A Combinatorial Interpretation of Lommel Polynomials and Their Derivatives

In this paper we present interpretations of Lommel polynomials and their derivatives. A combinatorial interpretation uses matchings in graphs. This gives an interpretation for the derivatives as well. Then Lommel polynomials are considered from the point of view of operator calculus. A step-3 nilpotent Lie algebra and finite-difference operators arise in the analysis

Generalization of matching extensions in graphs—combinatorial interpretation of orthogonal and q-orthogonal polynomials

AbstractIn this paper, we present generalization of matching extensions in graphs and we derive combinatorial interpretation of wide classes of orthogonal and q-orthogonal polynomials. Specifically, we assign general weights to complete graphs, cycles and chains or paths defining matching extensions in these graphs. The generalized matching polynomials of these graphs have recurrences defining various orthogonal polynomials—including classical and non-classical ones—as well as q-orthogonal polynomials. The Hermite, Gegenbauer, Legendre, Chebychev of the first and second kind, Jacobi and Pollaczek orthogonal polynomials and the continuous q-Hermite, Big q-Jacobi, Little q-Jacobi, Al Salam and alternative q-Charlier q-orthogonal polynomials appeared as applications of this study

Holonomic Bessel modules and generating functions

We have solved a number of holonomic PDEs derived from the Bessel modules which are related to the generating functions of classical Bessel functions and the difference Bessel functions recently discovered by Bohner and Cuchta. This $D$-module approach both unifies and extends generating functions of the classical and the difference Bessel functions. It shows that the algebraic structures of the Bessel modules and related modules determine the possible formats of Bessel's generating functions studied in this article. As a consequence of these $D$-modules structures, a number of new recursion formulae, integral representations and new difference Bessel polynomials have been discovered. The key ingredients of our argument involve new transmutation formulae related to the Bessel modules and the construction of $D$-linear maps between different appropriately constructed submodules. This work can be viewed as $D$-module approach to Truesdell's $F$-equation theory specialised to Bessel functions. The framework presented in this article can be applied to other special functions.Comment: 97 pages including one blank pag

A Combinatorial Interpretation of Lommel Polynomials and Their Derivatives

Abstract. In this paper we present interpretations of Lommel polynomials and their derivatives. A combinatorial interpretation uses matchings in graphs. This gives an interpretation for the derivatives as well. Then Lommel polynomials are considered from the point of view of operator calculus. A step-3 nilpotent Lie algebra and finite-difference operators arise in the analysis. I