Green's function for a Schroedinger operator and some related summation formulas

TL;DR

This paper derives summation formulas for associated Laguerre polynomials using the Green's function of a specific Schroedinger operator, introducing a new approach that could extend to broader generating functions.

Contribution

It presents a novel method employing Mercer-type theorems to construct Green's functions for a Schroedinger operator, enabling new summation formulas for special functions.

Findings

Derived summation formulas for associated Laguerre polynomials.

Introduced a new approach using Mercer-type theorems for Green's functions.

Potential to extend to wider classes of generating functions.

Abstract

Summation formulas are obtained for products of associated Lagurre polynomials by means of the Green's function K for the Hamiltonian H = -{d^2\over dx^2} + x^2 + Ax^{-2}, A > 0. K is constructed by an application of a Mercer type theorem that arises in connection with integral equations. The new approach introduced in this paper may be useful for the construction of wider classes of generating function.