In mathematics, the q -Laguerre polynomialsgeneralized Stieltjes–Wigert polynomials P (α) n x ;q ) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme introduced by Daniel S. Moak (1981 ). Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010 , 14) give a detailed list of their properties.
Definition
The q -Laguerre polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by
L
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{\displaystyle \displaystyle L_{n}^{(\alpha )}(x;q)={\frac {(q^{\alpha +1};q)_{n}}{(q;q)_{n}}}{}_{1}\phi _{1}(q^{-n};q^{\alpha +1};q,-q^{n+\alpha +1}x).}
Orthogonality
Orthogonality is defined by the unimono nature of the polynomials' convergence at boundaries in integral form.
References
Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series , Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press , 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), "Chapter 18: Orthogonal Polynomials" , in Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions ISBN 978-0-521-19225-5 MR 2723248 Moak, Daniel S. (1981), "The q-analogue of the Laguerre polynomials", J. Math. Anal. Appl. , 81 (1): 20– 47, doi :10.1016/0022-247X(81)90048-2 MR 0618759
