Summed Series Involving 1 F 2 Hypergeometric Functions

Summation of infinite series has played a significant role in a broad range of problems in the physical sciences and is of interest in a purely mathematical context. In a prior paper, we found that the Fourier–Legendre series of a Bessel function of the first kind J N k x and modified Bessel functions of the first kind I N k x lead to an infinite set of series involving F 2 1 hypergeometric functions (extracted therefrom) that could be summed, having values that are inverse powers of the eight primes 1 / 2 i 3 j 5 k 7 l 11 m 13 n 17 o 19 p 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 F 2 1 hypergeometric functions from Chebyshev polynomial expansions of Bessel functions, and trebly infinite sets of summed series involving F 2 1 hypergeometric functions from Gegenbauer polynomial expansions of Bessel functions. That the parameters in these new cases can be varied at will significantly expands the landscape of applications for which they could provide a solution.