Al-Salam–Chihara polynomials

In mathematics, the Al-Salam–Chihara polynomials Q_n(x;a,b;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Al-Salam and Chihara (1976). Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14.8) give a detailed list of the properties of Al-Salam–Chihara polynomials.

Definition

The Al-Salam–Chihara polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by

Q_{n}(x;a,b;q)={\frac {(ab;q)_{n}}{a^{n}}}{}_{3}\phi _{2}(q^{{-n}},ae^{{i\theta }},ae^{{-i\theta }};ab,0;q,q)

where x = cos(θ).

References

Al-Salam, W. A.; Chihara, Theodore Seio (1976), "Convolutions of orthonormal polynomials", SIAM Journal on Mathematical Analysis, 7 (1): 16–28, ISSN 0036-1410, MR 0399537, doi:10.1137/0507003
Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719, doi:10.2277/0521833574
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, ISBN 978-3-642-05013-8, MR 2656096, doi:10.1007/978-3-642-05014-5
Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Al-Salam–Chihara polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.