Integrals of K and E from lattice sums

Abstract

We give closed form evaluations for many families of integrals, whose integrands contain algebraic functions of the complete elliptic integrals K and E. Our methods exploit the rich structures connecting complete elliptic integrals, Jacobi theta functions, lattice sums, and Eisenstein series. Various examples are given, and along the way new (including 10-dimensional) lattice sum evaluations are produced.

Moments in Pearson's Four-Step Uniform Random Walk Problem and Other Applications of Very Well-Poised Generalized Hypergeometric Series

“…ð3:3Þ Wan and Zucker (2015) found both constants in the L-series evaluations discussed above, as did Wan and Zucker (2016) in certain eight-dimensional lattice sum evaluations. The first constant is a multiple of the constant c 4;0 ≔∫ ∞ 0 K 0 t ð Þ 4 dt that arose in evaluations by Bailey, Borwein, Broadhurst and Glasser (2008) of integrals that arise in Quantum Field Theory.…”

“…The paper also motivates questions for possible future work. By providing a formal sense in which the two constants in Section 3 were counterparts of each other, the paper goes farther than Rogers, Wan and Zucker (2016) in describing how the critical values of the L-series of two related, even-weight cusp forms are intertwined. It would be interesting to investigate whether the type of the symmetries exhibited here apply more widely in the interplay between modular forms and generalized hypergeometric series and, if so, whether they can be built up in a similar way.…”

“…Given for any weight 4 modular form f, the functional equation relates L(f, s) and L(f, 4 − s), the value at s = 4 may have special importance. Papanikolas, Rogers and Samart (2014) and Wan and Zucker (2016) have derived hypergeometric series expressions for the L-series of f 4, 8 evaluated at 4, one being…”

A multi-variable version of the completed Riemann zeta function and other $L$-functions

“…(4) Multiple Jacobi values of length one are related to 'lattice sums' which arise in a variety of contexts (see [22] for interesting examples and references) and to values of the Arakelov double zeta function Z Q (w, s) of [15].…”

The Exact Evaluation of Some New Lattice Sums

“…The origin of this is a recent paper, [10] which showed that any lattice sum involving θ-functions all having the same argument could be transformed in a simple manner into integrals of combinations of k, k , K, K . In particular (2.28) yield for s = 1, 2, 3 respectively the following neat results Remarkably this is not unique, for in a completely different manner [9] produced the following result .…”