Probabilistic derivation of a bilinear summation formula for the Meixner-Pollaczek polynominals
P. A. Lee
International Journal of Mathematics and Mathematical Sciences, 1980, vol. 3, 1-11
Abstract:
Using the technique of canonical expansion in probability theory, a bilinear summation formula is derived for the special case of the Meixner-Pollaczek polynomials { λ n ( k ) ( x ) } which are defined by the generating function ∑ n = 0 ∞ λ n ( k ) ( x ) z n / n ! = ( 1 + z ) 1 2 ( x − k ) / ( 1 − z ) 1 2 ( x + k ) , | z | < 1.
These polynomials satisfy the orthogonality condition ∫ − ∞ ∞ p k ( x ) λ m ( k ) ( i x ) λ n ( k ) ( i x ) d x = ( − 1 ) n n ! ( k ) n δ m , n , i = − 1 with respect to the weight function p 1 ( x ) = sech π x p k ( x ) = ∫ − ∞ ∞ … ∫ − ∞ ∞ sech π x 1 sech π x 2 … sech π ( x − x 1 − … − x k − 1 ) d x 1 d x 2 … d x k − 1 , k = 2 , 3 , …
Date: 1980
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:239895
DOI: 10.1155/S0161171280000555
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