  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.
