Abstract
The bilinear generating function for products of two Laguerre 2D polynomials Lm;n(z; z0) with different arguments is calculated. It corresponds to the formula of Mehler for the generating function of products of two Hermite polynomials. Furthermore, the generating function for mixed products of Laguerre 2D and Hermite 2D polynomials and for products of two Hermite 2D polynomials is calculated. A set of infinite sums over products of two Laguerre 2D polynomials as intermediate step to the generating function for products of Laguerre 2D polynomials is evaluated but these sums possess also proper importance for calculations with Laguerre polynomials. With the technique of SU(1; 1) operator disentanglement some operator identities are derived in an appendix. They allow to calculate convolutions of Gaussian functions combined with polynomials in one- and two-dimensional case and are applied to evaluate the discussed generating functions. Keywords: Laguerre and Hermite polynomials, Laguerre 2D polynomials, Jacobi polynomials, Mehler formula, SU(1; 1) operator disentanglement, Gaussian convolutions.
Highlights
Hermite and Laguerre polynomials play a great role in mathematics and in mathematical physics and can be found in many monographs of Special Functions, e.g., [1]-[4]
The alternative definitions of Hermite (1D and 2D) and Laguerre 2D polynomials extends the arsenal of possible approaches to problems of their application and one should have for disposal the new method in the same way as the former methods
We preferred the first definition of the Laguerre 2D polynomials (2) from the two alternative ones given in (1.2) which as it seems to us is advantageous for this purpose
Summary
Hermite and Laguerre polynomials play a great role in mathematics and in mathematical physics and can be found in many monographs of Special Functions, e.g., [1]-[4]. Special comprehensive representations of polynomials of two and of several variables are given in, e.g. Laguerre 2D polynomials Lm,n ( z, z′) with two, in general, independent complex variables z and z′ were introduced in [7]-[12] by (similar or more general objects with other names and notations were defined in [13]-[24]). How to cite this paper: Wünsche, A. (2015) Generating Functions for Products of Special Laguerre 2D and Hermite 2D Polynomials.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
