Superimposing theta structure on a generalized modular relation
Source
arXiv
Date Issued
2020-05-01
Author(s)
Dixit, Atul
Kumar, Rahul
Abstract
A 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.
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.