Lattice sums for double periodic polyanalytic functions

In 1892, Lord Rayleigh estimated the effective conductivity of rectangular arrays of disks and proved, employing the Eisenstein summation, that the lattice sum S2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_2$$\end{document} is equal to π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi $$\end{document} for the square array. Further, it became clear that such equality can be treated as a necessary condition of the macroscopic isotropy of composites governed by the Laplace equation. This yielded the description of two-dimensional conducting composites by the classic elliptic functions, including the conditionally convergent Eisenstein series. In 1935, Natanzon used a polyharmonic function to solve the plane elasticity problem. This paper is devoted to the extension of the classic lattice sums to the lattice sums for double periodic (pseudoperiodic) polyanalytic functions. The exact relations and computationally effective formulae between the polyanalytic and classic lattice sums are established. Polynomial representations of the lattice sums are obtained. They are a source of new exact formulae for the lattice sums.


Introduction
Various lattice sums are widely used to study the mechanical properties of crystals and the optical properties of regular lattices of atoms [1].The classical theory of Piotr Drygaś and Vladimir Mityushev contributed equally to this work.
elliptic functions [2] can be considered as the theory of doubly periodic analytic (meromorphic) functions or the theory of functions on torus [3].The 2D lattice sums are an element of this theory.This paper extends the lattice sums to doubly periodic polyanalytic functions.
Let a doubly periodic lattice on the complex plane be determined by two fundamental translation vectors expressed by the complex numbers ω 1 , ω 2 .Without loss of generality we may assume that ω 1 > 0 and Imτ > 0 where τ = ω 2 ω 1 .Let the area of each cell be normalized to unity, i.e., ω 2  1 Imτ = 1.Though the assumptions on ω 1 and the area partly restrict the application of the elliptic modular group SL(2, Z) [3, Sect.V.7] to the extension of the formulae derived in the present paper, they are natural in the theory of composites and explicitly demonstrate the role of the number π in the description of isotropic lattices.
The classic lattice sums are introduced as follows (1) The symbol m,n means that m and n run over all integer numbers, except the pair m = n = 0.The series S q (ω 1 , ω 2 ) we denote by S q (τ ) or S q for short.The series (1) are absolutely convergent for q > 2, and conditionally convergent for q = 1, 2 [4].
The classic Weierstrass functions are expanded into the Laurent series near zero The conditionally convergent sum S 2 can be defined by the Eisenstein summation method [4] m,n ( The Eisenstein functions of first and second order are introduced by the summation (3) and can be expanded into the series [4] Rayleigh [5] investigated doubly periodic problems for harmonic functions with one circular inclusion per periodicity cell and applied them to the calculation of the effective conductivity.Rayleigh [5] used the Eisenstein summation and proved that for rectangular lattices Rayleigh [5] demonstrated that S 2 = π for the square array when τ = i.The same formula S 2 = π holds for the hexagonal (triangular) array when τ = exp( πi 3 ).It is worth noting that only the square and hexagonal arrays form macroscopically isotropic conducting regular composites.
Historical remarks and various applications of polyanalytic functions can be found in [6,7].A polyanalytic version of the Weierstrass sigma function and its application to signal analysis was discussed in [6] and works cited therein.A theory of polyanalytic doubly periodic functions having applications to 2D elastic problems was developed in [8][9][10][11].The procedure of homogenization for double periodic elastic structures requires periodicity (pseudoperiodicity) of local elastic fields.This leads to the particular theory of periodic (pseudoperiodic) polyanalytic functions summarized in [10, Appendix 2].One of the fundamental notions of this theory is the lattice sums We use the following terminology: (1) is called the classic lattice sum and ( 6) the panalytic or polyanalytic lattice sums.Similar to the expansions (4) for the Weierstrass functions, the main bianalytic doubly periodic functions can be expanded into series, including the lattice sums S (1) The function (7) used to provide the solution for the plane elasticity problem is polyharmonic with p = 1 [8,10].
The sum S (1)  3 (ω 1 , ω 2 ) is conditionally convergent and can be defined by the Eisenstein summation, which yields [11, formula (2.10)] It was proved in [11] that S (1) 3 = π 2 for the hexagonal array, which is the unique regular periodic structure for 2D elastic composite.It is worth noting that the square array does not form an isotropic elastic structure, and S (1) [11, formula (4.19)].
In the present paper, we study the p-analytic lattice sums (6) associated with the panalytic functions.A computationally effective formula (18) for the p-analytic lattice sums (6) is derived.As the particular formulae ( 5) and (8), formula (18) is based on the trigonometric series, first arisen in the works due to Eisenstein [4] and further in [12], see for extended Refs.[1,11].Special attention is paid to the conditionally convergent sums S ( p) p+2 .A simple formula for some of S ( p) p+2 based on numerical computations is suggested.In particular cases, it coincides with the known formulae S q through the classic lattice sums S q is established.

Expression of p-analytic sums through the classic lattice sums
Consider the p-analytic lattice sums (6) for q − p ≥ 2. If q − p > 2, the series absolutely converges.In the case q = p + 2 we arrive at the conditionally convergent series S ( p) p+2 .The Eisenstein summation method will be applied to S ( p) p+2 .First, consider the function and the functions The n-th derivative of ε 0 satisfies equation [4] The following formula was proved in [13] S (1) We now proceed to extend this formula to the series (6).Due to assumptions about translation vectors ω 1 , ω 2 the expression (m + nτ ) p is equal to (m + nτ ) p .Using the binomial formula and adding and subtracting coefficient with nonconjugated τ , we have where p s denotes the binomial coefficient.Using the relation we obtain Introduction of the summation index t = s + r yields Selecting the terms with the same values of n n t m 1 (m+nτ ) q− p+t we obtain Using (11) we arrive at the formula for q − p ≥ 2 One can see that formula (18) for p = 1 coincides with formula (12) from [13].
The formula (18) is effective in numerical computations.It contains the classic lattice sums for which fast computational formulae are known [14].The sum n n t d q− p+t dτ q− p+t ε 0 (nτ ) contains the derivatives of the elementary function ( 9), hence, it can be also also quickly computed.The results of numerical computations are presented in Table 1 for hexagonal lattice and in Table 2 for square lattice (Appendix B).The values in the tables without the decimal points are exact.
3 Recurrence formula for S (1)  2l+1 hrough the classic lattice sums The following well-known recurrence formula is useful to compute the classic lattice sums beginning from S 4 and S 6 Fast formulae for S 4 and S 6 are known in the theory of elliptic functions [2].In the present section, we express the lattice sums S (1) 2l+3 through the classic sums by an analogous formula.It is worth noting that the lattice sums S 2l+1 and S (1)  2l+2 vanish for l = 0, 1, . . .[4,10].
We will use the expansion (7) of the Natanzon-Filshtinsky function and the following formula established in [10,13] It is assumed that the constants c and c 1 are undetermined.Substituting ( 2) and ( 7) into (20) we arrive at the formula We arrive at the following assertions by selecting coefficients at the same powers of z.First, the linear part Second, the sums S (1) m are equal zero for even m.Third, the lattice sums S (1) 2m+3 can be calculated through the classic sums S m by means of the following algebraic relations 2π S (1) Note that S 2 = π for the hexagonal and square lattices.Hence, in these cases ( 22) is reduced to 2π S (1) The relations ( 22) and ( 23) are useful for fast computations and can be applied to deduce new exact formulae.Consider, for instance, the square lattice for which S 6 = 0 and [15] where (z) stands for Euler's gamma-function [16, vol. I].It follows from ( 19) that Then, (23) yields the closed form expression for S (1) 7 = 10 7π S 2 4 after the substitution (24), see the value in the Table 5.

An useful function x derived from elliptic integrals
Let R denote the set of real numbers and R + the set of positive numbers.Following Akhiezer [2] and Erdélyi et al. [16, vol.II] introduce the complete elliptic integral of the first kind where the elliptic modulus k ∈ (0, 1) and the complimentary modulus can be considered as a one-to-one monotonically decreasing continuously differentiable function x : (0, 1) → R + .The previous papers implicitly used expressions for the derivative dx dk .Below, such a formula is explicitly written and proved for the completeness of the presentation.

Proposition 1 The derivative of the function x(k) can be calculated by the formula
Proof The complete elliptic integral K (k) satisfies Legendre's relation where E(k) is the complete elliptic integral of the second kind Its derivative can be calculated by the formula The pair of the functions K (k) and K (k ) satisfy the differential equation [17] The functions E(k) and E(k Then, the derivative of K (k) can be calculated by the formula It follows from Legendre's relation (28) that Differentiation of (26) and using the expressions on the derivatives of the elliptic integrals yield Application of Legendre's identity (28) reduces ( 35)-( 27).The proposition is proved.

The basic modular forms
Let r be an arbitrary natural number r ≥ 2. It will be convenient to consider the slightly modified lattice sums (1) called the modular form It follows from (1) that V (0) r (τ ) = 0 for r odd.Consider τ as a function τ = τ (x(k)) depending on k.Using ( 26) and ( 50)-( 51) we arrive at the formula The classic lattice sums S 4 and S 6 defined by (1) are expressed through the elliptic integrals [18, items 18.9.4 and 18.9.5].Using (36) we can write these expressions in the form Using ( 1) and (36) we rewrite relation (19) as the recursive formula to determine the values (36) The term (Imτ ) r /2 from (17) becomes i r /2 in the first case τ = i x and i 2 r /2 in the second case τ = 1+i x 2 .Consider the first case for definiteness.The expression contains K in the denominator with power r /2.But it is canceled r is a polynomial in k, K , K and E for any even r .The theorem is proved for p = 0.
The proof for p > 0 can be obtained from Faá di Bruno's formula.But we shall follow a constructive recursive approach preferable in computations.Looking at τ as a function τ = τ (x(k)) depending on k, it follows from the chain rule that The derivatives of

The extended modular forms
Using (41) introduce the functions The derivative dτ dx becomes the constant i in the first case τ = i x and i 2 in the second case τ = ±1+i x 2 .As we have already assumed before, consider the first case for definiteness.It follows from (27) that dτ dx dx dk The second formula (43) can be written in the form For example the first three functions V (1) r (k) are calculated by ( 43)-(45) by using (30) and ( 33) V (1) Equation (42) implies that (49)

A recurrence relation providing polyanalytic lattice sums from modular forms
We will use the following Rayleigh formulae [5] written in the form [11] Consider now (17) with q = p + 2 The formula (52) for p = 1 becomes Using (53) we can write (52) for p = 2 in the form 4 (τ ) = 2S (1) This recurrence can be continued and the values S ( p) p+2 (τ ) can be found by means of the previous ones S (l)  l+2 (τ ) (l = 0, 1, . . ., p − 1) and by the absolutely convergent series n n p m 1 (m+nτ ) p+2 .This series can be differentiated term by term p+r (τ ) for any integer p ≥ 0, r ≥ 2 can be written as a polynomial in four variables K , K , E and k.
Proof Using relation (49) formula ( 17) can be written in the form We now proceed to prove by induction that the recurrence formulae (45) determine the function V ( p) r (k) for any integer p ≥ 0 as a homogeneous polynomial in K and E of the total power in k, K at least p + 1 and at most 2( p + 1) for r = 2 and at least p + r and at most 2 p + r for r > 2.
Let us assume that the induction hypothesis is true for p − 1, i.e., V ( p−1) r (k) is a polynomial of the variables K , E and k of the total power in K of the total power in K at least p and at most 2 p for r = 2 and at least p + r − 1 and at most 2 p + r − 2 for r > 2. Consider the action of the operator determined by the right part of (45) up to a constant multiplier on the polynomial term where Here, we omit the argument in the elliptic integrals for short.Hence, after application of the operator from (45) the polynomial is presented as a polynomial with the multiplier K m , where from (56 The term (Imτ ) r 2 τ t Im(τ s ), (0 < s ≤ p, 0 ≤ t ≤ p −s) from (55) after substitution τ = i K K contains only K in the denominator with power less than p + r .But it is canceled with the multiplier K with power at least p + 1 for r = 2 and p + r for r > 2 in V ( p) r .Substitution of (49) into (52) yields the required polynomial representation of S ( p) p+r (τ ).
Remark 1 This yields a recursive method on V (t) q− p ( p ≥ 0, q ≥ p + 2) to compute the sums S ( p) q (τ ) as follows for τ = i x or τ = 1+i x 2 .

Explicit expressions of polyanalytical lattice sums using elliptic integrals
Appendix contains the Mathematica code to compute the lattice sums S ( p) q (τ ) for τ = i x (Listing 1) and τ = 1+i x 2 (Listing 2) where x is given by (26).The computed values S (0)
Below, we write typical new formulae obtained by application of Theorem 2 and by ( 57) All the results given above are expressed by using the elliptic modulus k that is related to the geometric parameter x by (26).For any value of x, k is given by inversion of (26).However for many cases of interest, k is a simple function of x, as shown in the following section, avoiding the tedious inversion giving x.
The value of α(r −1 ) is calculated by formula [19, p. 153] This yields Using the above and recursive formula (57) we calculate some S ( p) q .For example, if τ = i we have r = 1 and . The corresponding elliptic integrals of the first kind are calculated by the formulae The elliptic integrals of the second kind are calculated by ( 70) and ( 71) Substituting the above values into (58) we have Similarly, we can calculate some values of the S ( p) q .The results are summarized in Tables 5, 6, 7, 8 and 9 (Appendix B).

Discussion and conclusion
The paper systematically describes the p-analytic lattice sums (6).In particular, the computationally effective formula ( 18) is derived.The expressions ( 22)-( 23) of the panalytic lattice sums S (1) q through the classic lattice sums S q are established.Theorem 2 about polynomial representations is constructive and is the source of exact formulae with the number π selected in the tables.
After preparing our paper for submission, the paper [20] was published.The formulae of Chen et al. [20] containing the p-analytic lattice sums overlap with our results.Though our formulae (18) and the values from Tables 5, 6, 7, 8 and 9 coincide with the formulae (20), ( 21) and partially with Table 1 from Chen et al. [20], we give our alternative proofs in the present paper to completely present polynomial representations of lattice sums in Theorem 1.
The obtained formulae have fundamental applications in the theory of fibrous composites since the effective properties of composites are expressed in terms of the lattice sums [14,21].For example, consider the hexagonal array of unidirectional circular fibers embedded in a host material.Let φ denote the concentration of fibers.The effective shear modulus G e can be found by the following formula [21, formulas (3.83)- (3.84)] where G and κ denote the shear modulus and Muskhelishvili's constant of the host.The value A(φ) can be considered as a function expanded in the powers of φ written here in the shortened form Here, j ( j = 1, 2, 3) are some combination of elastic constants of the components of the considered composite.The theory of lattice sums for polyanalytic functions developed in the present paper can be helpful for further investigations of regular and random composites following the lines of Drygaś et al. [21].

Table 1
The values of S ( p) q for the hexagonal lattice calculated by (18) q \ p 1 The values of S ( p) q for the square lattice calculated by (18) q \ p 1 1.76622 1.13929

Table 5
The values of S

Table 7
The values of S ( p) p+2

Table 8
The values of S

Table 9
The values of S

Table 6
The values of S