Abstract
It is well known that generating functions play an important role in theory of the classical orthogonal polynomials. In this paper, we deal with systems of polynomials defined by generating functions and the following problems for them. (A) Derive a differential equation that each polynomial satisfies. (B) Derive the general solution for the differential equation obtained in (A). (C) Is the general solution obtained in (B) written as a linear combination of functions that are expressed by making use of generalized hypergeometric functions? The purpose of this paper is to give two examples that the problem (C) can be affirmatively solved. One is related to the Humbert polynomials, and its general solution is written by kFk-1-type hypergeometric functions. The other is related to a genelarization of the Hermite polynomials, and its general solution is written by kFl-type (k≤l) hypergeometric functions.
Highlights
IntroductionIt is interesting to define new polynomials by new generating functions, and important to study their properties
It is interesting to define new polynomials by new generating functions, and important to study their properties. Humbert (1921) defined the polynomials Πn,m(x), n = 0, 1, 2, ... , by the generating function∑∞ (1 − mtx + tm)− = Πn,m(x)tn. n=0Gould (1965) called Πn,m(x) the Humbert polynomial of degree n and gave its generalization. Milovanović and Djordjević (1987) gave a differential equation for the function Πn,m(x) using difference operators.ABOUT THE AUTHORSErika Suzuki and Nobuyuki Dobashi completed their master's theses at the University of Aizu in 2013, 2014, respectively, under Shigeru Watanabe
In Suzuki (2013), we considered defining the polynomials Qn(x;k, ), n = 0, 1, 2, ... , by the following generating function which is similar to that of the Humbert polynomials
Summary
It is interesting to define new polynomials by new generating functions, and important to study their properties. Humbert (1921) defined the polynomials Πn,m(x), n = 0, 1, 2, ... Gould (1965) called Πn,m(x) the Humbert polynomial of degree n and gave its generalization. Milovanović and Djordjević (1987) gave a differential equation for the function Πn,m(x) using difference operators Gould (1965) called Πn,m(x) the Humbert polynomial of degree n and gave its generalization. Milovanović and Djordjević (1987) gave a differential equation for the function Πn,m(x) using difference operators
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