Some Properties Of Hypergeometric Meixner-Pollaczek Polynomials

Year 2017, Volume: 7, 21 - 31, 19.12.2017

Nejla Özmen , Hasan Göksu

Abstract

Orthogonal polynomials appear in many areas of mathematics and have been the subject of interest of many mathematicians.The present study deals with some new properties for the Meixner-Pollaczek polynomials $P_{n}^{\left( \lambda\right) }\left( x;\phi \right) $. The results obtained here include various families of multilinear and multilateral generating functions, miscellaneous properties and also some special cases for these polynomials.Relevant connections of some of these families of generating functions with various known results are also indicated.

Keywords

Konhauser Biorthogonal Polynomials, generating function, multilinear and multilateral generating function

References

• Araaya, T.K., The Symmetric Meixner-Pollaczek Polynomials, Uppsala Dissertations in Mathematics, Uppsala University, 27, 2003.
• Askey, R, Wilson, J., Some Basic Hypergeometric Orthogonal Polynomials That Generalize Jacobi Polynomials, Mem. Am. Math. Soc., 54, 1985.
• Atakishiyev, N.M., Suslov S. K. , The Hahn and Meixner polynomials of an imaginary argument and some of their applications, J. Phys. A, Math. Gen., 18(1985).
• Chihara, T.S., An Introduction to Orthogonal Polynomials, Gorden and Breach, Science Publishers, pp 175-186, 1978.
• Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G., Higher Transcendental Functions, Vol. III, McGraw-Hill Book Company, New York, Toronto, London, 1955.
• Erkus, E., Srivastava, H.M., A unified presentation of some families of multivariable polynomials, Integral Transform Spec. Funct., 17(2006), 267–273.
• Koekoek, R., Lesky, P.A., Swarttouw, R.F., Hypergeometric Orthogonal Polynomials and Their q–Analogues, Springer-Verlag Berlin Heidelberg, 2010.
• Koekoek, R., Swarttouw, R.F., The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue, Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, pp. 37-38, 1998.
• Koornwinder, T.H., Meixner-Pollaczek polynomials and the Heisenberg algebra, J. Math. Phys., 30(4)(1989), 767–769.
• Li, X., Wong, R., On the asymptotics of the Meixner-Pollaczek polynomials and their zeros, Constr. Approx., 17(2001), 59–90.
• Meixner, J., Orthogonale polynomsysteme mit einer besonderen Gestalt der erzeugenden funktion, J. London Math. Soc., 9(1934), 6-13.
• Özmen, N., Erkus-Duman, E., Some results for a family of multivariable polynomials, AIP Conf. Proc., 1558 (2013), 1124-1127.
• Özmen, N., Erkus-Duman, E., On the Poisson-Charlier polynomials, Serdica Math. J., 41(2015), 457–470.
• Özmen, N., Erkus-Duman, E., Some families of generating functions for the generalized Cesa´ro polynomials, J. Comput. Anal. Appl., 25(4)(2018), Copyright 2018 Eudoxus Press, LLC, 670-683.
• Pollaczek, F., Sur une famille de polynomes orthogonaux qui contient les polynomes d’Hermite et de Laguerre comme cas limites, Ibid., 230(1950), 1563–1565.
• Szegö, G., An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.