end{equation*}
\notag
$$
Hence our problem reduces to an explicit computation of the following integral for arbitrary $t\in \mathbb{C}^\times$:
$$
\begin{equation*}
I_{N,M}:=\int_{0}^\infty \log\bigl|1-t\exp(2\pi i x(N+M\tau))\bigr|\,dx.
\end{equation*}
\notag
$$
We notice that the case $M<0$ is equivalent to this one because of the transformation $ x \leftrightarrow 1- x $ in the original integral. We return to the case $M=0$ below. For now consider the following lemma. Lemma 4. Suppose $M >0$ and define Proof. Consider the first case, that is, let $|t|\geqslant 1$. Then the integral can be split into two:
$$
\begin{equation*}
\begin{aligned} \, I_{N,M} &=\int_{0}^{{\log|t|}/(2\pi M\operatorname{Im}\tau)} \log\bigl|1-t\exp(2\pi i x(N+M\tau))\bigr|\,dx \\ &\qquad + \int_{\log|t|/(2\pi M\operatorname{Im}\tau)}^\infty \log\bigl|1-t\exp(2\pi i x(N+M\tau))\bigr|\,dx. \end{aligned}
\end{equation*}
\notag
$$
For the first term we notice that $0\leqslant x \leqslant \log|t|/(2\pi M\operatorname{Im}\tau)$. Therefore, $\log|t|-2\pi Mx\operatorname{Im}\tau \geqslant 0$ or, in other words,
$$
\begin{equation*}
\bigl|t\exp(2\pi i x(N+M\tau))\bigr|=|t|\exp(-2\pi Mx\operatorname{Im}\tau)\geqslant 1.
\end{equation*}
\notag
$$
Hence we can write
$$
\begin{equation*}
\begin{aligned} \, &\log\bigl|1-t\exp(2\pi i x(N+M\tau))\bigr| \\ &\qquad=\log|t|-2\pi Mx\operatorname{Im}\tau+\log\bigl|1-t^{-1} \exp(-2\pi i x(N+M\tau))\bigr| \end{aligned}
\end{equation*}
\notag
$$
and expand the logarithm on the right-hand side in a convergent Taylor series. After taking the termwise integrals with respect to $x$, this yields
$$
\begin{equation}
\begin{aligned} \, \nonumber &\int_{0}^{\log|t|/(2\pi M\operatorname{Im}\tau)} \log\bigl|1-t\exp(2\pi i x(N+M\tau))\bigr|\,dx \\ \nonumber &\qquad=\frac{\log^2|t|}{4\pi M\operatorname{Im}\tau}-\operatorname{Re}\sum_{n=1}^\infty \frac{t^{-n}}{n}\, \frac{1-\exp\bigl(-in(N+M\tau)\log|t|/(M\operatorname{Im}\tau)\bigr)}{2\pi in(N+M\tau)} \\ &\qquad=\frac{\log^2|t|}{4\pi M\operatorname{Im}\tau}-\operatorname{Re} \frac{\operatorname{Li}_2(t^{-1})}{2\pi i(N+M\tau)}+\operatorname{Re} \frac{\operatorname{Li}_2(\overline \mu) }{2\pi i(N+M\tau)}, \end{aligned}
\end{equation}
\tag{4.11}
$$
where $\overline\mu$ is the complex conjugate of the parameter defined in the statement of the lemma. The last term is well defined since the argument lies on the unit circle.
Similarly, for the second summand we have
$$
\begin{equation*}
|t\exp(2\pi i x(N+M\tau))|=|t|\exp(-2\pi Mx\operatorname{Im}\tau)\leqslant 1,
\end{equation*}
\notag
$$
so that we can expand the logarithm in a Taylor series and integrate termwise, which yields
$$
\begin{equation*}
\begin{aligned} \, &\int_{\log|t|/(2\pi M\operatorname{Im}\tau)}^\infty \log\bigl|1-t\exp(2\pi i x(N+M\tau))\bigr|\,dx \\ &\qquad =\operatorname{Re}\sum_{n=1}^\infty \frac{t^n}{n}\, \frac{\exp\bigl(i n(N+M\tau)\log|t|/(M\operatorname{Im}\tau)\bigr)}{2\pi i n(N+M\tau)} =\operatorname{Re} \frac{\operatorname{Li}_2(\mu)}{2\pi i (N+M\tau)}. \end{aligned}
\end{equation*}
\notag
$$
Thus, the full integral is indeed equal to
$$
\begin{equation*}
\begin{aligned} \, &\int_{0}^\infty \log\bigl|1-t\exp(2\pi i x(N+M\tau))\bigr|\,dx \\ &\qquad =\frac{\log^2|t|}{4\pi M\operatorname{Im}\tau}-\operatorname{Re} \frac{\operatorname{Li}_2(t^{-1})}{2\pi i(N+M\tau)} +\operatorname{Re} \frac{\operatorname{Li}_2(\overline{\mu}) +\operatorname{Li}_2(\mu)}{2\pi i(N+M\tau)}. \end{aligned}
\end{equation*}
\notag
$$
Consider the second case, that is, $|t|\leqslant 1$. Then $\log|t|/(2\pi M\operatorname{Im}\tau) \leqslant x$ for any ${0\leqslant x}$. In other words $|t\exp(2\pi i x(N+M\tau))|=|t|\exp(-2\pi Mx\operatorname{Im}\tau)\leqslant 1$. Hence the following holds:
$$
\begin{equation*}
\begin{aligned} \, &\int_0^\infty \log|1-t\exp(2\pi i x(N+M\tau))|\,dx \\ &\qquad=-\operatorname{Re}\sum_{n=1}^\infty \frac{t^n}{n} \frac{\exp(2\pi i nx(N+M\tau))}{2\pi i n(N+M\tau)}\bigg|_0^\infty=\operatorname{Re} \frac{\operatorname{Li}_2(t)}{2\pi i(N+M\tau)}. \end{aligned}
\end{equation*}
\notag
$$
The lemma is proved. Remark 5. By definition we have $|\mu|=1$ and $\operatorname{arg}\mu \in [0; 2\pi]$. Therefore, Lemma 4 implies that for $M\neq 0$ and any integer $N$ the integral $F_{N,M}(t)$ can always be computed. We have
$$
\begin{equation}
\begin{aligned} \, \nonumber &\int_{0}^\infty \log\bigl|\bigl(1-t\exp(2\pi i x(N+M\tau))\bigr) \bigl(1-t^{-1}\exp(2\pi i x(N+M\tau))\bigr)\bigr|\,dx \\ &\qquad=\frac{\log^2|t|}{4\pi M\operatorname{Im}\tau}-\frac{M\operatorname{Im}\tau \operatorname{Re}(\operatorname{Li}_2(\mu))}{\pi|N+M\tau|^2}, \qquad |t|\geqslant 1. \end{aligned}
\end{equation}
\tag{4.13}
$$
Since $\mu(t)$, as a function of $t$, satisfies $\mu(t^{-1})=\overline{\mu(t)}$, for $|t| \leqslant 1$ we have exactly the same answer. As a result, from the relation $\mu(tp^k)=\mu(t)\exp(-2\pi i k(N/M))$, for any $t \in \mathbb{C}^\times$ we obtain
$$
\begin{equation*}
\begin{aligned} \, &\int_{0}^{1} \log\bigl|\theta\bigl(t\exp(2\pi i x (N+M\tau)); p\bigr)\bigr|\,dx \\ &\qquad=\sum_{k=0}^{M-1} \biggl(\frac{\log^2|tp^k|}{4\pi M\operatorname{Im}\tau}-\frac{M\operatorname{Im}\tau \operatorname{Re}(\operatorname{Li}_2(\mu(tp^k)))}{\pi|N+M\tau|^2}\biggr). \end{aligned}
\end{equation*}
\notag
$$
Consider the greatest common divisor of $N$ and $M$, $D=\gcd(N,M)$, again, where we assume as usual that $\gcd(N,0)=N$ and $\gcd(0,M)=M$. Then for the second summand we have
$$
\begin{equation*}
\begin{aligned} \, &\sum_{k=0}^{M-1} \operatorname{Li}_2(\mu(tp^k)) =\sum_{k=0}^{M-1} \sum_{n=1}^\infty \frac{\exp(in\operatorname{arg}\mu(tp^k))}{n^2} \\ &\qquad=\sum_{k=0}^{M-1} \sum_{n=1}^\infty \frac{\exp(in\operatorname{arg}\mu-2\pi i knN/M)}{n^2} \\ &\qquad=\sum_{k=0}^{M-1} \sum_{\substack{n=1 \\ nN/M\in \mathbb{Z}}}^\infty \frac{\exp(in\operatorname{arg}\mu-2\pi i knN/M)}{n^2} \\ &\qquad=\sum_{k=0}^{M-1} \sum_{r=1}^\infty \frac{\exp\bigl(ir(M/D)\operatorname{arg}\mu-2\pi i kr(M/D)(N/M)\bigr)}{(rM/D)^2} =\frac{D^2}{M} \operatorname{Li}_2(\mu^{M/D}). \end{aligned}
\end{equation*}
\notag
$$
And the first summand is clearly equal to the following expression:
$$
\begin{equation*}
\frac{\log^2|t|}{4\pi \operatorname{Im}\tau} +\frac{\log|t|\log|p|}{4\pi \operatorname{Im}\tau}(M-1) +\frac{\log^2|p|}{4\pi \operatorname{Im}\tau}\,\frac{(M-1)(2M-1)}{6}.
\end{equation*}
\notag
$$
Combining them together we obtain the following result for the function $F_{N,M}(t)$ in (4.8) for $M>0$:
$$
\begin{equation}
\begin{aligned} \, \nonumber F_{N,M}(t) &=\frac{\log^2|t|}{4\pi\operatorname{Im}\tau}-\frac{(M-1)\log|t|}{2} +\frac{(M-1)(2M-1)\pi\operatorname{Im}\tau}{6} \\ &\qquad -\frac{D^2\operatorname{Im}\tau}{\pi|N+M\tau|^2} \operatorname{Re}(\operatorname{Li}_2(\mu^{M/D})). \end{aligned}
\end{equation}
\tag{4.14}
$$
The transformation $(N,M) \to (N/D,M/D)$ does not change the first and last terms. Hence this explicit representation can also be used to derive a formula for the difference $F_{N,M}(t)-F_{N/D,M/D}(t)$, which we have already computed using another, purely elementary strategy. It remains to understand the case $M=0$. Since $F_{N,0}(t)=F_{1,0}(t)$, we need to consider only the latter function. We have
$$
\begin{equation*}
\begin{aligned} \, &\int_{0}^{1} \log \bigl|\theta(\exp(2\pi i z); \exp(2\pi i\tau))\bigr|\,dx \\ &=\int_{0}^{1} \log \biggl|\exp\biggl(\!-\frac{\pi i}{6}(\tau\!+\!3\!-\!\widetilde \tau)+\pi i \widetilde\tau z^2+\pi i(1\!-\!\widetilde\tau)z\!\biggr)\theta(\exp(2\pi i \widetilde \tau z); \exp(2\pi i\widetilde \tau))\biggr|\,dx, \end{aligned}
\end{equation*}
\notag
$$
where $\exp(2\pi i z)=t\exp(2\pi i x)$ and $\widetilde \tau = -1/\tau$. Here we have used the modular transformation law for the theta function, which we describe in terms of the uniform variables $\tau:=\omega_1/\omega_2$ and $z:= u/\omega_2$:
$$
\begin{equation}
\begin{aligned} \, \nonumber &\theta\biggl(\exp\biggl(-2\pi{i}\frac{u}{\omega_1}\biggr); \exp\biggl(-2\pi{i}\frac{\omega_2}{\omega_1}\biggr)\biggr) \\ &\qquad =\exp(\pi{i}B_{2,2}(u;\mathbf{\omega})) \theta\biggl(\exp\biggl(2\pi{i}\frac{u}{\omega_2}\biggr); \exp\biggl(2\pi{i}\frac{\omega_1}{\omega_2}\biggr)\biggr), \end{aligned}
\end{equation}
\tag{4.15}
$$
where $B_{2,2}$ is a multiple Bernoulli polynomial of the second order:
$$
\begin{equation*}
B_{2,2}(u;\omega_1,\omega_2)=\frac{u^2}{\omega_1\omega_2}-\frac{u}{\omega_1} -\frac{u}{\omega_2}+\frac{\omega_1}{6\omega_2}+\frac{\omega_2}{6\omega_1}+\frac{1}{2}.
\end{equation*}
\notag
$$
Since $\exp(2\pi i \widetilde \tau z)=t^{\widetilde\tau}\exp(2\pi i \widetilde \tau x)$, from the previous computations we have
$$
\begin{equation*}
\begin{aligned} \, &\int_{0}^{1} \log \bigl|\theta(t^{\widetilde\tau} \exp(2\pi i \widetilde \tau x);\exp(2\pi i\widetilde \tau))\bigr|\,dx \\ &\qquad=\frac{|\tau|^2}{4\pi \operatorname{Im}\tau}\biggl(\operatorname{Re}\biggl(\frac{\log t}{\tau}\biggr)\biggr)^2-\frac{\operatorname{Im}\tau}{\pi}\operatorname{Re}(\operatorname{Li}_2(\mu)), \end{aligned}
\end{equation*}
\notag
$$
where by definition
$$
\begin{equation*}
\operatorname{arg}\mu=\operatorname{Im}\biggl(-\frac{\log t}{\tau}\biggr) +\frac{\operatorname{Re}(-1/\tau)}{\operatorname{Im}(-1/\tau)} \log \biggl|\exp\biggl(-\frac{\log t}{\tau}\biggr)\biggr| =\frac{\log|t|}{\operatorname{Im}\tau}.
\end{equation*}
\notag
$$
On the other hand
$$
\begin{equation*}
\begin{aligned} \, &\int_{0}^{1} \log \biggl|\exp\biggl(-\frac{\pi i}{6}(\tau+3-\widetilde \tau)+\pi i \widetilde\tau z^2+\pi i(1-\widetilde\tau)z\biggr)\biggr|\,dx \\ &\qquad=\frac{\pi}{6}\operatorname{Im}\tau+\frac{1}{2}\log|t| -\operatorname{Im}\biggl(\frac{\log^2 t}{4\pi\tau}\biggr). \end{aligned}
\end{equation*}
\notag
$$
Combining two pieces, after some simplifications, we finally obtain
$$
\begin{equation}
F_{1,0}(t)=\frac{\log^2|t|}{4\pi \operatorname{Im}\tau}+\frac{1}{2}\log|t|+\frac{\pi}{6}\operatorname{Im}\tau -\frac{\operatorname{Im}\tau}{\pi} \operatorname{Re}\biggl(\operatorname{Li}_2 \biggl(\exp\biggl(i\frac{\log|t|}{\operatorname{Im}\tau}\biggr)\biggr)\biggr),
\end{equation}
\tag{4.16}
$$
which coincides with the formal substitution $(N,M)=(1,0)$ into the expression (4.14) obtained before, since
$$
\begin{equation*}
\mu^{M/D}|_{M=0}=\exp\biggl(i\frac{\operatorname{Re}\log \overline{t}}{\operatorname{Im}\tau}\biggr).
\end{equation*}
\notag
$$
Thus, formula (4.14) is well defined and gives a correct answer for any $N,M \in \mathbb{Z}$. Now we return to our main problem of the evaluation of the integral in (4.7). Since the function (4.3) contains an equal number of theta functions in the numerator and denominator, the contributions from the second term in (4.9) cancel each other. The balancing condition (4.1) leads to the cancellation of the contributions from the third term in (4.9), since $\sum_{k=1}^r\log |t_k/w_k|=0.$ This means that the final expression does not depend on $D$:
$$
\begin{equation}
\int_{0}^{1} \log \bigl|H\bigl(\exp(2\pi i x (N+M\tau))\bigr)\bigr|\,dx =\int_{0}^{1} \log \biggl|H\biggl(\exp\biggl(2\pi i x \biggl(\frac{N}{D}+\frac{M}{D}\tau\biggr)\biggr)\biggr)\biggr|\,dx.
\end{equation}
\tag{4.17}
$$
Therefore, in this case we may assume that $D=(M,N)=1$. Then we see that the balancing condition eliminates the contributions from the second and third terms in the evaluation of (4.14). As a result, we conclude that the radius of convergence of the elliptic hypergeometric series under consideration has the following form:
$$
\begin{equation}
\begin{aligned} \, \nonumber \log r_{c}^{-1} &=\int_{0}^{1} \log \bigl|H\bigl(\exp(2\pi i x (N+M\tau))\bigr)\bigr|\,dx =\frac{1}{4\pi\operatorname{Im}\tau}\sum_{k=1}^r \bigl(\log^2|t_k|\!-\! \log^2|w_k|\bigr) \\ &\qquad +\frac{\operatorname{Im}\tau}{\pi|N+M\tau|^2} \operatorname{Re}\sum_{k=1}^r \biggl(\operatorname{Li}_2\biggl(\exp\biggl(i\operatorname{Re} \frac{(N+M\tau)\log \overline{w}_k}{\operatorname{Im}\tau}\biggr)\biggr) \nonumber \\ &\qquad\qquad -\operatorname{Li}_2\biggl(\exp\biggl(i\operatorname{Re} \frac{(N+M\tau)\log \overline{t}_k}{\operatorname{Im}\tau}\bigg)\biggr)\biggr). \end{aligned}
\end{equation}
\tag{4.18}
$$
In view of Remark 5 (see (4.12)) the second summand can be written as a combination of quadratic polynomials in the fractional parts of $\arg z/2\pi$ for $z$ being the dilogarithm arguments. As a result, we arrive at the following theorem. Theorem 6. The radius of convergence of the elliptic hypergeometric series $_{r+1} E_{r}$ for $t_0=q=\exp(2\pi i\chi (N+M\tau))$, where $\chi$ is an arbitrary real irrational number, the integers $N, M\in\mathbb{Z}$ are coprime, $(N,M)=1$, and no $t_i$ or $w_j$ lie on the line $q^{\mathbb{R}}$, is given by the explicit formula Now consider the radius of convergence of certain well-poised elliptic hypergeometric series appearing in the most interesting applications. Such series are defined by imposing the set of conditions so that there remains only $r$ free parameters, say, $t_1,\dots, t_r$, and $w_1$ constrained by the balancing condition $\prod_{j=2}^rt_j=\nu t_1^{(r-2)/2}w_1^{r/2}$, $\nu=\pm 1$, in addition to the qualitatively different variables $z$, $q$ and $p$. One can check that all the terms of the resulting $_{r+1}E_r$-series are elliptic functions of the parameters $t_j$ and $w_1$, that is, they are invariant under the transformations $t_j\to p^{n_j}t_j$ and $w_1\to p^m w_1$, $n_j, m\in\mathbb{Z}$, respecting the balancing condition. Namely, one has the constraint where for even $r$ and $\nu =1$ the discrete variables $n_1$ and $m$ can take any integer value, whereas for even $r$ and $\nu =-1$, as well as for odd $r$, the sum $n_1+m$ must be even. Since the invariance under the change $q\to pq$ is implemented from the very beginning, only the variable $z$ does not lie on the elliptic curve with modulus $p$, that is, it looks alien in the picture.The constraints (4.20), in combination with the balancing condition, force the first term in (4.19) to vanish. In order to analyze the second term it is convenient to use the following unique representations for the parameters:
$$
\begin{equation}
t_j=q^{h_j} \exp\biggl(\varphi_j \frac{2\pi \operatorname{Im}\tau}{N\,{+}\,M\overline\tau}\biggr) \quad\text{and}\quad w_j=q^{\widetilde{h}_j} \exp\biggl(\widetilde{\varphi}_j \frac{2\pi \operatorname{Im}\tau}{N\,{+}\,M\overline\tau}\biggr), \ \ h_j, \widetilde{h}_j, \varphi_j, \widetilde{\varphi}_j \,{\in}\, \mathbb{R}.
\end{equation}
\tag{4.21}
$$
As we see, the variables $\varphi_j$ and $\widetilde\varphi_j$ measure how far the poles and zeros determined by the parameters $t_j$ and $w_j$ lie from the line $q^\mathbb{R}$. The variables $h_j$, $\widetilde{h}_j$, $\varphi_j$ and $\widetilde{\varphi}_j$ are independent (similarly to the modulus and argument of a complex number) and satisfy the constraints
$$
\begin{equation*}
\widetilde{\varphi}_i+\varphi_i=\widetilde{\varphi}_j+\varphi_j, \quad \widetilde{h}_i+h_i=\widetilde{h}_j+h_j, \quad \sum_{k=1}^r\widetilde{\varphi}_k=\sum_{k=1}^r\varphi_k\quad\text{and} \quad \sum_{k=1}^r\widetilde{h}_k=\sum_{k=1}^r h_k.
\end{equation*}
\notag
$$
The variables $h_j$ and $\widetilde h_j$ do not contribute to the real parts under consideration, and the expression for the radius of convergence can be rewritten as
$$
\begin{equation}
\log r_{c}^{-1}=\frac{\pi\operatorname{Im}\tau}{|N+M\tau|^2} \sum_{k=1}^r (\{\widetilde{\varphi}_k\}-\{\varphi_k\})(\{\widetilde{\varphi}_k\}+\{\varphi_k\}-1).
\end{equation}
\tag{4.22}
$$
Clearly, if $\{\varphi_k\}=\varphi_k$ and $\{\widetilde{\varphi}_k\}=\widetilde{\varphi}_k$ for all $k$, then the sum on the right-hand side vanishes and $r_c=1$. Consider a particular example of the values of the parameters when $\{\varphi_k\}\neq\varphi_k$ for some $k$. Fix the following particular choice of the $\varphi_i$:
$$
\begin{equation*}
\varphi_1=1+\frac{\varepsilon r}{2} \quad\text{and}\quad \varphi_2=\dots=\varphi_r=1-\varepsilon, \qquad \varepsilon >0.
\end{equation*}
\notag
$$
The constraints imposed on the tilded variables can be resolved, which yields
$$
\begin{equation*}
\widetilde{\varphi}_j+\varphi_j=\frac{2}{r}\sum_{k=1}^r\varphi_k, \qquad \widetilde{\varphi}_1=1-\varepsilon \biggl(1-\frac{2}{r}+\frac{r}{2}\biggr) \quad\text{and}\quad \widetilde{\varphi}_2=\dots=\widetilde{\varphi}_r=1+\frac{2\varepsilon}{r}.
\end{equation*}
\notag
$$
Consider now a particular form of the $\varepsilon$-variable:
$$
\begin{equation*}
\varepsilon=\frac{k+1}{r/2+\lambda}, \qquad k \in \mathbb{Z}_{\geqslant 0},
\end{equation*}
\notag
$$
where $\lambda$ is a free real variable satisfying the constraints For this particular choice we have the bounds We also assume that $k+1 \leqslant {r}/{2}$, hence $\varepsilon < 1$ and $k+1 < {r}/{2}+\lambda$. Since $\lambda < 1$, we have ${r}/{2}-k\lambda > 0$. Adding $kr/2$ to both sides and dividing by $\lambda+ r/2$, we arrive at the key constraint ${\varepsilon r}/{2} > k$, or $\varphi_1>k+1$. The latter means that $\{\varphi_1\}={\varepsilon r}/{2}-k$. Consequently, $\widetilde{\varphi}_1 = 1-\varepsilon({r}/{2}+1-{2}/{r})$. We also have $0<\varepsilon (-\lambda+(1-{2}/{r})) < 1$. Hence $\{\widetilde{\varphi}_1\}=\widetilde{\varphi}_1+k+1$. Also, for $j>1$ we have $\{\varphi_j\}=\varphi_j$ and $\{\widetilde{\varphi}_j\}=\widetilde{\varphi}_j-1$. As a result, the straightforward computation yields
$$
\begin{equation}
\begin{aligned} \, \log r_{c}^{-1} =\frac{ \varepsilon\pi\operatorname{Im}\tau}{|N+M\tau|^2} \biggl(2\lambda+ \biggl(\frac{2}{r}-1\biggr)(2k+4-r)\biggr). \end{aligned}
\end{equation}
\tag{4.23}
$$
The choice of $k=[(r-2)/2]$ leads to a radius of convergence greater than $1$ for even $r$. Restricting the value of $\lambda$ by $0< 2\lambda < 1-{2}/{r}$ we see that $r_c>1$ for odd $r$ too. Thus we have shown that there are highly nontrivial examples of infinite well-poised elliptic hypergeometric series with radius of convergence larger than $1$. This is an important fact, since the values $z=\pm 1$ occur most often in applications (see below). Note that $r_c>1$ also for $q^b=p^M$ (see (4.6)), provided that $|R|<1$. Now suppose that $r$ is even, $r=2m$ and $\nu=1$. Then we get rid of the balancing condition by fixing $t_r=t_1^{m-1}w_1^m/\prod_{j=2}^{r-1}t_j$ and consider the resulting convergent well-poised series for fixed $z$ as an analytic function of the free parameters:
$$
\begin{equation*}
f(t_1,\dots,t_{r-1},w_1)=\sum_{k=0}^\infty \prod_{n=0}^{k-1} h(n)z^k, \qquad h(n)=\prod_{j=1}^{r} \frac{\theta(t_jq^n;p)}{\theta((t_1w_1/t_j)q^n;p)}.
\end{equation*}
\notag
$$
This is a $p$-shift invariant function of all of its variables
$$
\begin{equation*}
f(t_1,\dots, pt_j,\dots,w_1)=f(t_1,\dots, t_{r-1}, pw_1)=f(t_1,\dots,t_{r-1},w_1).
\end{equation*}
\notag
$$
However, it is not an elliptic function since it is not meromorphic. For example, as a function of $t_k$, for fixed $k\in\{2,\dots, r-1\}$ it has natural boundaries for $t_k=t_1w_1q^\mathbb{R}p^n$ and for $t_k=t_1^{m-2}w_1^{m-1}q^\mathbb{R}p^n/\prod_{j=2, \neq k}^{r-1}t_j$, where $n\in\mathbb{Z}$. As a function of $t_1$ or $w_1$, it has another, richer set of natural boundaries.
§ 5. The general case $t_0\neq q$Now we remove the constraint $t_0=q$ imposed before on the general $_{r+1}E_r$-series. As in the previous situation, the balancing condition implies that $H(u)$ with the argument $u=\exp(2\pi i x)$ is an elliptic function of $x$ with fundamental periods $\{1,\tau\}$ whose zeros are at the points $u=t_{i}^{-1}p^{\mathbb{Z}}$, $i=0,\dots,r$, and poles are at the points $u=w_{j}^{-1}p^{\mathbb{Z}}$, $j=1,\dots, r$. Consider a pair of integers $N$, $M$ such that the line from $x=0$ to $x=N+M\tau$ does not contain any zeros or poles of this function, together with a real number $\chi \in \mathbb{R}\setminus \mathbb{Q}$, and set The reason why we restrict our attention to irrational numbers $\chi$ is precisely the presence of the pole of $H(u)$ at the point $u=q^{-1}$. Since the singularity of $H(u)$ is of particularly nice type, the function $\log |H(\exp(2\pi i x (N+M\tau)))\sin(\pi (x+\chi ))|$, considered as a function on the unit interval, is continuous there. It follows that this function fulfills the requirements of Weyl’s equidistribution theorem, so that the following limit exists for arbitrary irrational $\chi $:
$$
\begin{equation}
\begin{aligned} \, \nonumber &\lim_{n\to\infty} \frac{1}{n} \sum_{k=0}^{n-1} \log \bigl|H\bigl(\exp(2\pi i k\chi (N+M\tau))\bigr) \sin\bigl(\pi ((k+1)\chi )\bigr)\bigr| \\ &\qquad =\int_{0}^{1} \log \bigl|H\bigl(\exp(2\pi i x (N+M\tau))\bigr)\sin\bigl(\pi (x+\chi )\bigr)\bigr| \,dx. \end{aligned}
\end{equation}
\tag{5.1}
$$
As follows from the considerations of Hardy and Littlewood in [5], this limit can be simplified further for those irrational numbers the denominators of whose Padé approximants satisfy the constraint
$$
\begin{equation}
\limsup_{k\to\infty} \frac{\log q_{k+1}}{q_k}=0 \quad\Longleftrightarrow\quad \liminf_{n \to \infty} \bigl|1-\exp(2\pi i n\chi)\bigr|^{1/n}=1.
\end{equation}
\tag{5.2}
$$
The latter condition holds for almost all irrational numbers. Therefore, for any such $\chi$ the sine-function multiplier in (5.1) can be removed, and we have
$$
\begin{equation*}
r_{c}^{-1}:=\lim_{n \to \infty} \bigl(|H(1)\dotsb H(q^{n-1})|\bigr)^{1/n} =\exp\biggl(\int_{0}^{1} \log\bigl |H\bigl(\exp(2\pi i x (N+M\tau))\bigr)\bigr|\,dx \biggr).
\end{equation*}
\notag
$$
Thus the computations of the previous section are applicable here as well — it is sufficient to extend summation to $k=0$ and set $w_0=q$. We have the following result, which extends that of Theorem 6
$$
\begin{equation}
\begin{gathered} \, \nonumber \log r_{c}^{-1}=\sum_{k=0}^r \biggl(\frac{\log^2|t_k|- \log^2|w_k|}{4\pi\operatorname{Im}\tau}+ \frac{\pi\operatorname{Im}\tau}{|N+M\tau|^2} \bigl(\alpha_k(\alpha_k-1)-\beta_k(\beta_k-1)\bigr)\biggr), \\ \alpha_k:=\biggl\{ \frac{\operatorname{Re}((N+M\tau)\log \overline{w}_k)}{2\pi\operatorname{Im}\tau}\biggr\}, \qquad \beta_k:=\biggl\{ \frac{\operatorname{Re}((N+M\tau)\log \overline{t}_k)}{2\pi\operatorname{Im}\tau}\biggr\}, \end{gathered}
\end{equation}
\tag{5.3}
$$
where $\{x\}$ means the fractional part of $x$. This formula in its turn can be simplified slightly further since $\operatorname{Re}((N+M\tau)\log \overline{q})=0$ implies that $\alpha_0=0$. Consider how our analysis simplifies in the limit as $p\to 0$. As $p$ approaches $0$, the expression we used for $q=\exp(2\pi i \chi (N+M\tau))$ becomes meaningless for $N, M>0$. Since we have assumed $(1/(2\pi i))\log q$ to lie on the straight line from $x=0$ to $x=N+M\tau$, the limit in question forces us to impose one of the following two conditions. Since $\operatorname{Im}\tau\to\infty$, for $M\neq 0$ our line becomes the vertical line $\operatorname{Re}x=0$ and the sequence $q^n$ should ‘walk’ along it. Therefore, $\operatorname{Re}((1/(2\pi i))\log q)=0$, that is, $q=\exp(-2\pi \chi)$ for some $\chi \in\mathbb{R}^\times$. For $M=0$ the sequence $q^n$ ‘walks’ along the line $\operatorname{Im}x=0$, that is, $q=\exp(2\pi i \eta)$ for some $\eta \in \mathbb{R}$. In the first case, we can relax the condition that the points $(1/(2\pi i))\log t_i$ and $(1/(2\pi i))\log w_i$, $i=1,\dots, r$, do not lie on the vertical ray containing the points $(x/(2\pi i))\log q$ for $x\leqslant 0$. It is enough to consider the standard condition
$$
\begin{equation}
t_j,w_j\neq q^l, \qquad l\in \mathbb{Z}_{\leqslant 0}, \quad j=0,\dots, r.
\end{equation}
\tag{5.4}
$$
And in the second case we impose the condition that the zeros and poles do not lie on the unit interval. Then the behaviour of the sequences under consideration (and therefore of integrals) as $p$ approaches $0$ can be described as follows. Since $H(u)$ itself has the limit
$$
\begin{equation*}
H_0(u) :=\lim_{p\to 0} H(u)=\frac{(1-ut_0)(1-ut_1)\dotsb(1-ut_r)}{(1-uq)(1-uw_1)\dotsb(1-uw_r)},
\end{equation*}
\notag
$$
in the case $M\neq 0$ the sequence $H_0(q^n)$ is bounded. The case when $M=0$ and $t_0=q=\exp(2\pi i \eta)$ for rational $\eta$ is also trivial. The case when $M=0$ and $t_0=q=\exp(2\pi i \eta)$ for an irrational $\eta$ leads to the limit
$$
\begin{equation*}
\lim_{n\to \infty} \frac{1}{n}\sum_{k=0}^{n-1} \log|H_0(q^k)|=\int_{0}^{1} \log|H_0(\exp(2\pi i x))|\,dx.
\end{equation*}
\notag
$$
Finally, in the case when $M=0$ and $t_0 \neq q=\exp(2\pi i \eta)$ the variable $\eta$ is supposed to be irrational by default because of the singularity of $H_0(u)$ at $u=q^{-1}$. For those irrational $\eta$ that satisfy the familiar constraint for the denominators of Padé approximants the same limit exists:
$$
\begin{equation*}
\lim_{n\to\infty} (|H_0(1)\cdots H_0(q^{n-1})|)^{1/n}=\exp\biggl(\int_{0}^{1} \log |H_0(\exp(2\pi i x))|\,dx\biggr).
\end{equation*}
\notag
$$
Since we restrict our attention to the case $M=0$, the only function $F_{N,M}(t)$ whose limiting behaviour is supposed to be examined is $F_{N,0}(t)=F_{1,0}(t)$. Set $C=[\log|t|/(2\pi \operatorname{Im}\tau)]$, where $[x]$ denotes the integer part of $x$. Using the remark (4.12) and the substitution
$$
\begin{equation*}
\biggl\{\frac{\log|t|}{2\pi \operatorname{Im}\tau}\biggr\} =\frac{\log|t|}{2\pi \operatorname{Im}\tau}-C
\end{equation*}
\notag
$$
into equality (4.16) we obtain an explicit expression:
$$
\begin{equation*}
F_{1,0}(t)=(C+1)\log|t|-C(C+1)\pi\operatorname{Im}\tau.
\end{equation*}
\notag
$$
For sufficiently large $\operatorname{Im} \tau$ the equality $C=0$ holds true if $\log|t| > 0$, and $C=-1$ if $\log|t|<0$, and we obtain
$$
\begin{equation*}
\begin{aligned} \, \lim_{p\to 0} F_{1,0}(t) =\begin{cases} 0 & \text{if}\ |t|\leqslant 1, \\ \log|t| & \text{if}\ |t|>1, \end{cases} \equiv\int_{0}^{1} \log |1-t\exp(2\pi ix)|\,dx. \end{aligned}
\end{equation*}
\notag
$$
Thus, the appearance of the function $H_0(u)$ under the integral sign is fully justified. Major applications of elliptic hypergeometric series appear in the form of the following very-well poised series
$$
\begin{equation}
{}_{r+1}V_{r}(t_0;t_1,\dots,t_{r-4};q,p):=\sum_{n=0}^\infty \frac{\theta(t_0q^{2n};p)}{\theta(t_0;p)}\prod_{m=0}^{r-4} \frac{\theta(t_m;p;q)_n}{\theta(qt_0t_m^{-1};p;q)_n}q^n
\end{equation}
\tag{5.5}
$$
(see [8]) with the balancing condition
$$
\begin{equation}
\prod_{k=1}^{r-4}t_k=\nu t_0^{(r-5)/2}q^{(r-7)/2}, \qquad \nu=\pm 1,
\end{equation}
\tag{5.6}
$$
where for odd $r$ it is assumed that $\nu=1$. This function can be obtained from the general elliptic hypergeometric $_{r+1}E_r$-series by imposing the condition of well-poisedness $qt_0=t_jw_j$, $j=1,\dots, r$, and the five additional constraints
$$
\begin{equation}
t_{r-3}=qt_0^{1/2}, \ \ t_{r-2}=-qt_0^{1/2}, \ \ t_{r-1}=q\biggl(\frac{t_0}p\biggr)^{1/2}, \ \ t_r=-q(pt_0)^{1/2}\ \ \text{and} \ \ z=-1,
\end{equation}
\tag{5.7}
$$
which result in the relation
$$
\begin{equation*}
\prod_{j=r-3}^r \frac{\theta(t_jq^n;p)}{\theta(t_0q^{n+1}/t_j;p)}z^n =\frac{\theta(t_0q^{2n};p)}{\theta(t_0;p)}q^n.
\end{equation*}
\notag
$$
Therefore, the function $H(u)$ determining the ratio $c_{n+1}/c_n=H(q^n)z$ of consecutive $_{r+1}E_r$-series coefficients, acquires the form
$$
\begin{equation}
H(u)=-q\frac{\theta(t_0q^2u^2;p)}{\theta(t_0u^2;p)} \prod_{j=0}^{r-4} \frac{\theta(t_ju;p)}{\theta(qt_0u/t_j;p)}
\end{equation}
\tag{5.8}
$$
with the balancing condition transformed into $q^4\prod_{k=0}^{r-4}t_k=\prod_{k=0}^{r-4}w_k$, where ${w_0\!=\!q}$, which reduces to relation (5.6) (a remark: the choice of $\nu=1$ for odd $r$ is a special convention [8]). Since the restrictions imposed on $t_j$ involve square roots of the variable $t_0$ and entail additional dependence on the $q$-variable, it is not obvious that all terms of the series (5.5) are elliptic functions of all of its parameters (including $q$), that is, that they are invariant under the transformations $t_j\to q^{n_j}t_j$, $j=0,\dots, r-4$, and $q\to p^m q$, where the integers $n_j$ and $m$ respect the balancing condition
$$
\begin{equation*}
\sum_{j=1}^{r-4}n_j=\frac{r-5}{2}\, n_0+\frac{r-7}{2}\, m.
\end{equation*}
\notag
$$
However, this is the case, as can be checked directly. In order to investigate the convergence of infinite $_{r+1}V_r$-series we do not substitute constraints (5.7) into the general formula for the radius of convergence (5.3), but we compute it anew using the function (5.8). Since the theta functions in the first ratio contain $u^2$ among their arguments, we should compute ${F_{N,M}(q^2t_0)-F_{N,M}(t_0)}$ using formula (4.14) where we choose $D=2$. However, we have $\operatorname{Re}\bigl((N+M\tau)\log \overline{q^2t_0}\,\bigr)=\operatorname{Re}\bigl((N+M\tau)\log \overline{t_0}\bigr)$ and the terms in (4.14) containing the function $\mathrm{Li}_2$ cancel out yielding
$$
\begin{equation*}
F_{N,M}(q^2t_0)-F_{N,M}(t_0)= \frac{\log^2|q^2t_0|-\log^2|t_0|}{4\pi\operatorname{Im}\tau}-\frac{M-1}{2}\log\frac{|q^2t_0|}{|t_0|}.
\end{equation*}
\notag
$$
The contributions of the other theta functions in (5.8) to the radius of convergence are
$$
\begin{equation*}
\begin{aligned} \, \sum_{k=0}^{r-4} (F_{N,M}(t_k)-F_{N,M}(w_k)) &=\frac{1}{4\pi\operatorname{Im}\tau} \log\biggl|\prod_{k=0}^{r-4}\frac{t_k}{w_k}\biggr| \log|qt_0| -\frac{M-1}{2}\log\biggl|\prod_{k=0}^{r-4}\frac{t_k}{w_k}\biggr| \\ &\qquad +\frac{\pi\operatorname{Im}\tau}{|N+M\tau|^2}\sum_{k=0}^{r-4} (\alpha_k(\alpha_k-1)-\beta_k(\beta_k-1)), \end{aligned}
\end{equation*}
\notag
$$
where the $\alpha_k$ and $\beta_k$ were defined in (5.3). Adding these two expressions and the contribution $\log|q|$ of the $q^n$-term in the $_{r+1}V_r$-series, applying the balancing condition (5.6) and the parametrization (4.21) we obtain
$$
\begin{equation}
\log r_{c}^{-1}=M\log|q|+\frac{\pi\operatorname{Im}\tau}{|N+M\tau|^2} \sum_{k=0}^{r-4} (\{\widetilde{\varphi}_k\}-\{\varphi_k\})(\{\widetilde{\varphi}_k\}+\{\varphi_k\}-1).
\end{equation}
\tag{5.9}
$$
In (5.9) we have the constraints $\widetilde \varphi_0=0$, following from the convention $w_0=q$, and also
$$
\begin{equation*}
\varphi_0=\varphi_k+\widetilde \varphi_k, \quad k=1,\dots, r-4,\quad\text{and} \quad \sum_{k=0}^{r-4} \varphi_k=\sum_{k=0}^{r-4}\widetilde \varphi_k.
\end{equation*}
\notag
$$
The latter equality follows from the relation $\prod_{k=0}^{r-4}t_k/w_k=q^{-4}$. Assume now that $\{\varphi_k\}=\varphi_k$, $k=0,\dots, r-4$, which is not forbidden by the constraints imposed. Then the sum of the terms depending on $\varphi_k$ and $\widetilde \varphi_k$ in (5.9) vanishes, and we obtain $\log r_{c}^{-1}=M\log|q|$. Because for $M>0$ we always have $|q|<1$, it follows that $r_c=|q|^{-M}>1$, that is, the infinite very-well poised elliptic hypergeometric $_{r+1}V_r$-series (5.5) does converge for the basic variable $q$ satisfying constraints (4.4) and (5.2) and the parameters $\varphi_k$ and $\widetilde \varphi_k$ (as fixed in (4.21)) taken from the domain $0\leqslant \varphi_k$, $\widetilde \varphi_k<1$ and $\varphi_k+\widetilde \varphi_k<1$, $k=1,\dots,r-4$. The very-well poised part $q^n\theta(t_0q^{2n};p)/\theta(t_0;p)$ of the $_{r+1}V_r$-series can be eliminated by a special choice of five parameters. Namely, one can assign to $w_{r-7},\dots, w_{r-4}$ the values $\pm qt_0^{1/2}$, $ q(t_0/p)^{1/2}$ and $-q(pt_0)^{1/2}$, respectively, and choose $t_{r-8}=q$. After denoting $qt_0=t_1w_1$, exactly the series considered in the previous section, for $z=-1$ and $r$ replaced by ${r-9}$, arises. Thus, the well-poised elliptic hypergeometric series with radius of convergence larger than $1$ (4.23) yields another example of a convergent infinite $_{r+1}V_r$-series. Alternatively, one can set only one restriction $t_0=q$ (so that $t_kw_k=q^2$, $k=1,\dots,r-4$), which makes the choice of $\chi$ as a rational number (see (4.6)) admissible, and then the resulting $_{r+1}V_r$-series is summed as a geometric series for $|R|<1$. All this opens the question of whether one can represent the results of exact computation of the elliptic beta integral and its multidimensional generalizations [11] as a consequence of some identities for infinite elliptic hypergeometric series (with appropriate restrictions on the parameters) which generalize the summation formulae for non-terminating very-well poised balanced $_8\varphi_7$-series [3] and their analogues for multiple sums. Our analysis of the convergence of elliptic hypergeometric series has used a special choice of the parameter $q$ guaranteeing that the coefficients of the series do not have singularities beyond certain particular lines on the complex plane which are determined by the parameters $t_k$ and $w_k$. For generic values of $p$ and $q$ for any choice of these parameters the sequence of points $q^n$, $n\to\infty$, approaches inevitably poles of the $H$-function arbitrarily closely. Then, the following question arises: is the two-dimensional version of Weyl’s equidistribution theorem applicable in such a situation? This question remains open and requires a separate detailed investigation. § 6. Special casesNow we restrict our attention to some special (namely, the simplest) theta-hypergeometric series and investigate if there are conditions that imply their convergence. We impose only one condition on the values of $q$, namely, $|q|\neq 1$. Considerations of the case $q=\exp(2\pi i \chi)$, $\chi\in\mathbb{R}$, require some deeper analysis in view of the recent results on the bound $\lim\inf_{n\to\infty}|(q;q)_n|>0$ established in [4] for $\chi $ equal to some algebraic numbers, in particular, to the golden ratio. $\bullet$ The $_{0} E_0$-series:
$$
\begin{equation*}
_{0} E_0( -; q, p\mid z)=\sum_{n=0}^\infty \frac{z^n}{\theta(q; p; q)_n}, \qquad 0 < |p| < 1.
\end{equation*}
\notag
$$
This series is interesting because it represents a direct theta-functional generalization of one of the Euler $q$-exponential functions
$$
\begin{equation*}
_{0} E_0(-;q, 0 \mid z)=\sum_{n=0}^\infty \frac{z^n}{(q; q)_n}=\frac{1}{(z;q)_\infty}, \qquad |z|<1
\end{equation*}
\notag
$$
(see [1] and [3]). The ratio of consecutive terms in $_{0} E_0$ is equal to $z/\theta(q^n; p)$, and we assume that $q^k\neq p^l$ for $k, l \in \mathbb{Z}$, that is, that the series is well defined. The standard theta function transformation (2.4) implies that it is enough to check the case $0 < |q| <1$ to understand the full picture. Consider the positive real number $\alpha = \log |q|/\log |p|$. For any positive integer $n$ consider also the integer $N_n=[n\alpha]$, so that the fractional part is $\{n\alpha\}=n\alpha - N_n$. Then
$$
\begin{equation*}
|\theta(q^n; p)|=|\theta(p^{N_n} q^n p^{-N_n}; p)| =| q^{-nN_n}p^{\binom{N_n+1}{2}} \theta(q^n p^{-N_n}; p)|.
\end{equation*}
\notag
$$
Now
$$
\begin{equation*}
|q^n p^{-N_n}|=|q|^n |p|^{-N_n}= \exp(n\log |q|- [n\alpha] \log |p|)=|p|^{\{n\alpha\}}.
\end{equation*}
\notag
$$
Also, since $|p|<1$, we have $\log |p| < 0$, hence $|p|< |p|^{\{n\alpha\}} \leqslant 1$. In addition, recall the trivial inequality $|1-u| \geqslant 1-|u|$. Hence we have the sequence of inequalities
$$
\begin{equation}
\begin{aligned} \, \nonumber |\theta(q^n; p)| & \geqslant |q|^{-nN_n} |p|^{\binom{N_n+1}{2}}(1-|p|^{\{n\alpha\}})(1-|p|^{1-\{n\alpha\}})(|p|; |p|)_\infty^2\\ \nonumber & \geqslant |p|^{-N_n^2-N_n\{n\alpha\}+\binom{N_n+1}{2}}(1-|p|^{\{n\alpha\}})(1-|p|^{1-\{n\alpha\}})(|p|; |p|)_\infty^2 \\ \nonumber & \geqslant |p|^{-N_n\{n\alpha\}-\binom{N_n}{2}}(1-|p|^{\{n\alpha\}})(1-|p|^{1-\{n\alpha\}})(|p|; |p|)_\infty^2 \\ & \geqslant |p|^{-\binom{N_n}{2}}(1-|p|^{\{n\alpha\}})(1-|p|^{1-\{n\alpha\}})(|p|; |p|)_\infty^2. \end{aligned}
\end{equation}
\tag{6.1}
$$
Suppose now that $\alpha$ is a rational number, $\alpha=a/b$. Recall that $q^b\neq p^a$, but it is still possible that $|q|^b=|p|^a$ for some coprime integers $a$ and $b$. If $n$ is coprime to $b$, then $(b-1)/b \geqslant \{n\alpha\}=k/b \geqslant 1/b$. Hence $|p|^{1-1/b}\leqslant |p|^{\{n\alpha\}} \leqslant |p|^{1/b}$, and the bound (6.1) can now be written as
$$
\begin{equation*}
|\theta(q^n; p)| \geqslant |p|^{-\binom{N_n}{2}}(1-|p|^{1/b})(1-|p|^{1-(1-1/b)})(|p|; |p|)_\infty^2 > (1-|p|^{1/b})^2(|p|; |p|)_\infty^2.
\end{equation*}
\notag
$$
If $b|n$, that is, $n=bk$, then $N_n=[bka/b]=ka$. Also, $\{ka\}=0$, so that $|q^n p^{-N_n}|=|q^{kb} p^{-ka}|=1$. Hence
$$
\begin{equation*}
\begin{aligned} \, |\theta(q^{bk}; p)| &=|p^{-\binom{ka}{2}} \theta(q^{kb} p^{-ka}; p)| \\ &=|p^{-\binom{ka}{2}} 2\sin(\pi k \sigma)(p\exp(2\pi i k\sigma); p)_\infty (p \exp(-2\pi i k\sigma); p)_\infty| \\ &\geqslant|p^{-\binom{ka}{2}} 2\sin(\pi k \sigma)|(|p|; |p|)_\infty^2, \end{aligned}
\end{equation*}
\notag
$$
where we have set $q^{b} p^{-a}= \exp(2\pi i \sigma)$, $\sigma \in \mathbb{R} \setminus \mathbb{Q}$ (recall that if $\sigma \in \mathbb{Q}$, then the series is undefined). Now, the function $|\sin(\pi k \sigma)|$ is $1$-periodic in $\sigma$ and, moreover, $|\sin(\pi k \sigma)|=|\sin(\pi \{k \sigma\})|=|\sin(\pi \{-k \sigma\})|$. Notice also, that $\{x\}+\{-x\}=1$ for $x \notin \mathbb{Z}$, hence $0 < \{x\} \leqslant 1/2$ if and only if $1/2 \leqslant \{-x\} < 1$. Choosing the appropriate sign and using the inequality $|\sin(\pi x)|\geqslant x/2$ for $0\leqslant x \leqslant 1/2$, we finally obtain the bound
$$
\begin{equation*}
|\theta(q^{bk}; p)| \geqslant|p|^{-\binom{ka}{2}}\{\pm k\sigma\}(|p|; |p|)_\infty^2.
\end{equation*}
\notag
$$
Now consider only those irrational numbers, that have the property
$$
\begin{equation*}
\exists\, C(\sigma)>0\colon \inf_{k\in \mathbb{Z}_{>0}} |p|^{-\binom{ka}{2}}\{\pm k\sigma\} > C(\sigma).
\end{equation*}
\notag
$$
For example, this can be any irrational algebraic number, or $\pi$, or any other number $\sigma$ such that the denominators of its Padé approximants satisfy the inequality
$$
\begin{equation*}
\limsup_{k\to \infty} \frac{\log q_{k+1}}{q_k^2} < -\frac{a^2}{2} \log |p|.
\end{equation*}
\notag
$$
Then the series converges for
$$
\begin{equation*}
|z|<\min(C(\sigma), (1-|p|^{1/b})^2)(|p|; |p|)_\infty^2.
\end{equation*}
\notag
$$
Suppose now that $\alpha$ is irrational. Then
$$
\begin{equation*}
\begin{aligned} \, |\theta(q^n; p)| &\geqslant |p|^{-\binom{N_n}{2}}(1-|p|^{\{n\alpha\}})(1-|p|^{\{-n\alpha\}})(|p|; |p|)_\infty^2 \\ &\geqslant |p|^{-\binom{N_n}{2}}(-\{n\alpha\}\log|p||p|^{\{n\alpha\}}) (-\{-n\alpha\}\log|p||p|^{\{-n\alpha\}})(|p|; |p|)_\infty^2 \\ &>|p|^{-\binom{n\alpha-1}{2}}\{n\alpha\}\{-n\alpha\}|p|\log^2|p|(|p|; |p|)_\infty^2, \end{aligned}
\end{equation*}
\notag
$$
where in the last line we used the trivial inequality $N_n > n\alpha-1$. Again, consider only those irrational numbers that have the property
$$
\begin{equation*}
\exists\, C(\alpha)>0\colon \inf_{n\in \mathbb{Z}_{>0}} |p|^{-\binom{n\alpha-1}{2}}\{n\alpha\}\{-n\alpha\} > C(\alpha),
\end{equation*}
\notag
$$
for example, any irrational algebraic number or $\pi$. Then the series converges in the domain $\bullet$ The $_{1} E_0$-series:
$$
\begin{equation*}
_{1} E_0\biggl(\begin{matrix} t_0 \\ - \end{matrix}; q, p\biggm|z\biggr)= \sum_{n=0}^\infty \frac{\theta(t_0; p; q)_n}{\theta(q; p; q)_n}z^n, \qquad 0 < |p| < 1.
\end{equation*}
\notag
$$
This is a theta-functional analogue of the series on the left-hand side of the $q$-binomial theorem
$$
\begin{equation*}
_{1} E_0\biggl(\begin{matrix} t_0 \\ - \end{matrix}; q, 0\biggm|z\biggr)= \sum_{n=0}^\infty \frac{(t_0;q)_n}{(q;q)_n}z^n=\frac{(t_0 z;q)_\infty}{(z;q)_\infty}, \qquad 0<|q|<1, \quad |z|<1
\end{equation*}
\notag
$$
(see [1] and [3]). The ratio of consecutive terms in $_{1} E_0$ is equal to $z\theta(t_0q^{n-1}; p)/\theta(q^n;p)$ which is well defined for $q^k\neq p^l$ for $k,l \in \mathbb{Z}$. Again, set $\alpha\!=\!\log|q|/\log|p|$. For any positive integer $n$ define $N_n=[n\alpha]$, so that $|q^n p^{-N_n}|=|p|^{\{n\alpha\}}$. Then from (2.4) we have the equality
$$
\begin{equation*}
\biggl|\frac{\theta(t_0q^{n-1};p)}{\theta(q^n;p)}\biggr|=|qt_0^{-1}|^{N_n} \biggl|\frac{\theta(t_0q^{n-1}p^{-N_n};p)}{\theta(q^np^{-N_n};p)}\biggr|.
\end{equation*}
\notag
$$
We proceed as in the previous example. Suppose that $\alpha$ is a rational number. Recall that $q^b \neq p^a$, but it is possible that $|q|^b=|p|^a$ for some coprime integers $a$ and $b$. If $n$ is coprime to $b$, then $(b-1)/b\geqslant \{n\alpha\} \geqslant 1/b$. Hence $|p|^{1-1/b} \leqslant |p|^{\{n\alpha\}} \leqslant |p|^{1/b}$. In addition, the absolute value of the theta function is clearly bounded above by some constant $S$ in the fundamental domain. Thus we arrive at the following bound
$$
\begin{equation*}
\biggl|\frac{\theta(t_0q^{n-1}p^{-N_n};p)}{\theta(q^np^{-N_n};p)}\biggr| \leqslant \frac{|\theta(t_0q^{n-1}p^{-N_n};p)|}{(1-|p|^{1/b})^2 (|p|;|p|)_\infty^2} \leqslant \frac{ S}{(1-|p|^{1/b})^2 (|p|;|p|)_\infty^2}.
\end{equation*}
\notag
$$
If $b|n$, that is, $n=bk$, then $N_n=ka$. Also $\{k\alpha\}=0$, so that $|q^n p^{-N_n}|=|q^{kb}p^{-ka}|=1$. Hence, we have
$$
\begin{equation*}
\biggl|\frac{\theta(t_0q^{kb-1};p)}{\theta(q^{kb};p)}\biggr| \leqslant \frac{1}{2}|qt_0^{-1}|^{ka}\frac{|\theta(t_0q^{-1}\exp(2\pi i \sigma k);p)|}{|\sin(\pi k\sigma)| (|p|;|p|)_\infty^2}.
\end{equation*}
\notag
$$
Here we write $q^bp^{-a}=\exp(2\pi i \sigma)$, for $\sigma \in \mathbb{R}\setminus \mathbb{Q}$. Now we choose $t_0$ such that
$$
\begin{equation*}
\exists\, C(\sigma, t_0) >0\colon \inf_{k\in\mathbb{Z}_{>0}} |qt_0^{-1}|^{-ka} |\{\pm k\sigma\}| > C(\sigma, t_0).
\end{equation*}
\notag
$$
Such $t_0$ does exist among ‘nice’ irrational numbers. As an example, we may take $t_0 = eq$ or $2q$ for an appropriate irrational number $\sigma$. Then any such pair $(t_0, \sigma)$ fits, and we have the bound
$$
\begin{equation*}
\biggl|\frac{\theta(t_0q^{kb-1};p)}{\theta(q^{kb};p)}\biggr| \leqslant \frac{|\theta(t_0q^{-1}\exp(2\pi i \sigma k);p)|}{C(\sigma, t_0)(|p|;|p|)_\infty^2} \leqslant \frac{S}{C(\sigma, t_0)(|p|;|p|)_\infty^2}.
\end{equation*}
\notag
$$
In other words the series converges for
$$
\begin{equation*}
|z| < \min ( (1-|p|^{1/b})^2, C(\sigma, t_0))S^{-1}(|p|;|p|)_\infty^2.
\end{equation*}
\notag
$$
Suppose now that $\alpha$ is irrational. We can write $q^n p^{-N_n}=|p|^{\{n\alpha\}} \exp(2\pi i r_n)$ for some real numbers $r_n$. The absolute value of the theta function in the numerator is bounded above, and also $1-|p|^{\{n\alpha\}} \geqslant |p|^{\{n\alpha\}}(-\log|p|)\{n\alpha\}$. Hence
$$
\begin{equation*}
\begin{aligned} \, \biggl|\frac{\theta(t_0q^{n-1};p)}{\theta(q^n;p)}\biggr| &\leqslant \frac{|qt_0^{-1}|^{N_n}|\theta(t_0q^{-1} |p|^{n\alpha} \exp(2\pi i r_n);p)|}{\{n\alpha\}\{-n\alpha\}|p|\log^2|p| (|p|;|p|)_\infty^2} \\ &\leqslant \frac{|qt_0^{-1}|^{N_n} S}{\{n\alpha\}\{-n\alpha\}|p|\log^2|p| (|p|;|p|)_\infty^2}. \end{aligned}
\end{equation*}
\notag
$$
Now we choose $t_0$ such that
$$
\begin{equation*}
\exists\, C(\alpha, t_0) >0\colon \inf_{n\in\mathbb{Z}_{>0}} |qt_0^{-1}|^{-N_n} |\{n\alpha\}\{-n\alpha\}| > C(\alpha, t_0).
\end{equation*}
\notag
$$
Then any such pair $(t_0, \alpha)$ fits, and the series converges for
$$
\begin{equation*}
|z| < C(\alpha, t_0) S^{-1}|p|\log^2|p|(|p|;|p|)_\infty^2.
\end{equation*}
\notag
$$
Citation in format AMSBIB
\Bibitem{KroSpi23} |
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