Biorthogonal polynomials suggested by the Jacobi polynomials

H. C. Madhekar and N. K. Thakare

Vol. 100 (1982), No. 2, 417–424

DOI: 10.2140/pjm.1982.100.417

Abstract

In this paper we introduce and study a pair of biorthogonal polynomials that are suggested by the classical Jacobi polynomials. Let α > −1, β > −1 and J_n(α,β,k;x) and K_n(α,β,k;x), n = 0,1,2,⋯ be respectively the polynomials of degree n in x^k and x, where x is real, k is a positive integer such that these two polynomial sets satisfy biorthogonality conditions with respect to the weight function (1 − x)^α(1 + x)^β, namely

∫ −11(1 − x)α(1 + x)βJ n(α,β,k;x)xi dx is
		(1)

and

∫ −11(1 − x)α(1 + x)βK n(α,β,k;x)(1 − x)ki dx is
		(2)

It follows from (1) and (2) that

∫ −11(1 − x)α(1 + x)βJ n(α,β,k;x)Km(α,β,k;x)dx is
		(3)

and conversely.

For k = 1 both these sets are reduced to the Jacobi polynomial sets. We obtain generating functions, recurrence relations for both these sets and explicitly show that they satisfy biorthogonality conditions.

Mathematical Subject Classification 2000

Primary: 33A65, 33A65

Secondary: 42C05

Milestones

Received: 22 July 1980

Published: 1 June 1982