Analogues of Jacobi's inversion formula for the incomplete elliptic integral of the first kind
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.
scientific article; zbMATH DE number 1895791
scientific article