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.

Not to be confused with Al-Salam–Carlitz polynomials.

Definition

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

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

where x = cos(θ).

Orthogonality

Recurrence and difference relations

Rodrigues formula

Generating function

Relation to other polynomials

References

• Al-Salam, W. A.; Chihara, Theodore Seio (1976), "Convolutions of orthonormal polynomials", SIAM Journal on Mathematical Analysis, 7 (1): 16–28, doi:10.1137/0507003, ISSN 0036-1410, MR 0399537
• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, 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), "Al-Salam–Chihara 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

Further reading

• Bryc, W., Matysiak, W., & Szabłowski, P. (2005). Probabilistic aspects of Al-Salam–Chihara polynomials. Proceedings of the American Mathematical Society, 133(4), 1127-1134.
• Floreanini, R., LeTourneux, J., & Vinet, L. (1997). Symmetry techniques for the Al-Salam-Chihara polynomials. Journal of Physics A: Mathematical and General, 30(9), 3107.
• Christiansen, J. S., & Koelink, E. (2008). Self-adjoint difference operators and symmetric Al-Salam–Chihara polynomials. Constructive Approximation, 28(2), 199-218.
• Ishikawa, M., & Zeng, J. (2009). The Andrews–Stanley partition function and Al-Salam–Chihara polynomials. Discrete Mathematics, 309(1), 151-175.
• Atakishiyeva, M. K., & Atakishiyev, N. M. (1997). Fourier-Gauss transforms of the Al-Salam-Chihara polynomials. Journal of Physics A: Mathematical and General, 30(19), L655.