Summed series involving \,_{1}F_{2} hypergeometric functions (2412.00019v1)

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.