Multivariate Laguerre polynomials: new results and insights

Abstract

In this paper, we study various properties of Erdélyi's multivariate Laguerre polynomials $L_{n_1,\cdots ,n_k}^{(\hat{I}\pm )}(x_1,\cdots ,x_k)$ including their generating functions, product formulas and fractional integral representations. Many useful consequences are derived. New insights into various classical results including the relationships of the multivariate Laguerre polynomials with Oshima's fractional calculus operator and with an integral formula of Srivastava and Niukkanen are also mentioned. We further present an interesting evaluation for a generating function of the main diagonal sequence $L_{n,\cdots ,n}^{(-{\hat{I}}^2-kn)}(x_1,\cdots ,x_k)$ which involves in a natural way the well-known Le Roy function ([Darboux Bull. 24 (2) (1899), 245--268]; [Toulouse Ann. 2 (2) (1900), 317--430]). The significance of the multivariate Laguerre polynomials $L_{n_1,\cdots ,n_k}^{(\hat{I}\pm )}(x_1,\cdots ,x_k)$ is demonstrated by observing that this class not only includes the generalized Hardy-Hille formula and the product formula but also contains the multiple Laguerre polynomials of the second kind as its important special cases. We briefly indicate also possible lines of future work.