Discrete analytic functions and Taylor series

History. The notion of discrete analytic function on the Gaussian lattice $\mathbb{G} = \mathbb{Z} + i\,\mathbb{Z}$ was given by R. F. Isaacs [1]. He classified these functions into functions of first and second kind and investigated those of first kind. Further J. Ferrand [2] and R. J. Duffin [3] created the theory of discrete analytic functions of second kind (from now on: discrete analytic functions). Important results which are connected with a behaviour of discrete analytic and harmonic functions at infinity was obtained by S. L. Sobolev [4]. New combinatorial and analytical ideas to the theory were input by D. Zeilberger [5]. They were generalized by A. D. Mednykh [6]. An advance of nonlinear theory of discrete analytic functions based on usage of circle patterns began by W. Thurston [7] and his students [8], [9]. In that way an approximation with rapid convergence was obtained in the theory of conformal maps of Riemann surfaces.

Definition. From now on, $\mathbb{G} = \{x+i\,y : x,y\in \mathbb Z\}$ is the Gaussian integer lattice and $\mathbb{G}^{+} = \{x+i\,y\in \mathbb{G} : x\geq0, y\geq0 \}$ is a part of Gaussian plane contained in the first quadrant. A complex function $f$ defined on some subset $E\subset \mathbb{G}$ is called discrete analytic on $E$ if for any square $\{z, z~+~1, z~+~1~+~i, z~+~i\} ~\subset~E$ there holds:
$$\frac{f(z+1+i)-f(z)}{i+1} = \frac{f(z+i)-f(z+1)}{i-1}$$
or equivalently
$${\bar \partial}f(z) = f(z) + if(z + 1) + i^{2}f(z + 1 + i) + i^{3}f(z + i) = 0.$$
A discrete analytic function on all $\mathbb{G}^{+}$ is called entire discrete. Let us denote the set of all discrete analytic functions on $E$ and on $\mathbb{G}^+$ by $\mathcal{D}(E)$ and $\mathcal{D}(\mathbb{G}^{+})$ correspondingly.

Theorem 1. Every discrete analytic function $f\in \mathcal{D}(\mathbb{G}^{+})$ has a Taylor expansion in terms of $\pi_{k}(z):$
$$f(z) = \sum_{0}^{\infty} a_{k}\pi_{k}(z), \quad z\in \mathbb{G}^{+}.$$


Theorem 2. Above mentioned expansion is not unique. More precisely,
$$f(z) = \sum_{0}^{\infty} a_{k}\pi_{k}(z)\equiv 0, \quad z\in \mathbb{G}^{+} \Leftrightarrow F(s) = 0, \quad s\in \mathbb{Z}.$$


Theorem 3. A homomorphism $\Theta: {\mathcal A}(U_{R})\rightarrow {\mathcal D}(Q_{R})$ is “onto” and $\Theta(F)\equiv 0 \Leftrightarrow F(s) = 0$, $s\in \mathbb Z$, $|s|< R.$ In this case
$$Ker \, \Theta = \langle F_{N}(\xi)\rangle = F_{N}\cdot\mathcal(U_{R})$$
is a principal ideal in $\mathcal{A}(U_{R})$ generated by $F_{N} (\xi) = \xi\prod_{k=1}^{N}(\xi^{2}-k^{2})$, where $N = [R]$, if $R$ is non-integer and $R-1$ otherwise.

Theorem 4. Let $f \in {\mathcal D}(\mathbb{G}^{+})$. Then there exists a function $F(\xi) = \sum_{|k| =0}^{\infty}a_{k}\frac{\xi^{k}}{(1 + i)^{|k|}}\in {\mathcal A}({\mathbb C}^{n})$ such that $f(z) = \sum_{|k| =0}^{\infty} a_{k}\pi_{k}(z)$ and this expansion converges absolutely for all $z\in\mathbb{G}^{+}.$ In addition, $\Theta F = 0 \Leftrightarrow F(s) = 0$ for all $s \in \mathbb{Z}^{n}$.

References

1. Isaacs R. F.,, “A Finite Difference Function Theory”, Univ. Nac. Tucuman. Revista A., 2 (1941), 177–201    
2. Ferrand J., “Fonctions Preharmoniques et Functions Preholomorphes'”, Bull. Sci. Math., 68 (1944), 152–180  
3. Duffin R. J., “Basic Properties of Discrete Analytic Functions”, Duke Math. J., 23 (1956), 335–363    
4. Sobolev S. L., “A difference analog of the polyharmonic equation”, Soviet Math. Dokl., 6 (1965), 1174–1178  
5. Zeilberger D. A., “A New Basis for Discrete Analytic Polynomials”, J. Austral. Math. Soc., 23 (1977), 95–104    
6. Mednykh A. D., “Discrete analytic functions and Taylor series”, Theory of mappings, its generalizations and applications, Naukova Dumka, Kiev, 1982, 137–144
7. Thurston W. P., The finite Riemann mapping theorem, Purdue University, West Lafayette, 1985
8. Stephenson K., “Circle packing and discrete analytic function theory”, Handbook of complex analysis: geometric function theory, North-Holland, Amsterdam, 2002, 333–370    
9. Schramm O., “Circle patterns with the combinatorics of the square grid”, Duke Math. J., 86 (1997), 347–389