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

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

Also known as alternative q-Charlier polynomials

K(x;a;q).

Orthogonality

\sum_^\left(\frac*q^*y_*(q^k;a;q)*y_*(q^k;a;q)\right)=(q;q)_*(-aq^n;q)_\frac\delta_

Roelof p527 where

(q;q)_n\text(-aq^n;q)_\infty

are

q-Pochhammer symbol In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer symb ...

Gallery

References

* * *{{dlmf, id=18, title=Orthogonal Polynomials, first1=Tom H. , last1=Koornwinder, first2=Roderick S. C., last2= Wong, first3=Roelof , last3=Koekoek, , first4=René F. , last4=Swarttouw, url=http://dlmf.nist.gov/18, archive-url=http://dlmf.nist.gov/18 Orthogonal polynomials Q-analogs Special hypergeometric functions