Little q-Laguerre polynomials
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In mathematics, the little q-Laguerre polynomials pn(x;a|q) or Wall polynomials Wn(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
- [math]\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) }[/math]
See also
References
- Chihara, Theodore Seio (1978), An introduction to orthogonal polynomials, Mathematics and its Applications, 13, New York: Gordon and Breach Science Publishers, Reprinted by Dover 2011, ISBN 978-0-677-04150-6, https://books.google.com/books?id=IkCJSQAACAAJ
- 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
- 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
- Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Chapter 18: Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/18
- 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, https://ir.cwi.nl/pub/1603
- 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
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