Tbilisi Mathematical Journal

Existence of a pair of new recurrence relations for the Meixner-Pollaczek polynomials

E. I. Jafarov, A. M. Jafarova, and S. M. Nagiyev

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We report on existence of pair of new recurrence relations (or difference equations) for the Meixner-Pollaczek polynomials. Proof of the correctness of these difference equations is also presented. Next, we found that subtraction of the forward shift operator for the Meixner-Pollaczek polynomials from one of these recurrence relations leads to the difference equation for the Meixner-Pollaczek polynomials generated via $\cosh$ difference differentiation operator. Then, we show that, under the limit $\varphi \to 0$, new recurrence relations for the Meixner-Pollaczek polynomials recover pair of the known recurrence relations for the generalized Laguerre polynomials. At the end, we introduced differentiation formula, which expresses Meixner-Pollaczek polynomials with parameters $\lambda>0$ and $0 \lt \varphi \lt \pi$ via generalized Laguerre polynomials.

Note

This work was supported by the Science Development Foundation under the President of the Republic of Azerbaijan Grant Nr EIF-KETPL-2-2015-1(25)-56/01/1 and Grant Nr EIF-KETPL-2-2015-1(25)-56/02/1. E.I. Jafarov kindly acknowledges support for visit to ICTP during July-September 2017, within the ICTP regular associateship scheme.

Article information

Source
Tbilisi Math. J., Volume 11, Issue 3 (2018), 29-39.

Dates
Received: 15 November 2017
Accepted: 15 June 2018
First available in Project Euclid: 3 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.tbilisi/1538532024

Subjects
Primary: 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]
Secondary: 39A10: Difference equations, additive 42C05: Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]

Keywords
Meixner-Pollaczek polynomials finite-difference equation recurrence relations

Citation

Jafarov, E. I.; Jafarova, A. M.; Nagiyev, S. M. Existence of a pair of new recurrence relations for the Meixner-Pollaczek polynomials. Tbilisi Math. J. 11 (2018), no. 3, 29--39. https://projecteuclid.org/euclid.tbilisi/1538532024


Export citation

References

  • L.D. Landau and E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 689 pages. Butterworth-Heinemann, Oxford (1981)
  • M. Moshinsky and Y.F. Smirnov, The Harmonic Oscillator in Modern Physics, 414 pages. Harwood Academic Publishers, Amsterdam (1996)
  • R. Koekoek, P.A. Lesky and R.F. Swarttouw, Hypergeometric orthogonal polynomials and their $q$-analogues, 578 pages. Springer-Verlag, Berlin (2010)
  • Y. Ohnuki and S. Kamefuchi, Quantum Field Theory and Parastatistics, 490 pages. Springer-Verlag, Berlin-Heidelberg (1982)
  • A. Erd´elyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw-Hill (1953)
  • S. Khan, A.A. Al-Gonah and G. Yasmin, Some properties of Hermite based Appell matrix polynomials, Tbilisi Math. J. 10 (2017), 121–131
  • N. Raza, S. Khan and M. Ali, Properties of certain new special polynomials associated with Sheffer sequences, Tbilisi Math. J. 9 (2016), 245–270
  • F.G. Abdullayev and G.A. Abdullayeva, The "algebraic zero" condition for orthogonal polynomials over a contour in the weighted Lebesgue spaces, Proc. IMM ANAS 42 (2016), 154–173
  • N.M. Atakishiev, R.M. Mir-Kasimov and Sh.M. Nagiev, Quasipotential models of a relativistic oscillator, Theor. Math. Phys. 44 (1980), 592–603
  • A.U. Klimyk, The $su(1,1)$-models of quantum oscillator, Ukr. J. Phys. 51 (2006), 1019–1027
  • N.M. Atakishiyev and S.K. Suslov, The Hahn and Meixner polynomials of an imaginary argument and some of their applications, J. Phys. A: Math. Gen. 18 (1985), 1583–1596
  • N.M. Atakishiyev, E.I. Jafarov, S.M. Nagiyev and K.B. Wolf, Meixner oscillators, Rev. Mex. Fis. 44 (1998), 235–244
  • E.I. Jafarov, J. Van der Jeugt, Discrete series representations for $sl(2|1)$, Meixner polynomials and oscillator models, J. Phys. A: Math. Theor. 45 (2012), 485201
  • N.M. Atakishiyev, A.M. Jafarova and E.I. Jafarov, Meixner Polynomials and Representations of the 3D Lorentz group $SO(2,1)$, Comm. Math. Anal. 17 (2014), 14–23
  • J. Meixner, Orthogonale Polynomsysteme Mit Einer Besonderen Gestalt Der Erzeugenden Funktion, J. London Math. Soc. s1-9 (1934), 6–13
  • M.V. Tratnik, Multivariable Meixner, Krawtchouk, and Meixner-Pollaczek polynomials, J. Math. Phys. 30 (1989), 2740–2749
  • K. Mimachi, Barnes-Type Integral and the Meixner-Pollaczek Polynomials, Lett. Math. Phys. 48 (1999), 365–373
  • Y. Chen and M.E.H. Ismail, Asymptotics of extreme zeros of the Meixner-Pollaczek polynomials, J. Comp. App. Math. 82 (1997), 59–78
  • X. Li and R. Wong, On the asymptotics of the Meixner-Pollaczek polynomials and their zeros, Constr. Approx. 17 (2001), 59–90
  • C. Ferreira, J.L. López and E.P. Sinusía, Asymptotic relations between the Hahn-type polynomials and Meixner-Pollaczek, Jacobi, Meixner and Krawtchouk polynomials, J. Comp. App. Math. 217 (2008), 88–109
  • S. Kanas and A. Tatarczak, Generalized Meixner-Pollaczek polynomials, Adv. Differ. Equ. 2013 (2013), 131
  • A. Kuznetsov, Integral representations for the Dirichlet L-functions and their expansions in Meixner-Pollaczek polynomials and rising factorials, Integral Transform. Spec. Funct. 18 (2007), 827–835
  • A. Kuznetsov, Expansion of the Riemann $\Xi$ Function in Meixner-Pollaczek Polynomials, Canad. Math. Bull. 51 (2008), 561–569
  • Z. Mouayn, A new class of coherent states with Meixner-Pollaczek polynomials for the Gol'dman-Krivchenkov Hamiltonian, J. Phys. A: Math. Theor. 43 (2010), 295201
  • D.D. Tcheutia, P. Njionou Sadjang, W. Koepf and M. Foupouagnigni, Divided-difference equation, inversion, connection, multiplication and linearization formulae of the continuous Hahn and the Meixner-Pollaczek polynomials, Ramanujan J. 44 (2017), 1–24
  • Sh.M. Nagiev, Dynamical symmetry group of the relativistic Coulomb problem in the quasipotential approach, Theor. Math. Phys. 80 (1989), 697–702
  • T. Koornwinder, Meixner-Pollaczek polynomials and the Heisenberg algebra, J. Math. Phys. 30 (1989), 767–769
  • E.P. Wigner, Do the equations of motion determine the quantum mechanical commutation relations?, Phys. Rev. 77 (1950), 711–712