Recurrence Relations of the Hypergeometric-type functions on the quadratic-type lattices

Year 2019, Volume: 3 Issue: 4, 201 - 219, 30.12.2019

Rezan Sevinik Adıgüzel

https://doi.org/10.31197/atnaa.629707

Abstract

The central idea of this article is to present a systematic approach to construct some recurrence relations for the solutions of the second-order linear difference equation of hypergeometric-type defined on the quadratictype lattices. We introduce some recurrence relations for such solutions by also considering their applications to polynomials on the quadratic-type lattices.

Keywords

Hypergeometric function on q-quadratic lattices, Second-order linear difference equation of hypergeometric-type on the q-quadratic lattices, Recurrence relations, q-Racah polynomials, dual Hahn polynomials, TTRR

References

• R. Alvarez-Nodarse, Polinomios hipergeometricos y q-polinomios, Monografias del Seminario Garcia Galdeano. Universidad de Zaragoza. Vol. 26. Prensas Universitarias de Zaragoza, Zaragoza, Spain, 2003. (In Spanish).
• R. Alvarez-Nodarse, On characterizations of classical polynomials, J. Comput. Appl. Math. 196 (2006), pp. 320-337.
• R. Alvarez-Nodarse, N. M. Atakishiyev and R. S. Costas-Santos, Factorization of hypergeometric type difference equations on nonuniform lattices: dynamical algebra, J. Phys. A: Math. Gen. 38 (2005), pp. 153-174.
• R. Alvarez-Nodarse, A La Carte recurrence relations for continuous and discrete hypergeometric functions, Sema Journal 55 (2011), pp. 41-57.
• R. Alvarez-Nodarse, J. L. Cardoso, Recurrence relations for discrete hypergeometric functions, J. Difference Eq. Appl. 11 (2005), pp. 829-850.
• R. Alvarez-Nodarse, J. L. Cardoso, On the Properties of Special Functions on the linear-type lattices, J. Math. Anal. Appl. 405 (2013), pp. 271-285.
• R. Alvarez-Nodarse, J. L. Cardoso and N. R. Quintero, On recurrence relations for radial wave functions for the N-th dimensional oscillators and hydrogen-like atoms: analytical and numerical study, Elect. Trans. Num. Anal. 24 (2006), pp. 7-23.
• R. Alvarez-Nodarse, J. C. Medem, q-Classical polynomials and the q-Askey and Nikiforov-Uvarov Tableaus, J. Comput. Appl. Math. 135 (2001), pp. 157-196.
• R. Alvarez-Nodarse, R. Sevinik Adiguzel, On the Krall type polynomials on q-quadratic lattices, Indagationes Mathematicae, 21 (2011), pp. 181-203, doi:10.1016/j.indag.2011.04.002.
• R. Alvarez-Nodarse, R. Sevinik Adiguzel, The q-Racah-Krall-type polynomials, Appl. Math. Comput., 218 (2012), pp. 11362-11369.
• R. Alvarez-Nodarse, Yu. F. Smirnov, R. S. Costas-Santos, A q-Analog of Racah Polynomials and q-Algebra SU q(2) in Quantum Optics, J.Russian Laser Research 27 (2006), 1-32.
• N. M. Atakishiyev, M. Rahman, S. K. Suslov, Classical Orthogonal Polynomials, Constr. Approx. 11 (1995), pp. 181-226.
• G. Gasper and M. Rahman, Basic Hypergeometric Series (2nd Ed.), Encyclopedia of Mathematics and its Applications 96, Cambridge University Press, Cambridge, 2004.
• A. Gil, J. Segura, and N.M. Temme, The ABC of hyper recursions, J. Comput. Appl. Math., 190 (2006), pp. 270-286.
• A. Gil, J. Segura, and N.M. Temme, Numerically satisfactory solutions of hypergeometric recursions, Math. Comp., 76 (2007), pp. 1449-1468.
• W.B. Jones, O. Njastad, Orthogonal Laurent Polynomials and strong moment theory: a survey, J. Comput. Appl. Math., 105 (1999), pp. 51-91.
• R. Koekoek, Peter A. Lesky, and R.F. Swarttouw, Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin-Heidelberg, 2010.
• A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer Ser. Comput. Phys., Springer-Verlag, Berlin, 1991.
• S. K. Suslov, The theory of difference analogues of special functions of hypergeometric type, Russian Math. Surveys 44:2 (1989), pp. 227-278.
• S. K. Suslov, A correction, Russian Math. Surveys 45:3 (1990), pp. 245-245.