International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 4, Pages 763-768
doi:10.1155/S0161171288000924

A pair of biorthogonal polynomials for the Szegö-Hermite weight function

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.