A PAIR OF BIORTHOGONAL POLYNOMIALS FOR THE SZEGO-HERMITE WEIGHT FUNCTION

ABSTRACT. A pair of polynomial sequences {S(x’k)} and {T(x;k)} where S(x;k) is n m n of degree n in xk and T(x;k) is of degree m in x, is constructed. It is shown m that this pair is biorthogonal with respect to the Szeg-Hermite weight function Ixl2exp(-x2), (>-I/2) over the interval (--,-) in the sense that f Ixl 2 exp(-x2) S(x;k) T(x;k)dx 0 if m # n 0, tfm--n where 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 I, both the above sets get reduced to the orthogonal polynomials introduced by professor Szeg