2020/05/17 by Dixit, Atul, Kumar, Rahul
#33E20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT) #Primary 11M06 #Secondary 33C10
paper · doi:10.48550/arxiv.2005.08316
A generalized modular relation of the form F(z, w, α)=F(z, iw,β), where αβ=1 and i=√(-1), is obtained in the course of evaluating an integral involving the Riemann Ξ-function. It is a two-variable generalization of a transformation found on page 220 of Ramanujan's Lost Notebook. This modular relation involves a surprising generalization of the Hurwitz zeta function ζ(s, a), which we denote by ζw(s, a). While ζw(s, 1) is essentially a product of confluent hypergeometric function and the Riemann zeta function, ζw(s, a) for 0-1 except for a simple pole at s=1. This is done by obtaining a generalization of Hermite's formula in the context of ζw(s, a). The theory of functions reciprocal in the kernel sin(πz) J2 z(2 √(xt)) -cos(πz) L2 z(2 √(xt)), where Lz(x)=-\frac2πKz(x)-Yz(x) and Jz(x), Yz(x) and Kz(x) are the Bessel functions, is worked out. So is the theory of a new generalization of Kz(x), namely, 1Kz,w(x). Both these theories as well as that of ζw(s, a) are essential to obtain the generalized modular relation.