International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 4, Pages 763-767
doi:10.1155/S0161171288000924
A pair of biorthogonal polynomials for the Szegö-Hermite weight function
1Mathematics Department, University of Poona, Pune 411 007, India
2Milind College of Science, Aurangabad 431 004, India
Received 1 April 1987; Revised 28 August 1987
Copyright © 1988 N. K. Thakare and M. C. Madhekar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
A pair of polynomial sequences {Snμ(x;k)} and {Tmμ(x;k)} where Snμ(x;k) is of degree n in xk and Tmμ(x;k) is of degree m in x, is constructed. It is shown that this pair is biorthogonal with respect to the Szegö-Hermite weight function |x|2μexp(−x2), (μ>−1/2) over the interval (−∞,∞) in the sense that∫−∞∞|x|2μexp(−x2)Snμ(x;k)Tmμ(x;k)dx=0, ifm≠n ≠0, ifm=nwhere m,n=0,1,2,… and k is an odd positive integer.
Generating functions, mixed recurrence relations for both these sets are obtained. For k=1, both the above sets get reduced to the orthogonal polynomials introduced by professor Szegö.