Summed series involving \,_{1}F_{2} hypergeometric functions

TL;DR

This paper extends previous work on series involving _{1}F_{2} hypergeometric functions by deriving new doubly and trebly infinite summed series from Chebyshev and Gegenbauer polynomial expansions of Bessel functions.

Contribution

It introduces methods to generate additional infinite series involving _{1}F_{2} hypergeometric functions from polynomial expansions of Bessel functions, expanding the scope of summable series.

Findings

Derived doubly infinite summed series from Chebyshev expansions

Established trebly infinite summed series from Gegenbauer expansions

Connected series values to inverse powers of primes

Abstract

In a prior paper we found that the Fourier-Legendre series of a Bessel function of the first kind J_{N}\left(kx\right) and of a modified Bessel functions of the first kind I_{N}\left(kx\right) lead to an infinite set of series involving \,_{1}F_{2} hypergeometric functions (extracted therefrom) that could be summed, having values that are inverse powers of the eight primes 1/\left(2^{i}3^{j}5^{k}7^{l}11^{m}13^{n}17^{o}19^{p}\right) multiplying powers of the coefficient k, for the first 22 terms in each series. The present paper shows how to generate additional, doubly infinite summed series involving \,_{1}F_{2} hypergeometric functions from Chebyshev polynomial expansions of Bessel functions, and trebly infinite sets of summed series involving \,_{1}F_{2} hypergeometric functions from Gegenbauer polynomial expansions of Bessel functions.