Vol. 100, No. 2, 1982

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Biorthogonal polynomials suggested by the Jacobi polynomials

H. C. Madhekar and N. K. Thakare

Vol. 100 (1982), No. 2, 417–424
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 Jn(α,β,k;x) and Kn(α,β,k;x), n = 0,1,2, be respectively the polynomials of degree n in xk 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
{
0 for i = 0,1,⋅⋅⋅ ,n− 1;
not 0 for i = n; (1)
and
11(1 x)α(1 + x)βK n(α,β,k;x)(1 x)ki dx is
{
0 for i = 0,1,⋅⋅⋅ ,n − 1;
not 0 for i = n. (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
{
0 for m,n = 0,1,⋅⋅⋅ ;m ⁄= n;
not 0 for m = n; (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
Authors
H. C. Madhekar
N. K. Thakare