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

N. K. Thakare and M. C. Madhekar

International Journal of Mathematics and Mathematical Sciences, 1988, vol. 11, 1-5

Abstract:

A pair of polynomial sequences { S n μ ( x ; k ) } and { T m μ ( x ; k ) } where S n μ ( x ; k ) is of degree n in x k and T m μ ( 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 | 2 μ exp ( − x 2 ) , ( μ > − 1 / 2 ) over the interval ( − ∞ , ∞ ) in the sense that ∫ − ∞ ∞ | x | 2 μ exp ( − x 2 ) S n μ ( x ; k ) T m μ ( x ; k ) d x = 0 ,       if m ≠n                                         ≠0 ,       if m = 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 = 1 , both the above sets get reduced to the orthogonal polynomials introduced by professor Szegö.

Date: 1988
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:292373

DOI: 10.1155/S0161171288000924

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