Superimposing theta structure on a generalized modular relation

A generalized modular relation of the form $F(z,w,\alpha )=F(z,iw,\beta )$, where $\alpha \beta =1$ and $i=\sqrt{-1}$, is obtained in the course of evaluating an integral involving the Riemann $\mathrm{\Xi }$-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 $\zeta (s,a)$, which we denote by ${\zeta }_w(s,a)$. While ${\zeta }_w(s,1)$ is essentially a product of confluent hypergeometric function and the Riemann zeta function, ${\zeta }_w(s,a)$ for $0<a<1$ is an interesting new special function. We show that ${\zeta }_w(s,a)$ satisfies a beautiful theory generalizing that of $\zeta (s,a)$ albeit the properties of ${\zeta }_w(s,a)$ are much harder to derive than those of $\zeta(s, a)$\mathrm{.}Inparticular,itisshownthatfor$0<a<1$and$w\in$\mathbb{C}$$,$\zeta_w(s, a)$canbeanalyticallycontinuedtoRe$(s)>-1$ except for a simple pole at $s=1$. This is done by obtaining a generalization of Hermite’s formula in the context of ${\zeta }_w(s,a)$. The theory of functions reciprocal in the kernel $\sin (\pi z)J_{2z}(2\sqrt{xt})-\cos (\pi z)L_{2z}(2\sqrt{xt})$, where $L_z(x)=-\frac{2}{\pi }{K}_z(x)-Y_z(x)$ and $J_z(x),Y_z(x)$ and $K_z(x)$ are the Bessel functions, is worked out. So is the theory of a new generalization of $K_z(x)$, namely, ${}_1{K}_{z,w}(x)$. Both these theories as well as that of ${\zeta }_w(s,a)$ are essential to obtain the generalized modular relation.