## Second structure relation for $$q$$-semiclassical polynomials of the Hahn Tableau.(English)Zbl 1113.33022

Only the first structure theorem for the $$q$$-semiclassical orthogonal polynomials exists. The authors propose a second structure relation. In doing so, the interplay of a general finite-type relation between a $$q$$-classical polynomial and the sequence of its $$q$$-differences is exploited.

### MSC:

 33D99 Basic hypergeometric functions

### Keywords:

q-classical orthogonal polynomials; q-Hahn Tableau
Full Text:

### References:

 [1] Al-Salam, W.A., Characterization theorems for orthogonal polynomials, (), 1-24 · Zbl 0133.32305 [2] Álvarez-Nodarse, R.; Arvesú, J., On the q-polynomials on the exponential lattice x(s)=c1qs+c3, Integral transforms spec. funct., 8, 299-324, (1999) · Zbl 0956.33009 [3] Álvarez-Nodarse, R.; Medem, J.C., The q-classical polynomials and the q-Askey and nikiforov – uvarov tableaus, J. comput. appl. math., 135, 197-223, (2001) · Zbl 1024.33013 [4] Chihara, T.S., An introduction to orthogonal polynomials, (1978), Gordon and Breach New York · Zbl 0389.33008 [5] García, A.G.; Marcellán, F.; Salto, L., A distributional study of discrete classical orthogonal polynomials, J. comput. appl. math., 57, 147-162, (1995) · Zbl 0853.33009 [6] Hahn, W., Über orthogonalpolynomen die q-differentialgleichungen genügen, Math. nachr., 2, 4-34, (1949) · Zbl 0031.39001 [7] Kheriji, L., An introduction to the $$H_q$$-semiclassical orthogonal polynomials, Methods appl. anal., 10, 387-412, (2003) · Zbl 1058.33018 [8] Kheriji, L.; Maroni, P., The $$H_q$$-classical orthogonal polynomials, Acta appl. math., 71, 49-115, (2002) · Zbl 1003.33008 [9] Koepf, W.; Schmersau, D., On a structure formula for classical q-orthogonal polynomials, J. comput. appl. math., 136, 99-107, (2001) · Zbl 1004.33008 [10] Koekoek, R.; Swarttouw, R.F., The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Rep. fac. of technical math. informatics, vol. 98-17, (1998), Delft University of Technology Delft, The Netherlands [11] Marcellán, F.; Salto, L., Discrete semi-classical orthogonal polynomials, J. difference equ. appl., 4, 5, 463-496, (1998) · Zbl 0916.33006 [12] F. Marcellán, R. Sfaxi, Second structure relation for semiclassical orthogonal polynomials, J. Comput. Appl. Math., in press · Zbl 1125.33008 [13] Maroni, P., Une théorie algébrique des polynômes orthogonaux. application aux polynômes orthogonaux semi-classiques, (), 95-130 · Zbl 0944.33500 [14] Maroni, P., Semiclassical character and finite-type relations between polynomial sequences, Appl. numer. math., 31, 295-330, (1999) · Zbl 0962.42017 [15] Maroni, P.; Sfaxi, R., Diagonal orthogonal polynomial sequences, Methods appl. anal., 7, 769-792, (2000) · Zbl 1025.42013 [16] J.C. Medem, Polinomios q-Semiclásicos, Doctoral Dissertation, Universidad Politécnica, Madrid, 1996 (in Spanish) [17] Medem, J.C.; Alvarez-Nodarse, R.; Marcellan, F., On the q-polynomials: A distributional study, J. comput. anal. math., 135, 157-196, (2001) · Zbl 0991.33007 [18] Nikiforov, A.F.; Suslov, S.K.; Uvarov, V.B., Classical orthogonal polynomials of a discrete variable, Springer ser. comput. physics, (1991), Springer-Verlag Berlin · Zbl 0743.33001
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.