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 ϕ 1 ( q − n − a q n 0 ; q , q x ) . {\displaystyle 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 ) . {\displaystyle K(x;a;q).}
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 {\displaystyle \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 and ( − a q n ; q ) ∞ {\displaystyle (q;q)_{n}{\text{ and }}(-aq^{n};q)_{\infty }} are q-Pochhammer symbols.
Gallery
References
Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719 Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096 Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
