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 q-Pochhammer symbol by

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

Orthogonality[]

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Recurrence and difference relations[]

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Rodrigues formula[]

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Generating function[]

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Relation to other polynomials[]

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See also[]

[1]

References[]

• Chihara, Theodore Seio (1978), An introduction to orthogonal polynomials, Mathematics and its Applications, vol. 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, vol. 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, 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 {{citation}}: |contribution-url= missing title (help)
• 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