Series over Bessel Functions as Series in Terms of Riemann’s Zeta Function

Abstract

Relying on the Hurwitz formula, we find closed-form formulas for the series over sine and cosine functions through the Hurwitz zeta functions, and using them and another summation formula for trigonometric series, we obtain a finite sum for some series over the Riemann zeta functions. We apply these results to the series over Bessel functions, expressing them first as series over the Riemann zeta functions.

Keywords

• Riemann’s zeta function
• Hurwitz’s zeta function
• gamma function
• harmonic numbers
• Bessel functions
• spherical Bessel functions