Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 044, 18 pages      arXiv:1511.06057
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

Polynomial Sequences Associated with the Moments of Hypergeometric Weights

Diego Dominici
Department of Mathematics, State University of New York at New Paltz, 1 Hawk Dr., New Paltz, NY 12561-2443, USA

Received November 23, 2015, in final form April 25, 2016; Published online April 29, 2016

We present some families of polynomials related to the moments of weight functions of hypergeometric type. We also consider different types of generating functions, and give several examples.

Key words: moments; hypergeometric functions; generating functions; Stieltjes transform.

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  1. Akhiezer N.I., The classical moment problem and some related questions in analysis, Hafner Publishing Co., New York, 1965.
  2. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
  3. Boelen L., Filipuk G., Van Assche W., Recurrence coefficients of generalized Meixner polynomials and Painlevé equations, J. Phys. A: Math. Theor. 44 (2011), 035202, 19 pages.
  4. Chihara T.S., An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York - London - Paris, 1978.
  5. Clarkson P.A., Recurrence coefficients for discrete orthonormal polynomials and the Painlevé equations, J. Phys. A: Math. Theor. 46 (2013), 185205, 18 pages, arXiv:1301.2396.
  6. Dominici D., Asymptotic analysis of the Askey-scheme. I. From Krawtchouk to Charlier, Cent. Eur. J. Math. 5 (2007), 280-304, math.CA/0501072.
  7. Dominici D., Asymptotic analysis of the Hermite polynomials from their differential-difference equation, J. Difference Equ. Appl. 13 (2007), 1115-1128, math.CA/0601078.
  8. Dominici D., Asymptotic analysis of generalized Hermite polynomials, Analysis (Munich) 28 (2008), 239-261, math.CA/0606324.
  9. Dominici D., Asymptotic analysis of the Krawtchouk polynomials by the WKB method, Ramanujan J. 15 (2008), 303-338, math.CA/0501042.
  10. Dominici D., Some properties of the inverse error function, in Tapas in Experimental Mathematics, Contemp. Math., Vol. 457, Amer. Math. Soc., Providence, RI, 2008, 191-203.
  11. Dominici D., Asymptotic analysis of the Bell polynomials by the ray method, J. Comput. Appl. Math. 233 (2009), 708-718, arXiv:0709.0252.
  12. Dominici D., Mehler-Heine type formulas for Charlier and Meixner polynomials, Ramanujan J. 39 (2016), 271-289, arXiv:1406.6193.
  13. Dominici D., Driver K., Jordaan K., Polynomial solutions of differential-difference equations, J. Approx. Theory 163 (2011), 41-48, arXiv:0902.0041.
  14. Dominici D., Knessl C., Asymptotic analysis of a family of polynomials associated with the inverse error function, Rocky Mountain J. Math. 42 (2012), 847-872, arXiv:0811.2243.
  15. Dominici D., Marcellán F., Discrete semiclassical orthogonal polynomials of class one, Pacific J. Math. 268 (2014), 389-411, arXiv:1211.2005.
  16. Dominici D., Van Assche W., Zero distribution of polynomials satisfying a differential-difference equation, Anal. Appl. (Singap.) 12 (2014), 635-666, arXiv:1312.0698.
  17. Filipuk G., Van Assche W., Recurrence coefficients of generalized Charlier polynomials and the fifth Painlevé equation, Proc. Amer. Math. Soc. 141 (2013), 551-562, arXiv:1106.2959.
  18. Hildebrandt E.H., Systems of polynomials connected with the Charlier expansions and the Pearson differential and difference equations, Ph.D. Thesis, University of Michigan, 1932.
  19. Ismail M.E.H., Stanton D., Classical orthogonal polynomials as moments, Canad. J. Math. 49 (1997), 520-542.
  20. Ismail M.E.H., Stanton D., More orthogonal polynomials as moments, in Mathematical Essays in Honor of Gian-Carlo Rota (Cambridge, MA, 1996), Progr. Math., Vol. 161, Birkhäuser Boston, Boston, MA, 1998, 377-396.
  21. Johnson N.L., Kotz S., Kemp A.W., Univariate discrete distributions, 2nd ed., Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New York, 1992.
  22. Kemp A.W., Studies in univariate discrete distribution theory based on the generalized hypergeometric function and associated differential equations, Ph.D. Thesis, The Queen's University of Belfast, 1968.
  23. Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
  24. Kreǐn M.G., Nudel'man A.A., The Markov moment problem and extremal problems, Translations of Mathematical Monographs, Vol. 50, Amer. Math. Soc., Providence, R.I., 1977.
  25. Maroni P., Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classiques, in Orthogonal Polynomials and their Applications (Erice, 1990), IMACS Ann. Comput. Appl. Math., Vol. 9, Baltzer, Basel, 1991, 95-130.
  26. Mohammad-Noori M., Some remarks about the derivation operator and generalized Stirling numbers, Ars Combin. 100 (2011), 177-192, arXiv:1012.3948.
  27. Njionou Sadjang P., Moments of classical orthogonal polynomials, Ph.D. Thesis, Universität Kassel, 2013.
  28. Njionou Sadjang P., Koepf W., Foupouagnigni M., On moments of classical orthogonal polynomials, J. Math. Anal. Appl. 424 (2015), 122-151.
  29. Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W. (Editors), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC, Cambridge University Press, Cambridge, 2010.
  30. Pearson K., Contributions to the mathematical theory of evolution. II. Skew variation in homogeneous material, Philos. Trans. Roy. Soc. London Ser. A 186 (1895), 343-414.
  31. Schoutens W., Stochastic processes and orthogonal polynomials, Lecture Notes in Statistics, Vol. 146, Springer-Verlag, New York, 2000.
  32. Shohat J.A., Tamarkin J.D., The problem of moments, American Mathematical Society Mathematical Surveys, Vol. 1, Amer. Math. Soc., New York, 1943.
  33. Wilf H.S., Generatingfunctionology, 3rd ed., A K Peters, Ltd., Wellesley, MA, 2006.

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