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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. give a detailed list of their properties.

Definition

The polynomials are given in terms of basic hypergeometric functions by ：

:y_{n}(x;a;q)=;{}_2\phi_1 \left(\begin{matrix} q^{-n} & -aq^{n} \ 0 \end{matrix} ; q,qx \right).

Also known as alternative q-Charlier polynomials K(x;a;q).

Orthogonality

\sum_{k=0}^{\infty}\left(\frac{a^k}{(q;q)n}*q^{k+1 \choose 2}*y{m}(q^k;a;q)y_{n}(q^k;a;q)\right)=(q;q)_{n}(-aq^n;q){\infty}\frac{ a^{n}*q^{n+1 \choose 2} }{1+aq^{2n}}\delta{mn} where (q;q)n\text{ and }(-aq^n;q)\infty are q-Pochhammer symbols.

Gallery

[[File:QBessel function abs complex 3D Maple plot.gif	thumb	QBessel function abs complex 3D Maple plot]]	[[File:QBessel function Im complex 3D Maple plot.gif	thumb	QBessel function Im complex 3D Maple plot]]	[[File:QBessel function Re complex 3D Maple plot.gif	thumb	QBessel function Re complex 3D Maple plot]]

[[File:QBessel function abs density Maple plot.gif	thumb	QBessel function abs density Maple plot]]	[[File:QBessel function Im density Maple plot.gif	thumb	QBessel function Im density Maple plot]]	[[File:QBessel function Re density Maple plot.gif	thumb	QBessel function Re density Maple plot]]

References

Roelof Koekoek, Peter Lesky Rene Swarttouw, Hypergeometric Orthogonal Polynomials and their q-Analogues, p526 Springer 2010
Roelof p527

Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

orthogonal-polynomials q-analogs special-hypergeometric-functions

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