Vol. 21, No. 2, 1967

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ISSN: 0030-8730
Biorthogonal polynomials suggested by the Laguerre polynomials

Joseph D. E. Konhauser

Vol. 21 (1967), No. 2, 303–314
Abstract

Let Y nc(x;k) and Znc(x;k),n = 0,1, , be polynomials of degree n in x and xk, respectively, where x is real, k is a positive integer and c > 1, such that

                       {
∫ ∞  c −x c      ki      0 for i = 0,1,⋅⋅⋅ ,n − 1;
x e  Yn(x;k)x  dx is not 0 for i = n;
0
(1)

and

∫ ∞                   {
xce−xZcn(x;k)xidx is   0 for i = 0,1,⋅⋅⋅ ,n− 1;
0                      not 0 for i = n.
(2)

For k = 1, conditions (1) and (2) reduce to the orthogonality requirement satisfied by the generalized Laguerre polynomials.

If (1) and (2) hold, then

∫                           {
∞  c −x  c     c           0 for i,j = 0,1,⋅⋅⋅ ;i ⁄= j;
0  x e  Yi (x;k)Zj(x;k)dx is not 0 for i = j;

and conversely.

For both sets of polynomials, we shall establish mixed recurrence relations from which we shall derive differential equations of order k + 1. From these mixed recurrence relations pure recurrence relations connecting k + 2 successive polynomials can also be obtained. For k = 1, the recurrence relations and the differential equations for both of the polynomial sets reduce to those for the generalized Laguerre polynomials.

Mathematical Subject Classification
Primary: 33.40
Milestones
Received: 9 September 1965
Published: 1 May 1967
Authors
Joseph D. E. Konhauser