On the functional properties of Bessel zeta-functions

On the functional properties of the confluent hypergeometric zeta-function

“…I (Theorem 1.1 (1.3), (1.4)) by following the method which was innovated in [7]. In addition, some relation formulas between contiguous functions are given (Theorem 1.2).…”

“…We introduced J -Bessel and K -Bessel zeta-functions in the recent work [7], where we derived integral representations, transformation formulas, a power series expansion involving the Riemann zeta-function and a recurrence formula. The inverse Laplace transform of Weber's first exponential integral of the Bessel function was essentially used to derive the integral representation of J -Bessel zeta-function, which leads to the transformation formula ([7, Theorem 1.1]).…”

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.…”

“…This fact strongly suggested that our zeta-functions should have a kind of functional equation. The inverse Laplace transform of Weber's first exponential integral was the key ingredient in the proof of the integral expression, which led to the expected transformation formula [16,Theorem 1.1 (1.4)]. Kaczorowski and Perelli treated more general zeta-functions twisted by hypergeometric or Bessel functions, and derived meromorphic continuations of these zeta-functions via the properties of the non-linear twists [9][10][11][12].…”

Some generating functions of the Riemann zeta function

“…In recent works [No1,No2] we introduced some new zeta-functions, Bessel zetafunctions and a confluent hypergeometric zeta-function, where we derived integral representations, transformation formulas, power series expansions involving the Riemann zeta-function, and recurrence formulas. The J-Bessel zeta-function introduced in [No1] 108 T. NODA appears in the Fourier series expansion of the Poincaré series attached to SL(2, Z) by applying the inverse Mellin transform. This fact strongly suggested that our zeta-functions should have a kind of functional equations.…”