Asymptotic analysis of the Askey-scheme I: from Krawtchouk to Charlier

Diego Dominici

Open Mathematics, Tome 5 (2007), p. 280-304 / Harvested from The Polish Digital Mathematics Library

Résumé

We analyze the Charlier polynomials C n(χ) and their zeros asymptotically as n → ∞. We obtain asymptotic approximations, using the limit relation between the Krawtchouk and Charlier polynomials, involving some special functions. We give numerical examples showing the accuracy of our formulas.

Détail

Publié le : 2007-01-01

Zbl 1124.33008

EUDML-ID : urn:eudml:doc:269797

Bibliographie

[1] M. Abramowitz and I.A. Stegun (Eds.): Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications Inc., New York, 1992. Reprint of the 1972 edition.

[2] N. Asai: “Integral transform and Segal-Bargmann representation associated to q-Charlier polynomials”, In: Quantum information, IV (Nagoya, 2001), World Sci. Publishing, River Edge, NJ, 2002.

[3] A.D. Barbour: “Asymptotic expansions in the Poisson limit theorem”, Ann. Probab., Vol. 15(2), (1987), pp. 748–766. | Zbl 0622.60049

[4] N. Barik: “Some theorems on generating functions for Charlier polynomials”, J. Pure Math., Vol. 3, (1983), pp. 111–114. | Zbl 0611.33017

[5] H. Bavinck and R. Koekoek: “On a difference equation for generalizations of Charlier polynomials”, J. Approx. Theory, Vol. 81(2), 1(995), pp. 195-206. | Zbl 0865.33006

[6] R. Bo and R. Wong: “Uniform asymptotic expansion of Charlier polynomials”, Methods Appl. Anal., Vol. 1(3), 1994, pp. 294–313. | Zbl 0846.41025

[7] V.V. Borzov and E.V. Damaskinskii: “Charlier polynomials and Charlier oscillator as discrete realization of the harmonic oscillator”, J. Math. Sci. (N. Y.), Vol. 128(5), (2005), 3161–3176. http://dx.doi.org/10.1007/s10958-005-0262-4 | Zbl 1099.33005

[8] C. Charlier: “ Über die Darstellung willkürlicher Funktionen”, Ark. Mat. Astron. Fys., Vol. 2(20), (1906), pp. 1–35. | Zbl 36.0460.01

[9] A. de Médicis, D. Stanton and D. White: “The combinatorics of q-Charlier polynomials”, J. Combin. Theory Ser. A, Vol. 69(1), (1995), pp. 87–114. http://dx.doi.org/10.1016/0097-3165(95)90108-6 | Zbl 0819.05061

[10] D. E. Dominici: “Asymptotic analysis of the Krawtchouk polynomials by the WKB method”, To appear in The Ramanujan Journal. | Zbl 1148.33006

[11] D. Dominici: “Asymptotic analysis of the Askey-scheme II: from Charlier to Hermite”, Submitted, 2005, arXiv: math.CA/0508264.

[12] T.M. Dunster: “Uniform asymptotic expansions for Charlier polynomials”, J. Approx. Theory, Vol. 112(1), (2001), pp. 93–133. http://dx.doi.org/10.1006/jath.2001.3595

[13] C. Ferreira, J.L. López and E. Mainar: “Asymptotic approximations of orthogonal polynomials”, In: Seventh Zaragoza-Pau Conference on Applied Mathematics and Statistics (Spanish) (Jaca, 2001), Vol. 27 of Monogr. Semin. Mat. García Galdeano, Univ. Zaragoza, Zaragoza, 2003. | Zbl 1046.33007

[14] C. Ferreira, J.L. Lopez and E. Mainar: “Asymptotic relations in the Askey scheme for hypergeometric orthogonal polynomials” Adv. in Appl. Math., Vol. 31(1), (2003), pp. 61–85. http://dx.doi.org/10.1016/S0196-8858(02)00552-3 | Zbl 1029.33004

[15] W.M.Y. Goh: “Plancherel-Rotach asymptotics for the Charlier polynomials”, Constr. Approx., Vol. 14(2), (1998), pp. 151–168. http://dx.doi.org/10.1007/s003659900067 | Zbl 0894.33004

[16] M.N. Hounkonnou, C. Hounga and A. Ronveaux: “Discrete semi-classical orthogonal polynomials: generalized Charlier”, J. Comput. Appl. Math., Vol. 114(2), (2000), pp. 361–366. http://dx.doi.org/10.1016/S0377-0427(99)00275-7 | Zbl 0971.42015

[17] L.C. Hsu: “Certain asymptotic expansions for Laguerre polynomials and Charlier polynomials”, Approx. Theory Appl. (N.S.), Vol. 11(1), (1995), pp. 94–104. | Zbl 0827.41024

[18] D.L. Jagerman: “Nonstationary blocking in telephone traffic”, Bell System Tech. J., Vol. 54, (1975), pp. 625–661.

[19] G.C. Jain and R.P. Gupta: “On a class of polynomials and associated probabilities”, Utilitas Math., Vol. 7, (1975), pp. 363–381. | Zbl 0311.60010

[20] R. Koekoek and R.F. Swarttouw: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Technical Report 98-17, Delft University of Technology, 1998, http://aw.twi.tudelft.nl/ koekoek/askey/.

[21] H.T. Koelink: “Yet another basic analogue of Graf’s addition formula” J. Comput. Appl. Math., Vol. 68(1-2), (1996), pp. 209–220. http://dx.doi.org/10.1016/0377-0427(95)00257-X

[22] I. Krasikov: “Bounds for zeros of the Charlier polynomials”, Methods Appl. Anal., Vol. 9(4), (2002), pp. 599–610. | Zbl 1166.33301

[23] L. Larsson-Cohn: “L p -norms and information entropies of Charlier polynomials”, J. Approx. Theory, Vol. 117(1), (2002), pp. 152–178. http://dx.doi.org/10.1006/jath.2002.3691

[24] P.A. Lee: “Some generating functions involving the Charlier polynomials”, Nanta Math., Vol. 8(1), (1975), pp. 83–87. | Zbl 0327.33003

[25] J. Letessier: “Some results on co-recursive associated Meixner and Charlier polynomials”, J. Comput. Appl. Math., Vol. 103(2), (1999), pp. 323–335. http://dx.doi.org/10.1016/S0377-0427(98)00269-6 | Zbl 0941.33004

[26] J.L. López and N.M. Temme: “Convergent asymptotic expansions of Charlier, Laguerre and Jacobi polynomials”, Proc. Roy. Soc. Edinburgh Sect. A, Vol. 134(3), (2004), pp. 537–555. | Zbl 1059.33010

[27] M. Maejima and W. Van Assche: “Probabilistic proofs of asymptotic formulas for some classical polynomials”, Math. Proc. Cambridge Philos. Soc., Vol. 97(3), (1985), pp. 499–510. http://dx.doi.org/10.1017/S0305004100063088 | Zbl 0557.33006

[28] E.B. McBride: Obtaining generating functions, Springer Tracts in Natural Philosophy, Vol. 21, Springer-Verlag, New York, 1971.

[29] M.L. Mehta and E.A. van Doorn: “Inequalities for Charlier polynomials with application to teletraffic theory”, J. Math. Anal. Appl., Vol. 133(2), (1988), pp. 449–460. http://dx.doi.org/10.1016/0022-247X(88)90414-3 | Zbl 0648.60094

[30] C. Micu and E. Papp: “Discrete analogs of quantum mechanical systems. Kravchuk and Charlier polynomials”, In: Proceedings of the Tenth Symposium of Mathematics and its Applications (Timişoara, Nov.6–9, 2003), 2003, pp. 458-464.

[31] J. Negro and L.M. Nieto: “Symmetries of the wave equation in a uniform lattice” J. Phys. A, Vol. 29(5), (1996), pp. 1107–1114. http://dx.doi.org/10.1088/0305-4470/29/5/023

[32] N. Privault: “Multiple stochastic integral expansions of arbitrary Poisson jump times functionals”, Statist. Probab. Lett., Vol. 43(2), (1999), pp. 179–188. http://dx.doi.org/10.1016/S0167-7152(98)00257-0 | Zbl 0938.60077

[33] B. Roos: “Poisson approximation of multivariate Poisson mixtures”, J. Appl. Probab., Vol. 40(2), (2003), pp. 376–390. http://dx.doi.org/10.1239/jap/1053003550 | Zbl 1028.60011

[34] A. Ruffing, J. Lorenz and K. Ziegler: „Difference ladder operators for a harmonic Schrödinger oscillator using unitary linear lattices”, In: Proceedings of the Sixth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Rome, 2001), Vol. 153, 2003, pp. 395–410. | Zbl 1023.39015

[35] W. Schoutens: Lévy-Sheffer and IID-Sheffer polynomials with applications to stochastic integrals”, In: Proceedings of the VIIIth Symposium on Orthogonal Polynomials and Their Applications (Seville, 1997), Vol. 99, 1998, pp. 365–372.

[36] B. Şefik: “Coherent structures in nonlinear dynamical systems. Method of the random point functions”, In: Nonlinear evolution equations and dynamical systems (Baia Verde, 1991), World Sci. Publishing, River Edge, NJ, 1992, pp. 385–394.

[37] R. M. Shreshtha: “On generalised Charlier polynomials”, Nepali Math. Sci. Rep., 7(2), (1982), pp. 65–69. | Zbl 0537.33009

[38] J. Spanier and K.B. Oldham: An Atlas of Functions, Hemisphere Pub. Corp., 1987.

[39] F.H. Szafraniec: “Charlier polynomials and translational invariance in the quantum harmonic oscillator”, Math. Nachr., Vol. 241, (2002), pp. 163–169. http://dx.doi.org/10.1002/1522-2616(200207)241:1<163::AID-MANA163>3.0.CO;2-W

[40] G. Szegő: Orthogonal polynomials, 4th ed., American Mathematical Society, Providence, R.I., 1975; American Mathematical Society, Colloquium Publications, Vol. XXIII.

[41] N.M. Temme and J.L. López: “The Askey scheme for hypergeometric orthogonal polynomials viewed from asymptotic analysis”, In: Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999), Vol. 133, 2001, pp. 623–633. | Zbl 0990.33010

[42] T.T. Truong: “On a class of inhomogeneous Ising quantum chains”, J. Phys. A, Vol. 28(24), 1995, pp. 7089–7096. http://dx.doi.org/10.1088/0305-4470/28/24/009

[43] W. Van Assche and M. Foupouagnigni: “Analysis of non-linear recurrence relations for the recurrence coefficients of generalized Charlier polynomials”, J. Nonlinear Math. Phys., Vol. 10(suppl. 2), (2003), pp. 231–237. http://dx.doi.org/10.2991/jnmp.2003.10.s2.19

[44] J. Zeng: “The q-Stirling numbers, continued fractions and the q-Charlier and q-Laguerre polynomials”, J. Comput. Appl. Math., Vol. 57(3), (1995), pp. 413–424. http://dx.doi.org/10.1016/0377-0427(93)E0211-4