Little q-Laguerre polynomials

In mathematics, the little q-Laguerre polynomials p_n(x;a|q) or Wall polynomials W_n(x; b,q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme closely related to a continued fraction studied by Wall (1941). (The term "Wall polynomial" is also used for an unrelated Wall polynomial in the theory of classical groups.) 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 and the Pochhammer symbol by

\displaystyle p_{n}(x;a|q)={}_{2}\phi _{1}(q^{-n},0;aq;q,qx)={\frac {1}{(a^{-1}q^{-n};q)_{n}}}{}_{2}\phi _{0}(q^{-n},x^{-1};;q,x/a)

Orthogonality

Recurrence and difference relations

Rodrigues formula

Generating function

Relation to other polynomials

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References

Chihara, Theodore Seio (1978), An introduction to orthogonal polynomials, Mathematics and its Applications, 13, New York: Gordon and Breach Science Publishers, ISBN 978-0-677-04150-6, MR 0481884, Reprinted by Dover 2011
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), http://dlmf.nist.gov/18 |contribution-url= missing title (help), 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
Van Assche, Walter; Koornwinder, Tom H. (1991), "Asymptotic behaviour for Wall polynomials and the addition formula for little q-Legendre polynomials", SIAM Journal on Mathematical Analysis, 22 (1): 302–311, doi:10.1137/0522019, ISSN 0036-1410, MR 1080161
Wall, H. S. (1941), "A continued fraction related to some partition formulas of Euler", The American Mathematical Monthly, 48 (2): 102–108, doi:10.1080/00029890.1941.11991074, ISSN 0002-9890, JSTOR 2303599, MR 0003641

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