On a power series involving classical orthogonal polynomials. (English) Zbl 1231.81097
Summary: 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.
MSC:
81V80 | Quantum optics |
33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
33C90 | Applications of hypergeometric functions |