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

[摘要] A pair of polynomial sequences{Snμ(x;k)}and{Tmμ(x;k)}whereSnμ(x;k)is of degreeninxkandTmμ(x;k)is of degreeminx, 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=nwherem,n=0,1,2,…andkis an odd positive integer.Generating functions, mixed recurrence relations for both these sets are obtained. Fork=1, both the above sets get reduced to the orthogonal polynomials introduced by professor Szegö.

[学科分类] 数学（综合）