On a power series involving classical orthogonal polynomials

TL;DR

This paper explores power series involving classical orthogonal polynomials like Gegenbauer, Laguerre, and Hermite, using a novel method based on generating functions and complex analysis to evaluate their sums and derivatives.

Contribution

It introduces a new approach employing generating functions and functional equations to evaluate sums of power series with orthogonal polynomial coefficients, including higher-order derivatives and asymptotic cases.

Findings

Derived sums as higher-order derivatives of generating functions.

Unified treatment of Gegenbauer, Laguerre, and Hermite polynomial series.

Applied results to quantum mechanics, specifically the harmonic oscillator propagator.

Abstract

We investigate a class of power series occurring in some problems in quantum optics. Their coefficients are either Gegenbauer or Laguerre polynomials multiplied by binomial coefficients. Although their sums have been known for a long time, we employ here a different method to recover them as higher-order derivatives of the generating function of the given orthogonal polynomials. The key point in our proof consists in exploiting a specific functional equation satisfied by the generating function in conjunction with Cauchy's integral formula for the derivatives of a holomorphic function. Special or limiting cases of Gegenbauer polynomials include the Legendre and Chebyshev polynomials. The series of Hermite polynomials is treated in a straightforward way, as well as an asymptotic case of either the Gegenbauer or the Laguerre series. Further, we have succeeded in evaluating the sum of a similar power series which is a higher-order derivative of Mehler's generating function. As a prerequisite, we have used a convenient factorization of the latter that enabled us to employ a particular Laguerre expansion. Mehler's summation formula is then applied in quantum mechanics in order to retrieve the propagator of a linear harmonic oscillator.