Dixit, AtulAtulDixitKumar, RahulRahulKumar2025-08-282025-08-282020-05-01http://arxiv.org/abs/2005.08316https://d8.irins.org/handle/IITG2025/20055A generalized modular relation of the form F(z, w, ?) = F(z, iw, ?), where ?? = 1 and i = ? ?1, is obtained in the course of evaluating an integral involving the Riemann ?-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 ?(s, a), which we denote by ?w(s, a). While ?w(s, 1) is essentially a product of confluent hypergeometric function and the Riemann zeta function, ?w(s, a) for 0 < a < 1 is an interesting new special function. We show that ?w(s, a) satisfies a beautiful theory generalizing that of ?(s, a) albeit the properties of ?w(s, a) are much harder to derive than those of ?(s, a). In particular, it is shown that for 0 < a < 1 and w ? C, ?w(s, a) can be analytically continued to Re(s) > ?1 except for a simple pole at s = 1. This is done by obtaining a generalization of Hermite�s formula in the context of ?w(s, a). The theory of functions reciprocal in the kernel sin(?z)J2z(2? xt) ? cos(?z)L2z(2? xt), where Lz(x) = ? 2 ? Kz(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 ?w(s, a) are essential to obtain the generalized modular relation.en-USe-Printe-Print123456789/555