A new family of orthogonal polynomials in three variables

Rabia Aktaş,Iván Area,Esra Güldoğan

doi:10.1186/s13660-020-02434-5

Abstract

In this paper we introduce a six-parameter generalization of the four-parameter three-variable polynomials on the simplex and we investigate the properties of these polynomials. Sparse recurrence relations are derived by using ladder relations for shifted univariate Jacobi polynomials and bivariate polynomials on the triangle. Via these sparse recurrence relations, second order partial differential equations are presented. Some connection relations are obtained between these polynomials. Also, new results for the four-parameter three-variable polynomials on the simplex are given. Finally, some generating functions are derived.

Highlights

The classical univariate Jacobi polynomials Pn(a,b)(x) defined by the Rodrigues formula, Pn(a,b)(x) = (–1)n 2nn! (1 x)–a(1 +x)–b dn dxn (1 – x)n+a(1 + x)n+b, n ≥ 0, (1)are orthogonal with respect to the weight function wa,b(x) = (1 – x)a(1 + x)b on the interval (–1, 1) for a, b > –1
The Jacobi polynomials on the interval (0, 1), referred as shifted univariate Jacobi polynomials [8], which we denote by Pn(a,b)(x) := Pn(a,b)(2x – 1), Pn(a,b)(x) x)–ax–b dn dxn
By using the techniques of reference [9] entirely, we present 36 sparse recurrence relations for a six-parameter variant of the three-variable polynomials

Summary

Introduction

The classical univariate Jacobi polynomials Pn(a,b)(x) defined by the Rodrigues formula, Pn(a,b)(x). Jacobi polynomials multiplied by x, while the fifth, sixth, ninth, and tenth relations give the second order differential equation for the univariate shifted Jacobi polynomials multiplied by 1 – x By using these 12 relations, in [9] 24 relations for the bivariate orthogonal polynomials Pn(a,k,b,c,d)(x, y) defined in (7) are derived. 2 for the univariate shifted Jacobi polynomials Pn(a,b)(x) defined in (2) and the 24 differential relations given in Sect.

From the relation

From these relations we have

Proof The first equation is obtained from

Proof The first relation comes from the equality

The third relation holds from the relation