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Q Bessel polynomials

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Q-Bessel polynomials

In mathematics, the q-Bessel polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.

Contents

Definition

The polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by ：

y n ( x ; a ; q ) = 2 ϕ 1 ( q − N − a q n 0 ; q , q x )

Orthogonality

∑ k = 0 ∞ ( a k ( q ; q ) n ∗ q ( k + 1 2 ) ∗ y m ∗ ( q k ; a ; q ) ∗ y n ∗ ( q k ; a ; q ) = ( q ; q ) n ∗ ( − a q n ; q ) ∞ a n ∗ q ( n + 1 2 ) 1 + a q 2 n δ m n

References

Q-Bessel polynomials Wikipedia
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