Exact summation of special function series: Arbitrary parameter dependence and a new analytic structure from generalized Laguerre polynomials

Abstract

We present closed-form evaluations for a class of infinite series involving Bessel functions, Struve functions, and generalized Laguerre polynomials, each expressed in terms of arbitrary parameter values. For the Bessel and Struve cases, the resulting expressions reduce to combinations of classical functions, notably Gamma functions and power-law terms in the free parameters. These results are valid for all parameter values $\nu > 0$, and have been verified through high-precision analytical and numerical evaluations using both Maxima and Mathematica. The most significant contribution arises in the Laguerre case, where we construct a novel analytic function defined on the complex plane. This function, derived from the series involving generalized Laguerre polynomials, exhibits uniform convergence on every compact disk in $\mathbb{C}$, and represents a previously undocumented structure in the theory of orthogonal polynomials. The findings open new avenues for functional analysis and complex-variable techniques in the study of special functions.