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This article is cited in 2 scientific papers (total in 2 papers) A generalization of the discrete Rodrigues formula for Meixner polynomials V. N. Sorokin Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
Russian version:
Matematicheskii Sbornik, 2022, Volume 213, Number 11, DOI: https://doi.org/10.4213/sm9765 
Bibliographic databases:
    § 1. Introduction1.1Let the discrete probability measure $\mu(x)=\mu(x;c,\beta)$ with support on the set of nonnegative integers $\mathbb Z_+$ be defined by putting the masses
$$
\begin{equation*}
\mu(\{x\})=(1-c)^\beta c^x \frac {\Gamma (x+\beta)}{\Gamma (x+1)\Gamma (\beta)}
\end{equation*}
\notag
$$
at the points $x\in \mathbb Z_+$, where $\Gamma$ is the gamma function (the Euler integral of the second kind). The measure involves two parameters, We denote by $P_n(x)$ the Meixner polynomials (see [1] and also [2]), that is, the polynomials orthogonal with respect to $\mu$. In other words, $P_n$ is a nonzero polynomial of degree $\leqslant n$ satisfying the orthogonality relations which define $P_n$ uniquely (up to normalization). The degree of $P_n$ is $n$, all of its zeros are simple and lie on the interval $\Delta=[0,+\infty)$, on which they are separated by integer points by the Chebyshev-Markov-Stieltjes theorem; this means that each interval $[m,m+1]$, $m\in \mathbb Z_+$, contains at most one zero of $P_n$.In what follows we consider only the case $\beta=1$. In this case is a geometric distribution.Meixner obtained the following discrete analogue of Rodrigues’s formula:
$$
\begin{equation}
P_n(x)c^x=\frac{1}{n!}\boldsymbol\Delta^n\{c^x(x+1)_n\}, \qquad n\in \mathbb Z_+;
\end{equation}
\tag{1.2}
$$
here is the Pochhammer symbol, and $\boldsymbol\Delta$ is the backward difference operator, that is, By formula (1.2) we fix a normalization of the polynomials, namely, $P_n(0)=1$. Various generalizations of Meixner polynomials related to Hermite-Padé approximations (see [11], [12]) were studied in [3]–[10]. General problems of convergence of sequences of discrete measures and their potentials were considered in the recent paper [13]. Apart from Hermite-Padé approximations, Diophantine approximation theory also uses extensively other constructions related, for example, to generalizations of Rodrigues’s formula. In this paper we study the polynomials defined by the formula
$$
\begin{equation}
A_n(x)c^x=\frac{1}{n!}\boldsymbol\Delta^n\{c^x(x+1)_nQ_n(x)\},
\end{equation}
\tag{1.3}
$$
where
$$
\begin{equation}
Q_n(x)c^x=\frac{1}{n!\,n!}\boldsymbol\Delta^n\{c^x((x+1)_n)^2\}, \qquad n\in \mathbb Z_+.
\end{equation}
\tag{1.4}
$$
The polynomial $A_n(x)$ has degree $3n$ and is normalized by the condition $A_n(0)=1$. In § 3 we describe fully the asymptotic behaviour of these polynomials as $n\to \infty$. We also formulate there the main results of the paper. For us, the main impetus for introducing the polynomials $A_n$ stems from their relations to Diophantine approximation theory. Namely, these polynomials are a new example of special functions providing the Apéry approximations to $\zeta(3)$, the value of the Riemann zeta function at $3$ (for more details, see § 1.3). The polynomials $Q_n$ are also related to Diophantine approximations. In [3] we obtained, inter alia, weak asymptotics of these polynomials. Below, in § 2, we quote from [3] to provide basic formulae for the asymptotic behaviour of the polynomials $Q_n$ in the form required in the proof of our main results. 1.2Let discuss the concept of weak asymptotics using the example of the Meixner polynomials. The masses (1.1) decay exponentially as $x\to\infty$, and so, following Rakhmanov [14], we scale by where $C_n$ is the normalizing constant such that the leading coefficient of the polynomial $P^\ast_n$ is $1$. The limit
$$
\begin{equation}
V(x)=\lim_{n\to\infty}\biggl({-\frac{1}{n}}\biggr)\log|P^\ast_n(x)|
\end{equation}
\tag{1.6}
$$
is known as the weak asymptotics of the sequence $\{P^\ast_n\}^\infty_{n=0}$. The function $V(x)$ is defined for those $x\in \mathbb {C}$ for which this limit exists. We denote the set of zeros of the polynomial $P^\ast_n$ by $\mathfrak {Z}(P^\ast_n)$. Consider the measures
$$
\begin{equation}
\lambda_n=\frac{1}{n}\sum_{\xi\in\mathfrak {Z}(P^\ast_n)}\delta_\xi,
\end{equation}
\tag{1.7}
$$
where $\delta_\xi$ is the Dirac function (the unit measure concentrated at the point $\xi$). So $\lambda_n$ is the normalized measure counting the zeros of the polynomial $P^\ast_n$. In [3] the author of this paper showed that the measures (1.7) converge in the weak${}^\ast$ topology of the dual space (pointwise convergence of functionals), that is,
$$
\begin{equation*}
\lambda_n\xrightarrow{\ast}\lambda, \qquad n\to\infty.
\end{equation*}
\notag
$$
The support $\mathsf{S}(\lambda)$ of the finite positive Borel measure $\lambda$ lies in the interval $\Delta$, and its total variation $\|\lambda\|$ is $1$. The measure $\lambda$ is called the limit measure of the zero distribution of the polynomials $P^\ast_n$. It was also shown in [3] that the limit (1.6) exists outside the support of this measure. Note also that
$$
\begin{equation*}
V(x)=V^\lambda(x), \qquad x\in \mathbb {C}\setminus\mathsf{S}(\lambda),
\end{equation*}
\notag
$$
where $V^\lambda$ is the logarithmic potential of $\lambda$ (see [15]), that is, the following (finite or infinite) Lebesgue integral
$$
\begin{equation*}
V^\lambda(x)=\int \log\frac{1}{|x-t|}\,d\lambda(t), \qquad x\in \mathbb {C}.
\end{equation*}
\notag
$$
The measure $\lambda$ is a solution of the following equilibrium problem in the theory of logarithmic potential (see [16]). Problem 1. Find a finite positive Borel measure $\lambda$ such that: 1) the support of this measure lies in the interval $\Delta$, $\mathsf{S}(\lambda) \subset \Delta;$ 2) the total variation of the measure is $1$, $\|\lambda\|=1$; 3) the constraint holds, where $\chi$ is the classical Lebesgue measure on $\Delta$;4) the equilibrium conditions
$$
\begin{equation*}
W=2V^\lambda+\Phi \begin{cases} \leqslant w &\text{on } \mathsf{S}(\lambda), \\ \geqslant w &\text{on } \Delta \setminus \mathsf{Z}(\lambda) \end{cases}
\end{equation*}
\notag
$$
are met, where $w$ is some equilibrium constant, and
$$
\begin{equation*}
\mathsf{Z}(\lambda)=\mathsf{S}(\lambda)\setminus\mathsf{S}(\chi-\lambda)
\end{equation*}
\notag
$$
is the saturation zone of the measure $\lambda$, that is, the subset of the support on which constraint (1.8) is attained; here $\Phi$ is the external field, The external field (1.9) appears after the scaling (1.5) due to the exponential decay of the measure $\mu$ at infinity. The constraint also appears after scaling in view of the theorem on the separation of zeros. Inequality (1.8) means that the signed measure $\chi-\lambda$, which is the difference of two measures, is also a positive measure. Rakhmanov [17] was the first to consider equilibria with constraints. We set
$$
\begin{equation*}
x_-=\frac{1-\sqrt c}{1+\sqrt c }\quad\text{and} \quad x_+=\frac{1+\sqrt c}{1-\sqrt c}.
\end{equation*}
\notag
$$
The measure $\lambda$ is supported on $[0,x_+]$ its saturation zone is $[0,x_-]$. 1.3We conclude this section by noting the following interesting fact on the relation of the polynomials $A_n$ to Diophantine approximation theory, or, more precisely, to one of its directions dealing with the arithmetical properties of values of the Riemann-Euler zeta-function. Recall that
$$
\begin{equation*}
\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}=\prod_p \frac{1}{1-1/{p^s}}, \qquad \operatorname{Re}s>1,
\end{equation*}
\notag
$$
where the product is taken over all prime numbers. The polynomials $A_n$ depend analytically on the parameter $c$; more precisely, they are polynomials of $q=1/c$. Setting $c=1$ and results in strong degeneracy: the degree of the polynomial $\mathring{A}_n$ is $n$. Consider the numbers These numbers are the denominators of the Apéry Diophantine approximations to $\zeta(3)$ (see [18]). The numbers (1.10) satisfy the three-term recurrence relation where $n=1, 2, 3, \dots$, with the initial conditions $a_0=1$ and $a_1=5$. This result is a direct consequence of (1.3) and (1.4).Prévost (see [19]) found the Apéry approximations to $\xi(2)$ by using the Touchard polynomials (see [20]), which coincide with the polynomials of degree $n$ that are orthogonal with respect to a certain linear functional $\mathfrak{S}$:
$$
\begin{equation*}
\mathfrak{S}\{\mathring{Q}_n(x)x^l\}=0, \qquad l=0, \dots, n-1, \quad n\in \mathbb Z_+.
\end{equation*}
\notag
$$
The power moments of this functional are the Bernoulli numbers, Recall that the Bernoulli numbers are defined by the following generating function (the Planck distribution):
$$
\begin{equation*}
\frac{z}{e^z-1}=\sum_{n=0}^{\infty}\frac{B_n}{n!}z^n, \qquad |z|<2\pi.
\end{equation*}
\notag
$$
The Apéry approximations to $\zeta(3)$ were obtained by Prévost by considering the polynomials orthogonal with respect to the functional
$$
\begin{equation*}
\mathfrak{S}'\{\varphi(x)\}=-\mathfrak{S}\{\varphi'(x)\}, \qquad \varphi(x)\in \mathbb C[x];
\end{equation*}
\notag
$$
these polynomials are related to the Wilson polynomials [21]. The above polynomials $\mathring{A}_n$ are different, so that they provide a new tool for constructing the Apéry approximations.
§ 2. Asymptotics of the polynomials $Q_n$2.1The polynomials $Q_n$, as defined by the discrete Rodrigues formula (1.4), were studied by this author in [3]. These are polynomials of degree $2n$ normalized by $Q_n(0)=1$ and satisfying the orthogonality conditions
$$
\begin{equation*}
\int{Q_n(x)x^l\,d\mu(x)=0}\quad\text{and} \quad \int{Q'_n(x)x^l\,d\mu(x)=0},
\end{equation*}
\notag
$$
where $l=0, \dots, n-1$, $n\in\mathbb Z_+$. Consider the following scaling: where $C^\ast_n$ is the normalizing constant such that the leading coefficient of the polynomial $Q^\ast_n$ is $1$. We let
$$
\begin{equation}
V_\ast(x)=\lim_{n\to\infty}\biggl({-\frac{1}{n}}\biggr)\log|Q^\ast_n(x)|
\end{equation}
\tag{2.1}
$$
denote the corresponding weak asymptotics. We set
$$
\begin{equation}
\lambda^Q_n=\frac{1}{n}\sum_{\xi\in\mathfrak {Z}(Q^\ast_n)}\delta_\xi.
\end{equation}
\tag{2.2}
$$
Then $\lambda^Q_n$ is the normalized measure counting the zeros of $Q^\ast_n$. It was shown in [3] that the sequence of measures (2.2) has a weak${}^\ast$ limit, where $\lambda_\ast$ is a finite Borel positive measure (on the complex plane) with compact support $\mathsf{S}(\lambda_\ast)$ and total variation 2 ($\|\lambda_\ast\|=2)$. This is the limit measure of the zero distribution of the polynomials $Q^\ast_n$. We also have
$$
\begin{equation*}
V_\ast(x)=V^{\lambda_\ast}(x), \qquad x \in \mathbb {C}\setminus\mathsf{S}(\lambda_\ast).
\end{equation*}
\notag
$$
The measure $\lambda_\ast$ is a solution of a certain vector Angelesco-type equilibrium problem (see [12]) with external field and a constraint. Before we can formulate this problem, we introduce some notation. We fix two (arbitrary for now) complex conjugate points $\zeta_+$ and $\zeta_-$, and a point ${x_\ast>0}$. Assume that $\zeta_+$ ($\zeta_-$) lies in the open upper (lower, respectively) half-plane: ${\zeta_\pm\!\in\!\mathbb {C}_\pm}$. We connect $\zeta_+$ and $x_\ast$ by an (arbitrary for now) analytic Jordan arc $\gamma_+$ lying in the open upper half-plane (except the endpoint $x_\ast)$. Next we connect $\zeta_-$ and $x_\ast$ by a similar arc $\gamma_-$, which is symmetric to $\gamma_+$ with respect to the real axis. We set The class of all such curves $\gamma$ is denoted by $\gimel$. The set will be referred to as a beetle.Problem 2. Find a curve $\Gamma \in \gimel$ and two finite Borel (in $\mathbb {C}$) positive measures $\lambda_\Delta$ and $\lambda_\Gamma$ such that: 1) the support of $\lambda_\Delta$ lies in the interval $\Delta=[0,+\infty)$: $\mathsf{S}(\lambda_\Delta)\subset \Delta$, and the support of $\lambda_\Gamma$ is the beetle $\Gamma_\ast$: $\mathsf{S}(\lambda_\Gamma)=\Gamma_\ast;$ 2) the total variation of each measure is $1$, 3) the constraint
$$
\begin{equation}
\lambda_\ast|_\Delta\leqslant2\chi, \qquad \lambda_\ast=\lambda_\Delta+\lambda_\Gamma
\end{equation}
\tag{2.3}
$$
holds, where $\chi$ is the classical Lebesgue measure on $\Delta$; 4) the following equilibrium conditions are met:
5) the curve $\Gamma$ is extremal, that is, it has the $S$-property (symmetry property), namely, where $\vec{n}_\pm$ are the unit normal vectors to the two sides of the cut $\Gamma$.2.2The following scheme of the proof of the above result has been taken from [3]. Rewriting the discrete Rodrigues formula (1.4) using Cauchy’s formula, scaling, and changing the variable we obtain
$$
\begin{equation*}
\begin{aligned} \, &Q_n(nx)c^{nx} \\ &\qquad =\frac{1}{n!}\,\frac{1}{n^n}\,\frac{1}{2\pi i}\int_{l_\ast} {c^{nt}\biggl(\frac{\Gamma(nt+n+1)}{\Gamma(nt+1)}\biggr)^2\frac{dt}{(t-x)(t-x+1/n)\cdots(t-x+n/n)}}, \end{aligned}
\end{equation*}
\notag
$$
where the closed contour $l_\ast$ separates the poles of the Cauchy kernel (namely, $x$, $x-1/n, \dots, x-n/n$) from those of the gamma function $\Gamma (nt+n+1)$ (namely, the points $-1-m/n$, $m \in \mathbb Z_+$). In place of the limit (2.1) it is more convenient to consider the limit We have
$$
\begin{equation}
V_Q(x)+\operatorname{Re} x \log c=\lim_{n\to\infty}\frac{1}{n} \log \biggl|\int_{l_\ast}\exp\{n{S}(t;x)\}\,dt\biggr|,
\end{equation}
\tag{2.4}
$$
where
$$
\begin{equation}
\begin{aligned} \, \notag S(t;x) &=t\log c +2\bigl((t+1)\log(t+1)-t\log t\bigr) \\ &\qquad +\bigl((t-x)\log(t-x)-(t-x+1)\log(t-x+1)\bigr). \end{aligned}
\end{equation}
\tag{2.5}
$$
The asymptotics of the integral in (2.4) will be examined using the saddle-point method. The critical points of the function (2.5) can be found from the equation
$$
\begin{equation*}
\frac{\partial{S}}{\partial t}=\log\frac{c(t+1)^2(t-x)}{t^2(t-x+1)}=0,
\end{equation*}
\notag
$$
which is equivalent to the cubic equation The discriminant of (2.6) is
$$
\begin{equation}
4(1-c)^2x^4-12(1-c^2)x^3+4(3-5c+3c^2)x^2-4(1-c^2)x+c.
\end{equation}
\tag{2.7}
$$
For any $c \in (0,1)$ the polynomial (2.7) has two complex conjugate zeros $\zeta_\pm\in\mathbb {C}_\pm$ and two positive zeros $a$ and $b$, $0<a<b$. These zeros are second-order branch points of the algebraic function $t(x)$. To identify its single-valued branches we draw cuts as follows: we connect the points $a$ and $b$ by the closed interval $\mathsf{E}$, and connect the points $\zeta_+$ and $\zeta_-$ by an arbitrary (for now) curve in the class $\gimel$. We let $t_0(x)$ denote the branch such that The main contribution to the asymptotics (at least, in some neighbourhood of the point at infinity) comes from this critical point $t_0$. Let denote the corresponding critical value. Consider the function Differentiating, we obtain where Eliminating the variable $t_0$, which satisfies equation (2.6), we obtain the following cubic equation for $\theta=\theta_0(x)$:
$$
\begin{equation}
\begin{aligned} \, \notag &c^2x^2\theta^3+(-c^2+2c(1-c)x-c(2+c)x^2)\theta^2 \\ &\qquad\qquad +(1-2(1-c)x+(1+2c)x^2)\theta-x^2=0. \end{aligned}
\end{equation}
\tag{2.9}
$$
The discriminant of this equation is the polynomial (2.7). Its zeros are second-order branch points of the algebraic function $\theta(x)$, and there are no other branch points. The branch $\theta_0(x)$ behaves at infinity as follows:
$$
\begin{equation*}
\theta_0(x)=1+\frac{2}{x}+O\biggl(\frac{1}{x^2}\biggr), \qquad x\to\infty.
\end{equation*}
\notag
$$
In (2.8) the principal branch of the logarithm is taken. The other two branches behave at the point at infinity as follows:
$$
\begin{equation*}
\theta_\pm(x)=\frac{1}{c}\biggl(1-\frac{1}{x}-\frac{a_\pm}{x^2}\biggr) +O\biggl(\frac{1}{x^3}\biggr), \qquad x\to\infty,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
a_+=\frac{\sqrt c}{1-\sqrt c}, \qquad a_-=-\frac{\sqrt c}{1+\sqrt c}.
\end{equation*}
\notag
$$
The function $h(x)$ is the Markov function of some measure $\lambda_\ast$:
$$
\begin{equation*}
h(x)=\int \frac{d\lambda_\ast(t)}{x-t}, \qquad x \in \mathbb {\overline C} \setminus \mathsf{S}(\lambda_\ast).
\end{equation*}
\notag
$$
The support of $\lambda_\ast$ is the set ${\Gamma_\ast \cup \mathsf{E}}$. In general, this is a complex measure, which can be recovered from $h$ using the Sokhotski-Plemelj formulae. The measure $\lambda_\ast$ is positive if the curve $\Gamma$ is extremal. We have where the measures $\lambda_\Delta$ and $\lambda_\Gamma$ have the following Markov functions:
$$
\begin{equation*}
-\log(c\theta_\pm(x)) \begin{cases} x \in \mathbb {\overline C} \setminus \mathsf{E}, \\ x \in \mathbb {\overline C} \setminus \Gamma_\ast. \end{cases}
\end{equation*}
\notag
$$
A detailed analysis of equation (2.9) leads to Problem 2 of equilibrium. Let us summarize the above findings. We have
$$
\begin{equation*}
\mathsf{S}(\lambda_\Delta)=\mathsf{E}, \qquad\mathsf{S}(\lambda_\Gamma)=\Gamma_\ast\quad\text{and} \quad \mathsf{Z}=[0,x_\ast].
\end{equation*}
\notag
$$
The point $x_\ast$ satisfies some transcendental equation. The interval $\mathsf{E}$ and the beetle $\Gamma_\ast$ can either be disjoint or intersect depending on the value of the parameter $c$ (Figures 1 and 2). We introduce the following more convenient notation for single-valued branches of the algebraic function $\theta$ near the point at infinity:
$$
\begin{equation*}
\theta_\ast=\theta_0, \qquad \theta_\Delta=\theta_+, \qquad \theta_\Gamma=\theta_-.
\end{equation*}
\notag
$$
The Riemann surface $\mathfrak{K}$ of this function is schematically shown in Figures 3 and 4 (in the cases when the compact sets $\mathsf{E}$ and $\Gamma_\ast$ are disjoint or intersect, respectively). This surface has genus $0$. Lifted to its Riemann surface, $\theta$ becomes a meromorphic function with the following divisor. The function $\theta$ has a second-order zero at the point $x=0$ on the sheet $\mathfrak{K}_\ast$, namelyThe function $\theta$ has a second-order pole at the point $x=0$ on the sheet $\mathfrak{K}_\Gamma$, namely
$$
\begin{equation*}
\theta_\Gamma(x)\thicksim\frac{1}{x^2}, \qquad x\to 0,
\end{equation*}
\notag
$$
and there are no other poles. The normalization condition is as follows: $ \theta=1$ at the point $x=\infty$ on $\mathfrak{K}_\ast$. Figure 5 shows the cross-section of the graph of $\theta(x)$ by the real plane $(\operatorname{Re}x, \operatorname{Re}\theta)$ in the case when the compact sets $\mathsf{E}$ and $\Gamma_\ast$ are disjoint (the changes necessary in the other case are clear). As $c\to 0$, the closed interval $\mathsf{E}$, the beetle $\Gamma_\ast$ and the saturation zone alike shrink to the interval $[0,1]$. As $c\to 1$, the closed interval $\mathsf{E}$ goes off to infinity, and the beetle $\Gamma_\ast$ shrinks to the interval $[-i/2,+i/2]$ of the imaginary axis, while the saturation zone disappears. § 3. Asymptotics of the polynomials $A_n$3.1Recall that the polynomials $A_n$ are defined by the discrete Rodrigues formula (1.3). Consider the scaling
$$
\begin{equation*}
A^\ast_n(x)=\widetilde C_nA_n(nx), \qquad n\in \mathbb {Z_+},
\end{equation*}
\notag
$$
where $\widetilde C_n$ is the normalizing constant such that the leading coefficient of the polynomial $A^\ast_n$ is $1$. We are interested in the weak asymptotics of the sequence $\{A^\ast_n\}^\infty_{n=0}$, that is, we need to find the limit
$$
\begin{equation}
\widetilde V(x)=\lim _{n\to\infty}\biggl(-\frac{1}{n}\biggr)\log|A^\ast_n(x)|.
\end{equation}
\tag{3.1}
$$
Let
$$
\begin{equation*}
\lambda^A_n=\frac{1}{n}\sum_{\xi\in\mathfrak{Z}(A^\ast_n)}\delta_\xi
\end{equation*}
\notag
$$
be the normalized measure counting the zeros of $A^\ast_n$. We show that there exists a limit measure of the zero distribution of these polynomials, that is, we show that the limit
$$
\begin{equation*}
\lambda^A_n\xrightarrow{\ast}\lambda_\star, \qquad n\to\infty
\end{equation*}
\notag
$$
exits. Here $\lambda_\star$ is a finite Borel positive measure (on the complex plane) with compact support $\mathsf{S}(\lambda_\star)$ and total variation 3 ($\|\lambda_\star\|=3)$. This measure is a solution of the vector equilibrium problem of the theory of logarithmic potential. Problem 3. Find a curve $\Gamma\in \gimel$ and four finite Borel positive measures $\lambda_\Delta$, $\lambda_\Gamma$, $\lambda_\alpha$ and $\lambda_\beta$ (in $\mathbb {C}$) such that: 1) the support of $\lambda_\Delta$ lies in the interval $\Delta=[0,\infty)$: $\mathsf{S}(\lambda_\Delta)\subset \Delta$, the support of the measure $\lambda_\Gamma$ is the beetle $\Gamma_\ast$: $\mathsf{S}(\lambda_\Gamma)=\Gamma_\ast$, the supports of the measures $\lambda_\alpha$ and $\lambda_\beta$ lie in the interval $\mathsf{F}=(-\infty,0]$: $\mathsf{S}(\lambda_\alpha)\subset \mathsf{F}$ and $\mathsf{S}(\lambda_\beta)\subset \mathsf{F}$; 2) the total variation of $\lambda_\Delta$ is 2, $\|\lambda_\Delta\|=2$, and the total variations of $\lambda_\Gamma$, $\lambda_\alpha$ and $\lambda_\beta$ are equal to $1$,
$$
\begin{equation*}
\|\lambda_\Gamma\|=\|\lambda_\alpha\|=\|\lambda_\beta\|=1;
\end{equation*}
\notag
$$
3) the following constraints are satisfied:
$$
\begin{equation*}
\lambda_\star|_\Delta\leqslant\chi_c\quad\text{and} \quad \lambda_\star=\lambda_\Delta+\lambda_\Gamma,
\end{equation*}
\notag
$$
where $\chi_c$ is a measure on $\Delta$ with density
$$
\begin{equation}
\chi'_c= \begin{cases} 1 &\text{on } [0,1), \\ 2 &\text{on } [1,+\infty], \end{cases}
\end{equation}
\tag{3.2}
$$
and also
$$
\begin{equation*}
\lambda_\alpha\leqslant\chi_{\mathsf{F}}\quad\text{and} \quad \lambda_\beta\leqslant\chi_{\mathsf{F}},
\end{equation*}
\notag
$$
where $\chi_{\mathsf{F}}$ is the classical Lebesgue measure on the interval $\mathsf{F}$; 4) the following equilibrium conditions are satisfied:
5) the curve $\Gamma$ is extremal, that is, it has the $S$-property (symmetry property), namely where $\vec{n}_\pm$ are the unit normal vectors to opposite sides of the cut $\Gamma$.The main result of the present paper is as follows. Theorem. There exists a limit measure $\lambda_\star$ of the zero distribution of the polynomials $A^\ast_n$. This measure is a solution of Problem 3. Moreover, The proof of this theorem is presented in the rest of § 3. 3.2Rewriting the discrete Rodrigues formula (1.3) using the Cauchy formula, scaling and changing the variable we have
$$
\begin{equation*}
A_n(nz)c^{nz}=\frac{1}{n^n}\,\frac{1}{2\pi i}\int_{l_\ast} {c^{nx}\frac{\Gamma(nx+n+1)}{\Gamma(nx+1)}\,\frac{Q_n(nx)\,dx}{(x-z)(x-z+1/n)\cdots(x-z+n/n)}},
\end{equation*}
\notag
$$
where the integration contour is the same as in (2.4). Here, in place of the limit (3.1) it is more convenient to introduce the limit Hence we have
$$
\begin{equation}
V_A(z)+\operatorname{Re} z \log c=\lim_{n\to\infty}\frac{1}{n} \log \biggl|\int_{l_\ast}\exp\bigl\{n\Sigma(x;z)\bigr\}\,dx\biggr|,
\end{equation}
\tag{3.3}
$$
where
$$
\begin{equation*}
\begin{aligned} \, \Sigma(x;z) &=\bigl((x+1)\log(x+1)-x\log x\bigr)+{S}_0(x) \\ &\qquad+\bigl((x-z)\log(x-z)-(x-z+1)\log(x-z+1)\bigr). \end{aligned}
\end{equation*}
\notag
$$
It should be noted that, although in our results we always present (and are interested in) weak asymptotics for polynomials, the saddle-point method delivers strong asymptotics in the actual fact. Of course, in the derivation of (3.3) we plug the strong asymptotics of the polynomials $Q_n$ under the integral sign. Moreover, the strong asymptotics hold uniformly on compact sets, and so, applying the saddle-point method to the residual term we obtain the same asymptotic formula for it as for the principal term, but with a factor of the form $O(1/\sqrt{n})$. To find the asymptotics of the integral in (3.3) we use the saddle-point method. The critical points of the function $\Sigma$ can be found from the equation
$$
\begin{equation}
\frac{\partial \Sigma}{\partial x}=\log\frac{(x+1)(x-z)}{x(x-z+1)}+{S}'_0(x)=0.
\end{equation}
\tag{3.4}
$$
We have
$$
\begin{equation*}
{S}'_0(x)=\frac{\partial {S}} {\partial x}\Big|_{t=t_0(x)}=\log\frac{t-x+1}{t-x},
\end{equation*}
\notag
$$
where $t=t_0(x)$. So (3.4) is equivalent to the equation Eliminating the variable $t$, which satisfies equation (2.6), we arrive at an algebraic equation for the critical points:
$$
\begin{equation}
\begin{aligned} \, \notag &(1-c)x^5+(1-c)(3-z)x^4+(3(1-c)+(3c-4)z)x^3 \\ &\qquad\qquad +((1-c)+(3c-5)z+2z^2)x^2+z((c-2)+3z)x+z^2(1-z)=0. \end{aligned}
\end{equation}
\tag{3.5}
$$
We consider the branch $x_\star(z)$ of the algebraic function $x(z)$ at infinity such that The main contribution to the asymptotics near the point at infinity comes from this critical point. We let denote the corresponding critical value. We set
$$
\begin{equation}
h_\star(z)=-\frac{d}{dz}\{\Phi(z)+\Sigma_\star(z)\}=\log\frac{1}{c}+ \frac{\partial \Sigma}{\partial z}\Big|_{x=x_\star(z)}=\log\phi(z),
\end{equation}
\tag{3.6}
$$
where Eliminating the variable $x$, which satisfies (3.5), we arrive at the following algebraic equation:
$$
\begin{equation}
\begin{aligned} \, \notag &c^4z^4\phi^5+c^3z(-c+(2-3c)z+(4-3c)z^2-(4+c)z^3)\phi^4 \\ \notag &\qquad +c^2((1-c)+2(2-c)z+3cz^2-4(3-2c)z^3+2(3+2c)z^4)\phi^3 \\ \notag &\qquad +cz((3c-4)+3(c-2)z+6(2-c)z^2-2(2+3c)z^3)\phi^2 \\ &\qquad +z^2((4-3c)-4z+(1+4c)z^2)\phi+z^3(1-z)=0. \end{aligned}
\end{equation}
\tag{3.7}
$$
3.3We consider (3.7) and the algebraic function $\phi(z)$ defined by this equation. Let us examine the behaviour of $\phi$ at infinity. In a neighbourhood of the point at infinity that is cut along the negative part of the real line, the multivalued function $\phi$ splits into five single-valued branches such that, as $z\to \infty$,
$$
\begin{equation*}
\begin{gathered} \, \phi_\star(z)=1+\frac{3}{z}+O\biggl(\frac{1}{z^2}\biggr), \\ \phi^+_\pm(z)=\frac{1}{c}\biggl(1-\frac{1}{z}\biggl(1+\frac{a_\pm}{\sqrt z}\biggr)\biggr)+O\biggl(\frac{1}{z^2}\biggr), \\ \phi^-_\pm(z)=\frac{1}{c}\biggl(1-\frac{1}{z}\biggl(1-\frac{a_\pm}{\sqrt z}\biggr)\biggr)+O\biggl(\frac{1}{z^2}\biggr), \end{gathered}
\end{equation*}
\notag
$$
where here we choose the branch of the square root $\sqrt z$ that is positive for $z>0$. We also note that the branch $\phi_\star$ corresponds to the critical point $x_\star$. Near the origin these three branches behave as
$$
\begin{equation*}
\phi(z)\thicksim \delta_{(j)}z, \qquad z \to 0, \quad j=1, 2, 3,
\end{equation*}
\notag
$$
where $\delta_{(1)}$, $\delta_{(2)}$ and $\delta_{(3)}$ are the roots of the cubic equation For all $c \in (0,1)$ this equation has one negative and two positive roots,
$$
\begin{equation*}
-\infty<\delta_{(1)}<0<\delta_{(2)}<\delta_{(3)}<+\infty.
\end{equation*}
\notag
$$
One branch goes off to infinity as
$$
\begin{equation*}
\phi(z)\thicksim\frac{1-c}{c^2}\,\frac{1}{z}, \qquad z\to 0,
\end{equation*}
\notag
$$
and the other as At the point $z=1$ one of the branches has a first-order zero. The discriminant of equation (3.7) is
$$
\begin{equation*}
\begin{aligned} \, Q(z;c) &=c(c-1)(32-27c)-4(32-127c+117c^2-27c^3)z \\ &\qquad -2(416-800c+453c^2-81c^3)z^2-4(152-123c+9c^2-27c^3)z^3 \\ &\qquad +(2320-3032c+813c^2+27c^3)z^4-32(1-c)(44+27c)z^5 \\ &\qquad+256(1-c)^2z^6. \end{aligned}
\end{equation*}
\notag
$$
For all $c \in (0,1)$ the discriminant has two negative zeros, two positive zeros and two complex conjugate ones, which we denote by
$$
\begin{equation*}
-\infty < \beta<\alpha<0<a<b<+\infty\quad\text{and} \quad \zeta_\pm \in \mathbb {C}_\pm.
\end{equation*}
\notag
$$
At each zero of the discriminant the function $\phi$ has a second-order branch point and three regular points. At the point at infinity $\phi$ has two second-order branch points and a regular point corresponding to the branch $\phi_\star$. This function has no other branch points. By the Riemann-Hurwitz formula the genus of the Riemann surface $\mathfrak {R}$ of $\phi$ is zero. Consequently, $\mathfrak {R}$ is a sphere. To identify single-valued branches we draw cuts as follows: we connect the points $a$ and $b$ by the closed interval $\mathsf{E}$, and connect the points $\zeta_+$ and $\zeta_-$ by an (arbitrary for now) curve $\Gamma$ in the class $\gimel$. We also draw the cuts $(-\infty,\beta]$ and $(-\infty,\alpha]$. This singles out the following branches:
$$
\begin{equation*}
\begin{aligned} \, \phi_\star \quad&\text{on the sheet } \mathfrak {R}_\star= \overline {\mathbb{C}}\setminus(\mathsf{E}\cup\Gamma), \\ \phi_\Delta=\phi^+_- \quad&\text{on the sheet } \mathfrak {R}_\Delta=\mathbb{C}\setminus(\mathsf{E}\cup(-\infty,\alpha]), \\ \phi_\Gamma=\phi^+_+ \quad&\text{on the sheet } \mathfrak {R}_\Gamma=\mathbb{C}\setminus(\Gamma\cup(-\infty,\beta]), \\ \phi_\alpha=\phi^-_- \quad&\text{on the sheet } \mathfrak {R}_\alpha=\mathbb{C}\setminus(-\infty,\alpha], \\ \phi_\beta=\phi^-_+ \quad&\text{on the sheet } \mathfrak {R}_\beta=\mathbb{C}\setminus(-\infty,\beta]. \end{aligned}
\end{equation*}
\notag
$$
Here and in what follows we consider the case when $x_\ast<a$ (the arguments in the other case are similar; see § 2.2). The Riemann surface $\mathfrak{R}$ is shown schematically in Figure 6. If $\phi$ is lifted to its Riemann surface, then the meromorphic function is uniquely defined by its divisor and the normalization condition. This function has a first-order zero at the point $z=1$ on the sheet $\mathfrak{R}_\Gamma$ and three first-order zeros at the point $z=0$ on the sheets $\mathfrak{R}_\star$, $\mathfrak{R}_\alpha$ and $\mathfrak{R}_\beta$, there being no other zeros. The function $\phi$ has a first-order pole at the point $z=0$ on the sheet $\mathfrak{R}_\Delta$ and a third-order pole at the point $z=0$ on the sheet $\mathfrak{R}_\Gamma$, there being no other poles. The normalization condition is as follows: $\phi=1$ at the point $z=\infty$ on $\mathfrak{R}_\star$. The cross-section of the graph of $\phi$ by the real plane is shown in Figure 7.3.4We present a solution to Problem 3. If $\nu$ is a finite Borel complex (in general) measure (in $\mathbb{C}$) with support $\mathsf{S}(\nu)$, then we let
$$
\begin{equation*}
\mathsf{h}_\nu(z)=\int\frac{d\nu(t)}{z-t}, \qquad z\in \mathbb{C}\setminus \mathsf{S}(\nu),
\end{equation*}
\notag
$$
denote its Markov function. The function $\phi_\alpha$ is holomorphic in the domain $\mathfrak{R}_\alpha$. We set here we take the principal branch of the logarithm for $z\to +\infty$. The function $h_\alpha$ is holomorphic in the domain $\mathbb{C}\setminus \mathsf{F}$ and is the Markov function of some positive measure $\lambda_\alpha$, that is, The support of this measure is the interval $\mathsf{F}$, and its total variation $\|\lambda_\alpha\|$ is $1$. If ${z\in [\alpha,0]}$, then $\operatorname{Im}h_\alpha(z)=\mp\pi i$ (on the upper and lower edges of this interval, respectively). Therefore, the measure $\lambda_\alpha$ is distributed uniformly with density $1$ on the interval $[\alpha,0]$. This closed interval is the saturation zone $\mathsf{Z}_\alpha$ of $\lambda_\alpha$.The function $\phi_\beta$ is holomorphic in the domain $\mathfrak {R}_\beta$. We set
$$
\begin{equation*}
h_\beta=-\log(c\phi_\beta)\quad\text{and} \quad h_\beta(\infty)=0.
\end{equation*}
\notag
$$
The function $h_\beta$ is holomorphic in the domain $\mathbb {C}\setminus\mathsf{F}$. We also have where $\lambda_\beta$ is a positive measure with support $\mathsf{F}$, total variation 1 ($\|\lambda_\beta\|=1)$ and saturation zone $\mathsf{Z_\beta}=[\beta,0]$. The function $\phi_\Delta$ is meromorphic in the domain $\mathfrak {R}_\Delta$. We set
$$
\begin{equation*}
h_\Delta=-\log(c\phi_\Delta), \qquad h_\Delta(\infty)=0.
\end{equation*}
\notag
$$
The function $h_\Delta$ is holomorphic in the domain $\mathbb {C}\setminus(\mathsf{E}\cup\mathsf{F})$. We also have
$$
\begin{equation}
h_\Delta=\mathsf{h}_{\lambda_\Delta}-\mathsf{h}_{\lambda_\alpha}.
\end{equation}
\tag{3.10}
$$
(The measure $\lambda_\alpha$ was defined above.) The support of the positive measure $\lambda_\Delta$ is the closed interval $\mathsf{E}$, and its total variation $\|\lambda_\Delta\|$ is $2$. The function $\phi_\Gamma$ is meromorphic in the domain $\mathfrak {R}_\Gamma$. We set
$$
\begin{equation*}
h_\Gamma(z)=-\log\biggl(c\phi_\Gamma(z)\frac{z}{z-1}\biggr),\qquad h_\Gamma(\infty)=0.
\end{equation*}
\notag
$$
The function $h_\Gamma$ is holomorphic in the domain $\mathbb {C}\setminus(\Gamma_\ast\cup\mathsf{F})$. We also have
$$
\begin{equation}
h_\Gamma=\mathsf{h}_{\lambda_\Gamma}-\mathsf{h}_{\lambda_\beta}.
\end{equation}
\tag{3.11}
$$
(The measure $\lambda_\beta$ was defined above.) The measure $\lambda_\Gamma$ has support on the beetle $\Gamma_\ast$; its total variation $\|\lambda_\Gamma\|$ is $1$. On $\Gamma$ this measure is in general complex. We draw a curve $\Gamma$ on which the measure $\lambda_\Gamma$ is positive. The function $\phi_\star$ is holomorphic in the domain $\mathfrak {R}_\star$. As in (3.6), we set $h_\star=\log \phi_\star$. The function $h_\star$ is holomorphic in the domain $\mathbb {C}\setminus(\mathsf{E}\cup\Gamma_\ast)$. We also have
$$
\begin{equation}
h_\star=\mathsf{h}_{\lambda_\star}\quad\text{and} \quad \lambda_\star=\lambda_\Delta+\lambda_\Gamma.
\end{equation}
\tag{3.12}
$$
Next, $\operatorname{Im} h_\star=\mp\pi i$ on the upper and lower sides of the interval $[0,x_\ast]$, respectively. Therefore, on this interval the measure $\lambda_\star$ is uniformly distributed with density $1$. The saturation zone $\mathsf{Z}$ of $\lambda_\star|_\Delta$ is the interval $[0,x_\ast]$. (Recall that we carry out the proof in the case when $x_\ast<a$.) According to the saddle-point method, outside the compact set $\mathsf {E} \cup \Gamma_\ast$ the polynomials $A^\ast_n$ have geometric asymptotics, and this compact set is the limit set for the zeros of these polynomials. This is just the condition for the positivity of $\lambda_\Gamma$. From (3.8)–(3.12) it follows that
$$
\begin{equation}
\begin{gathered} \, \mathsf{h}_{\lambda_\alpha}=h_\alpha, \qquad \mathsf{h}_{\lambda_\beta}=h_\beta, \\ \mathsf{h}_{\lambda_\Delta}=h_\Delta+h_\alpha, \qquad \mathsf{h}_{\lambda_\Gamma}=h_\Gamma+h_\beta\quad\text{and} \quad \mathsf{h}_{\lambda_\star}=h_\star. \end{gathered}
\end{equation}
\tag{3.13}
$$
We let $\mathcal {W}_J$, where $J=\alpha, \beta, \Delta, \Gamma$, denote the complexification of the harmonic (outside the relevant cuts) function $W_J$. From (3.13) we obtain
$$
\begin{equation*}
\mathcal {W}'_\alpha\,{=}\,\log \frac{\phi_\alpha}{\phi_\Delta}, \qquad\mathcal {W}'_\beta=\log \frac{\phi_\beta}{\phi_\Gamma}, \qquad\mathcal {W}'_\Delta=\log \frac{\phi_\Delta}{\phi_\star}\quad\text{and} \quad \mathcal {W}'_\Gamma=\log \frac{\phi_\Gamma}{\phi_\star};
\end{equation*}
\notag
$$
here we take the principal branch of the logarithm at infinity. The generalized potentials $W_J$ are continuous in the whole complex plane and piecewise differentiable on the real axis. We have
$$
\begin{equation*}
W'_\alpha=\log\biggl|\frac{\phi_\alpha}{\phi_\Delta}\biggr| \begin{cases} =0 &\text{on } (-\infty,\alpha), \\ <0 &\text{on } (\alpha,0). \end{cases}
\end{equation*}
\notag
$$
Therefore, for some constant $w_\alpha$ we have
$$
\begin{equation*}
W_\alpha \begin{cases} =w_\alpha &\text{on } (-\infty,\alpha], \\ \leqslant w_\alpha &\text{on } [\alpha,0], \end{cases}
\end{equation*}
\notag
$$
which proves the equilibrium condition ($\alpha$). Next, we have
$$
\begin{equation*}
W'_\beta=\log\biggl|\frac{\phi_\beta}{\phi_\Gamma}\biggr| \begin{cases} =0 &\text{on } (-\infty,\beta), \\ <0 &\text{on } (\beta,0). \end{cases}
\end{equation*}
\notag
$$
Therefore, for some constant $w_\beta$,
$$
\begin{equation*}
W_\beta \begin{cases} = w_\beta &\text{on } (-\infty,\beta], \\ \leqslant w_\beta &\text{on } [\beta,0], \end{cases}
\end{equation*}
\notag
$$
which proves the equilibrium condition ($\beta$) . Further, we have
$$
\begin{equation*}
W'_\Delta=\log\biggl|\frac{\phi_\Delta}{\phi_\star}\biggr| \begin{cases} <0 &\text{on } (x_\ast, a), \\ =0 &\text{on } (a,b), \\ >0 &\text{on } (b,+\infty). \end{cases}
\end{equation*}
\notag
$$
Therefore, for some constant $w_\Delta$,
$$
\begin{equation*}
W_\Delta \begin{cases} \geqslant w_\Delta&\text{on } [x_\ast,a], \\ = w_\Delta &\text{on } [a,b], \\ \geqslant w_\Delta &\text{on } [b,+\infty), \end{cases}
\end{equation*}
\notag
$$
which proves the equilibrium condition ($\Delta$). Next, we have
$$
\begin{equation*}
W'_\Gamma=\log\biggl|\frac{\phi_\Gamma}{\phi_\star}\biggr| >0 \quad \text{on } (0,x_\ast).
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
W_\Gamma \leqslant w_\Gamma \quad \text{on } [0,x_\ast],
\end{equation*}
\notag
$$
where $w_\Gamma=W_\Gamma(x_\ast)$. It remains to verify the equilibrium condition on the curve $\Gamma$. Let $\tau$ be the unit tangent vector to $\Gamma$ (represented by a complex number). The derivative of the generalized potential $W_\Gamma$ with respect to the natural parameter on this curve is
$$
\begin{equation*}
\dot W_\Gamma=\operatorname{Re}\biggl\{\tau \log\frac{\phi_\Gamma}{\phi_\star}\biggr\}=\operatorname{Re}\{\tau ((\mathsf{h}_{\lambda_\Gamma})_+-(\mathsf{h}_{\lambda_\Gamma})_-)\};
\end{equation*}
\notag
$$
here the values of the Markov function $\mathsf{h}_{\lambda_\Gamma}$ are taken on the opposite sides of the cut $\Gamma$. This function can be expressed as a Cauchy-type integral:
$$
\begin{equation*}
\mathsf{h}_{\lambda_\Gamma}(z)=\int_{\Gamma} \frac{\omega(\zeta)\,d\zeta}{z-\zeta}+\widetilde h(z);
\end{equation*}
\notag
$$
here $\omega$ is some complex function on $\Gamma$, and the function $\widetilde h$ is holomorphic on the set $\mathring\Gamma_+ \cup \mathring\Gamma_-$. Hence by the Sokhotski-Plemelj formulae
$$
\begin{equation*}
\dot W_\Gamma=\operatorname{Re}\{2\pi i \omega \tau\},
\end{equation*}
\notag
$$
where $ \omega \tau$ is the density of a positive measure. Therefore, $\dot W_\Gamma=0$ on $\mathring\Gamma_+ \cup \mathring\Gamma_-$, and thus $W_\Gamma=w_\Gamma$ on $\Gamma$. This proves the equilibrium condition ($\Gamma$). The equilibrium condition on $\Gamma$ is equivalent to the $S$-property of this curve. This proves the theorem. 3.5In this concluding subsection we consider the limit cases. Let $c\downarrow0$. This is the case of a strong external field. We have
$$
\begin{equation*}
a\downarrow0, \qquad b\downarrow2\quad\text{and}\quad \zeta_\pm\to2.
\end{equation*}
\notag
$$
The interval $\mathsf{E}$ and the beetle $\Gamma_\ast$ shrink to the interval $[0,2]$. In the limit, equation (3.7) degenerates to where $\mathring\phi=\phi|_{c=0}$. The function $\mathring\phi$ has simple zeros at $z=0$ and $z=1$, and a second-order pole at $z=2$, there being no other zeros and poles. At infinity
$$
\begin{equation*}
\mathring\phi(z)=1+\frac{3}{z}+O\biggl(\frac{1}{z^2}\biggr).
\end{equation*}
\notag
$$
We set
$$
\begin{equation*}
\mathring h =\log\mathring\phi, \qquad \mathring h(\infty)=0.
\end{equation*}
\notag
$$
The function $\mathring h$ is holomorphic in the domain $\overline{\mathbb{C}}\setminus[0,2]$. We also have where $\mathring\lambda=\chi_c$ is a positive measure with density (3.2) on the interval $[0,2]$. There are no equilibrium conditions, and the limit measure coincides with its constraint. Let $c\uparrow1$. This is the case of a weak external field. We have
$$
\begin{equation*}
a\uparrow +\infty, \qquad b\uparrow +\infty, \qquad \beta\downarrow\beta^\circ\quad\text{and} \quad \zeta_\pm\to\zeta_\pm^\circ,
\end{equation*}
\notag
$$
where $\beta^\circ$ and $\zeta_\pm^\circ$ are the zeros of the cubic polynomial namely, $\beta^\circ\approx-0.35$ and $\zeta_\pm^\circ\approx 0.35 \pm 0.56i$. The interval $\mathsf{E}$ goes off to infinity. The beetle $\Gamma_\ast$ approaches some limiting beetle $\Gamma^{\circ}_\ast$. In the limit (3.7) turns to the cubic equation
$$
\begin{equation*}
z^3\varphi^3-(3z^3-z^2+z+1)\varphi^2+z(3z^2-2z+1)\varphi+z^2(1-z)=0,
\end{equation*}
\notag
$$
where $\varphi=\phi|_{c=1}$. The roots of this equation behave as follows at infinity:
$$
\begin{equation*}
\begin{gathered} \, \varphi_0(z)=1+\frac{1}{z}+O\biggl(\frac{1}{z^2}\biggr), \\ \varphi_\pm(z)=1-\frac{1}{z}\biggl(1\pm\frac{1}{\sqrt{2z}}\biggr)+O\biggl(\frac{1}{z^2}\biggr); \end{gathered}
\end{equation*}
\notag
$$
here $\sqrt{2z}>0 $ as $z\to+\infty$. For the function $\varphi$ the points $\zeta^\circ_\pm$, $\beta^\circ$ and $\infty$ are second-order branch points, there being no other branch points. The Riemann surface of $\varphi$, which is glued of the three sheets
$$
\begin{equation*}
\begin{gathered} \, \mathfrak{N}_\circ=\mathbb{\overline C}\setminus\Gamma^\circ, \\ \mathfrak{N}_+=\mathbb{C}\setminus(\Gamma^\circ\cup(-\infty,\beta^\circ]) \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mathfrak{N}_-=\mathbb{C}\setminus(-\infty,\beta^\circ],
\end{equation*}
\notag
$$
is of genus $0$. The branch $\varphi_J$ is meromorphic on the sheet $\mathfrak{N}_J$, where $J=0,\pm$. The function $\varphi$ (as lifted to $\mathfrak{N}$) has a simple zero at the point $z=1$ on the sheet $\mathfrak{N}_+$ and two simple zeros at the point $z=0$ on the sheets $\mathfrak{N}_-$ and $\mathfrak{N}_0$. Note also that
$$
\begin{equation*}
\varphi_-(z)\thicksim\frac{1}{\upsilon}z, \qquad z\to 0,
\end{equation*}
\notag
$$
and where $\upsilon=(\sqrt 5 -1)/2$ is the golden section. The function $\varphi$ has a third-order pole at the point $z=0$ on the sheet $\mathfrak{N}_+$, namely, This function, which has no other zeros and poles, is normalized by the condition $\varphi=1$ at the point $z=\infty$ on the sheet $\mathfrak{N}_0$. The graph of the function $\varphi$ on the real axis is shown in Figure 8. We set
$$
\begin{equation*}
h_-=-\log\varphi_-\quad\text{and} \quad h_-(\infty)=0.
\end{equation*}
\notag
$$
This function is holomorphic in the domain $\mathbb{C}\setminus\mathsf{F}$. Moreover, where $\lambda_-$ is a positive measure with support $\mathsf{F}$, total variation 1 ($\|\lambda_-\|=1)$ and saturation zone $[\beta^\circ,0]$. We set
$$
\begin{equation*}
h_0=\log\varphi_0\quad\text{and} \quad h_0(\infty)=0.
\end{equation*}
\notag
$$
This function is holomorphic in the domain $\overline{\mathbb{C}}\setminus\Gamma^\circ_\ast$. We also have where $\lambda_0$ is the positive unit measure with support $\Gamma^\circ_\ast$. We set
$$
\begin{equation*}
h_+(z)=-\log\biggl(\varphi_+(z)\frac{z}{z-1}\biggr), \qquad h_+(\infty)=0.
\end{equation*}
\notag
$$
The function $h_+$ is holomorphic in the domain $\mathbb{C}\setminus(\Gamma^\circ_\ast\cup\mathsf{F})$. We also have The following equilibrium conditions are satisfied for some constants $w_0$ and $w_-$ under the constraints $\lambda_-\leqslant \chi_{\mathsf{F}}$ and $\lambda_0\leqslant\chi_c$:
$$
\begin{equation*}
\begin{gathered} \, W_0=2V^{\lambda_0}-V^{\lambda_-}+V^{\chi_0} \begin{cases} = w_0 &\text{on } \Gamma^\circ, \\ \leqslant w_0 &\text{on } [0,x_\ast], \end{cases} \\ W_-=-V^{\lambda_0}+2V^{\lambda_-}-V^{\lambda_{\chi_0}} \begin{cases} = w_- &\text{on } (-\infty,\beta^\circ], \\ \leqslant w_- &\text{on } [\beta^\circ,0]. \end{cases} \end{gathered}
\end{equation*}
\notag
$$
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\Bibitem{Sor22} |
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