On the non homogeneous quadratic Bessel zeta function

TL;DR

This paper investigates the non-homogeneous quadratic Bessel zeta function, providing explicit formulas for key zeta invariants like poles, residues, and derivatives at zero, enhancing understanding of its mathematical properties.

Contribution

It offers explicit formulas for the poles, residues, and derivatives at zero of the non-homogeneous quadratic Bessel zeta function, a novel contribution to special function analysis.

Findings

Explicit formulas for poles and residues of the zeta function

Calculation of the zeta function's derivative at zero

Enhanced understanding of Bessel zeta function properties

Abstract

We study the non homogeneous quadratic Bessel zeta function $\zeta_{RB}(s,\nu,a)$ defined as the sum of the square of the positive zeros of the Bessel function $J_\nu(z)$ plus a positive constant. In particular, we give explicit formulas for the main associated zeta invariants, namely poles and residua, $\zeta_{RB}(0,\nu,a)$ and $\zeta'_{RB}(0,\nu,a)$.