Analogues of Jacobi's inversion formula for the incomplete elliptic integral of the first kind
type: Item
instance of: scholarly article
title: Analogues of Jacobi's inversion formula for the incomplete elliptic integral of the first kind
zbMATH Open document ID: 1018.33015
DOI: 10.1016/S0001-8708(02)00054-3
author: Heng Huat Chan
author: Zhi-Guo Liu
published in: Advances in Mathematics
publication date: 2003-04-09T00:00:00Z
review text: Let \(Im(\tau)> 0\), \(q= \exp(2\pi i\tau)\), and define \[ S_{2k} = S_{2k}(q) = \sum^\infty_{n=1} \frac{n^{2k} q^n}{1+ q^n+ q^{2n}},\quad k= 1,2,3,\dots\;. \] The purpose of this paper is to study properties of \(S_{2k}(q)\). The authors first show that the generating function \[ L= L(u|\tau)= \sum^\infty_{k=1} (-1)^k S_{2k}(q) \frac{(2u)^{2k}}{(2k)!} = \sum^\infty_{n=1} \frac{q^n}{1+ q^n+ q^{2n}}(\cos 2nu-1) \] is an elliptic function with periods \(\pi\) and \(3\pi\tau\). They establish that \(L\) satisfies the differential equation \[ \frac{d^2L}{du^2} = -4S_2 - 4a^2L - 24aL^2 - 32L^3, \] where \(a= a(q)= \frac 16 + \sum^\infty_{n=1} \frac{q^n}{1+ q^n+ q^{2n}}\). Expanding both sides of the differential equation in powers of \(u\), an attractive recurrence relation (Corollary 2.4) which expresses \(S_{2k+2}(q)\) in terms of \(S_{2k}(q),S_{2k-2}(q),\dots, S_2(q)\) and \(a(q)\) is obtained. The effect of the transformation \(\tau\to -1/3\tau\) is studied, and this is used to show (eq. 4.15) that if \(q= \exp(2\pi i\tau)\), \(p= \exp(-2\pi i/3\tau)\), then \[ S_{2k}(p) = -\frac{(i\tau)^{2k+1}}{2^{2k+1}} \cot^{(2k)}\Biggl(\frac\pi 3\Biggr) E_{2k}(\chi_3; q),\quad k= 1,2,3,\dots, \] where \[ E_{2k}(\chi_3;q) = 1+ \frac{(-1)^k 2^{2k+1}\sqrt{3}}{ \cot^{(2k)}(\pi/3)} \sum^\infty_{n=1} \frac{\chi_3(n) n^{2k} q^n}{1-q^n}. \] The reviewer believes that the statements of the main theorems may be simplified slightly by defining \[ S_0(q)= \frac 16 + \sum^\infty_{k=1} \frac{q^n}{ 1+ q^n+ q^{2n}}, \] observing that \(S_0(q)= a(q)= E_0(\chi_3;q)\), and using \(\sum^\infty_{k=0} (-1)^k S_{2k}(q) \frac{(2n)^{2k}}{(2k)!}\) in place of \(L(u|\tau)\). But the results in this paper, however they are stated, are beautiful.
reviewed by: Shaun Cooper
Mathematics Subject Classification ID: 33E05
Mathematics Subject Classification ID: 33C60
Mathematics Subject Classification ID: 33C05
zbMATH DE Number: 1895791
zbMATH Keywords: incomplete elliptic integrals
MaRDI profile type: Publication
cites work: Q3995823
cites work: Q4000402
cites work: Ramanujan's Theories of Elliptic Functions to Alternative Bases
cites work: On Ramanujan's quartic theory of elliptic functions
cites work: Representations of integers as sums of 32 squares
cites work: Q5641206
cites work: Q4460860
cites work: Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions
cites work: Representations of integers as sums of squares
cites work: Q4040264
cites work: Q4771024
cites work: On a Definite Integral of a Hypergeometric Function
cites work: On an identity of Ramanujan based on the hypergeometric series \(_2F_1(\frac{1}{3},\frac{2}{3};\frac{1}{2};x)\)
cites work: Q3844537
description: scientific article; zbMATH DE number 1895791