Existence of a pair of new recurrence relations for the Meixner-Pollaczek polynomials. (English) Zbl 1435.33009

Summary: 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 \(\varphi \rightarrow 0\), 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 \(0 < \varphi < \pi\) via generalized Laguerre polynomials.


33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
39A10 Additive difference equations
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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[1] L.D. Landau and E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 689 pages. Butterworth-Heinemann, Oxford (1981)
[2] M. Moshinsky and Y.F. Smirnov, The Harmonic Oscillator in Modern Physics, 414 pages. Harwood Academic Publishers, Amsterdam (1996) · Zbl 0865.00015
[3] R. Koekoek, P.A. Lesky and R.F. Swarttouw, Hypergeometric orthogonal polynomials and their \(q\)-analogues, 578 pages. Springer-Verlag, Berlin (2010) · Zbl 1200.33012
[4] Y. Ohnuki and S. Kamefuchi, Quantum Field Theory and Parastatistics, 490 pages. Springer-Verlag, Berlin-Heidelberg (1982) · Zbl 0566.46041
[5] A. Erd´elyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw-Hill (1953)
[6] S. Khan, A.A. Al-Gonah and G. Yasmin, Some properties of Hermite based Appell matrix polynomials, Tbilisi Math. J. 10 (2017), 121-131 · Zbl 1364.33004
[7] N. Raza, S. Khan and M. Ali, Properties of certain new special polynomials associated with Sheffer sequences, Tbilisi Math. J. 9 (2016), 245-270 · Zbl 1342.33027
[8] 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 · Zbl 1364.30008
[9] N.M. Atakishiev, R.M. Mir-Kasimov and Sh.M. Nagiev, Quasipotential models of a relativistic oscillator, Theor. Math. Phys. 44 (1980), 592-603 · Zbl 0582.33004
[10] A.U. Klimyk, The \(su(1,1)\)-models of quantum oscillator, Ukr. J. Phys. 51 (2006), 1019-1027
[11] 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 · Zbl 0582.33006
[12] N.M. Atakishiyev, E.I. Jafarov, S.M. Nagiyev and K.B. Wolf, Meixner oscillators, Rev. Mex. Fis. 44 (1998), 235-244 · Zbl 1291.81128
[13] 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 · Zbl 1278.81067
[14] 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 · Zbl 1323.33010
[15] J. Meixner, Orthogonale Polynomsysteme Mit Einer Besonderen Gestalt Der Erzeugenden Funktion, J. London Math. Soc. s1-9 (1934), 6-13 · JFM 60.0293.01
[16] M.V. Tratnik, Multivariable Meixner, Krawtchouk, and Meixner-Pollaczek polynomials, J. Math. Phys. 30 (1989), 2740-2749
[17] K. Mimachi, Barnes-Type Integral and the Meixner-Pollaczek Polynomials, Lett. Math. Phys. 48 (1999), 365-373 · Zbl 0935.33010
[18] Y. Chen and M.E.H. Ismail, Asymptotics of extreme zeros of the Meixner-Pollaczek polynomials, J. Comp. App. Math. 82 (1997), 59-78 · Zbl 0902.33004
[19] X. Li and R. Wong, On the asymptotics of the Meixner-Pollaczek polynomials and their zeros, Constr. Approx. 17 (2001), 59-90 · Zbl 0971.41016
[20] 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 · Zbl 1151.33007
[21] S. Kanas and A. Tatarczak, Generalized Meixner-Pollaczek polynomials, Adv. Differ. Equ. 2013 (2013), 131 · Zbl 1392.33006
[22] 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 · Zbl 1134.11033
[23] A. Kuznetsov, Expansion of the Riemann \(\Xi\) Function in Meixner-Pollaczek Polynomials, Canad. Math. Bull. 51 (2008), 561-569 · Zbl 1173.41003
[24] 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 · Zbl 1193.81051
[25] 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 · Zbl 1382.33008
[26] Sh.M. Nagiev, Dynamical symmetry group of the relativistic Coulomb problem in the quasipotential approach, Theor. Math. Phys. 80 (1989), 697-702
[27] T. Koornwinder, Meixner-Pollaczek polynomials and the Heisenberg algebra, J. Math. Phys. 30 (1989), 767-769 · Zbl 0672.33011
[28] E.P. Wigner, Do the equations of motion determine the quantum mechanical commutation relations?, Phys. Rev. 77 (1950), 711-712 · Zbl 0036.14301
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