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.

Definition

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

$y_n(x;a;q)={}_2{\varphi }_1(\begin{array}{cc}{q}^{-n}& -a{q}^n\\ 0\end{array};q,qx).$

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

Orthogonality

$\sum _{k=0}^{\mathrm{\infty }}(\frac{a^k}{(q;q{)}_n}*q^{(\genfrac{}{}{0ex}{}{k+1}{2})}*y_m*(q^k;a;q)*y_n*(q^k;a;q))=(q;q{)}_n*(-a{q}^n;q{)}_{\mathrm{\infty }}\frac{a^n*q^{(\genfrac{}{}{0ex}{}{n+1}{2})}}{1+a{q}^{2n}}{\delta }_{mn}$

where $(q;q{)}_n\text{ and }(-a{q}^n;q{)}_{\mathrm{\infty }}$ are q-Pochhammer symbols.

Gallery

QBessel function abs complex 3D Maple plot

QBessel function Im complex 3D Maple plot

QBessel function Re complex 3D Maple plot

File:QBessel function abs density Maple plot.gif QBessel function abs density Maple plot

File:QBessel function Im density Maple plot.gif QBessel function Im density Maple plot

File:QBessel function Re density Maple plot.gif QBessel function Re density Maple plot

References

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