Some generating functions of the Riemann zeta function

Abstract

In this survey paper, functional relations between some generating functions of the Riemann zeta function are formulated. These generating functions themselves are zetafunctions (Bessel type, confluent-hypergeometric type and hypergeometric type), introduced by M. Katsurada and the author. The explicit formula of the special values of these zeta functions at non-positive integers are also given.

The exponential-type generating function of the Riemann zeta-function revisited

“…In recent works [16][17][18], we derived some functional properties of Bessel zetafunctions and a confluent hypergeometric zeta-function. The J -Bessel zeta-function appears in the Fourier series expansion of the Poincaré series attached to SL(2, Z) by the inverse Mellin transform.…”