# zbMATH — the first resource for mathematics

$$q$$-difference operators for orthogonal polynomials. (English) Zbl 1185.39005
The families of orthogonal polynomials appearing as solutions of Sturm-Liouville differential equations support the operation of two adjoint linear differential operators which respectively raise and lower the degree. A similar theory has been developped for $$q$$-orthogonal polynomials and many results of the “classical” (differential) theory have been extended to the “basic” ($$q$$-analogue) theory, in particular by the authors of the paper reviewed here, which tackles indeterminate moment problems.

##### MSC:
 39A13 Difference equations, scaling ($$q$$-differences) 33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
Full Text:
##### References:
 [1] Chen, Y.; Ismail, M.E.H., Ladder operators and differential equations for orthogonal polynomials, J. phys. A, 30, 7818-7829, (1997) · Zbl 0927.33011 [2] Bauldry, W., Estimates of asymptotic freud polynomials on the real line, J. approx. theory, 63, 225-237, (1990) · Zbl 0716.42019 [3] Bonan, S.S.; Clark, D.S., Estimates of the Hermite and freud polynomials, J. approx. theory, 63, 210-224, (1990) · Zbl 0716.42018 [4] Mhaskar, H.L., Bounds for certain freud polynomials, J. approx. theory, 63, 238-254, (1990) · Zbl 0716.42020 [5] Ismail, M.E.H., Difference equations and quantized discriminants for $$q$$-orthogonal polynomials, Adv. appl. math., 30, 562-589, (2003) · Zbl 1032.33012 [6] Ismail, M.E.H.; Nikolova, I.; Simeonov, P., Difference equations and discriminants for discrete orthogonal polynomials, Ramanujan J., 8, 475-502, (2004) · Zbl 1081.33014 [7] Ismail, M.E.H., Classical and quantum orthogonal polynomials in one variable, (2005), Cambridge University Press Cambridge [8] Chen, Y.; Ismail, M.E.H., Jacobi polynomials from compatibility conditions, Proc. amer. math. soc., 133, 465-472, (2004) · Zbl 1057.33004 [9] Chen, Y.; Ismail, M.E.H., Ladder operators for $$q$$-orthogonal polynomials, J. math. anal. appl., 345, 1-10, (2008) · Zbl 1154.33010 [10] Akhiezer, N.I., The classical moment problem and some related questions in analysis,, (1965), Oliver and Boyed Edinburgh, English translation · Zbl 0135.33803 [11] Szegő, G., Orthogonal polynomials, (1975), Amer. Math. Soc. Providence · JFM 65.0278.03 [12] Christiansen, J.S., The moment problem associated with the stieltjes – wigert polynomials, J. math. anal. appl., 277, 218-245, (2003) · Zbl 1019.44005 [13] R. Koekoek, R. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its $$q$$-analogues, Reports of the Faculty of Technical Mathematics and Informatics no. 98-17, Delft University of Technology, Delft, 1998
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.