On a generalization of polynomials in the ballot problem

Abstract

This work gives two generalizations of polynomials in the ballot problem based on a dual relation for addition formulas by Watanabe (1984, 1985). One is given by solving an interpolation problem in the case of several variables. In the case of one variable, Niederhausen (1981) developed a theory of some rank order statistic based on this problem. On the other hand, Takács' urn model gives a multinomial generalization of the Gould polynomials. Since the Gould polynomials are the special case of basic polynomials, by obtaining a theory of multinomial basic polynomials with several variables, we give a generalization of polynomials in Takács' urn model.