Abstract
Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of (\(q=1\)) classical orthogonal polynomials, and study those cases in which the exponential generating function has a nice form. In the opposite direction, we show that the generalized Dumont–Foata polynomials with six parameters are the moments of rescaled continuous dual Hahn polynomials. Finally, we show that one of our methods can be applied to deal with the moments of Askey–Wilson polynomials.
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Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and its Applications, vol. 71, pp. xvi+664. Cambridge University Press, Cambridge (1999)
Askey, R.: Beta integrals and the associated orthogonal polynomials. In: Alladi, K. (ed.) Number Theory. Lecture Notes in Mathematics, vol. 1395, pp. 84–121. Springer, New York (1989)
Askey, R., Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Memoir AMS. 54 (1985)
Bigeni, A.: A generalization of the Kreweras triangle through the universal \({\rm sl}_2\) weight system. J. Comb. Theory Ser. A 161, 309–326 (2019)
Carlitz, L.: Some polynomials of Touchard connected with the Bernoulli numbers. Can. J. Math. 9, 188–190 (1957)
Carlitz, L.: A conjecture concerning Genocchi numbers. Koninkl. norske Vidensk. Selsk. Sk. 9, 1–4 (1972)
Carlitz, L.: Explicit formulas for the Dumont–Foata polynomials. Discret. Math. 30, 211–225 (1980)
Chapoton, F.: Ramanujan-Bernoulli numbers as moments of Racah polynomials. J. Théor. Nombres Bordeaux 32(1), 205–215 (2020)
Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, New-York/London/Paris (1978)
Corteel, S., Kim, J.S., Stanton, D.: Moments of orthogonal polynomials and combinatorics. In: Recent Trends in Combinatorics. The IMA Volumes in Mathematics and its Applications, vol. 159 (2016)
Corteel, S., Stanley, R., Stanton, D., Williams, L.: Formulae for Askey–Wilson moments and enumeration of staircase tableaux. Trans. Am. Math. Soc. 364(11), 6009–6037 (2012)
De Branges, L.: Gauss spaces of entire functions. J. Math. Anal. Appl. 37, 1–41 (1972)
Dominici, D.: Polynomial sequences associated with the moments of hypergeometric weights. SIGMA Symmetry Integrability Geom. Methods Appl. 12(44), 18 (2016)
Dumont, D., Foata, D.: Une propriété de symétrie des nombres de Genocchi. Bull. Soc. Math. France 104(4), 433–451 (1976)
Dumont, D.: Conjectures sur des symétries ternaires liées aux nombres de Genocchi. Discret. Math. 139, 469–472 (1995)
Dumont, D., Zeng, J.: Further results on the Euler and Genocchi numbers. Aequationes Math. 47, 31–42 (1994)
Erdély, A., et al.: Higher Transcendental Functions. Bateman Manuscript Project, vol. 1. McGraw-Hill Book Company, New-York (1953)
Feigin, E.: The median Genocchi numbers, \(q\)-analogues and continued fractions. Eur. J. Comb. 33, 1913–1918 (2012)
Fulmek, M., Krattenthaler, C.: The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis II. Eur. J. Comb. 21, 601–640 (2000)
Gandhi, J.M.: A conjectured representation of Genocchi numbers. Am. Math. Monthly 77, 505–506 (1970)
Gasper, G., Rahman, M.: Basic Hypergeometric Series. With a Foreword by Richard Askey. Encyclopedia of Mathematics and its Applications, vol. 96, 2nd edn. Cambridge University Press, Cambridge (2004)
Gessel, I.M.: Applications of the classical umbral calculus. Algebra Universalis 49(4), 397–434 (2003)
Guo, V.J.W., Ishikawa, M., Tagawa, T., Zeng, J.: A quadratic formula for basic hypergeometric series related to Askey–Wilson polynomials. Proc. Am. Math. Soc. 143(5), 2003–2015 (2015)
Hetyei, G.: Alternation acyclic tournaments. Eur. J. Comb. 81, 1–21 (2019)
Ismail, M.: Classical and quantum orthogonal polynomials in one variable. With two chapters by Walter Van Assche. With a foreword by Richard A. Askey. Encyclopedia of Mathematics and its Applications, vol. 98. Cambridge University Press, Cambridge (2005)
Josuat-Vergès, M.: Generalized Dumont–Foata polynomials and alternative tableaux. Sém. Lothar. Comb. 64, 17 (2011)
Kim, J.S., Stanton, D.: Moments of Askey–Wilson polynomials. J. Comb. Theory Ser. A 125, 113–145 (2014)
Knuth, D.E.: Two notes on notation. Am. Math. Monthly 99(5), 403–422 (1992)
Koekoek, R., Lesky, P., Swarttouw, R.: Hypergeometric orthogonal polynomials and their q-analogues. With a foreword by Tom H. Koornwinder, pp. xx+578. Springer Monographs in Mathematics. Springer, Berlin (2010)
Krattenthaler, C.: Advanced determinant calculus. Sém. Lothar. Comb. 42, 67 (1999)
Lazar, A., Wachs, M.L.: On the homogenized Linial arrangement: intersection lattice and Genocchi numbers. Sém. Lothar. Comb. 82B, 12 (2020)
Milne-Thomson, L.M.: The Calculus of Finite Differences. Macmillan and Co. Ltd, London (1951)
Nicole, F.: Méthode pour sommer une infinité de Suites nouvelles, dont on ne peut trouver les Sommes par les Méthodes connuës, Mémoires de l’Academie Royale des Sciences 257–268 (1727)
Randrianarivony, A.: Sur une extension des polynômes de Dumont–Foata. Sém. Lothar. Comb. 32, 12 (1994)
Riordan, J., Stein, P.: Proof of a conjecture on Genocchi numbers. Discret. Math. 5, 381–388 (1973)
Sadjang, P.N., Koepf, W., Foupouagnigni, M.: On moments of classical orthogonal polynomials. J. Math. Anal. Appl. 424(1), 122–151 (2015)
Seidel, L.: Über eine einfache entstehungsweise der Bernoullischen zahlen und einiger verwandten reihen. Sitzungsber. Münch. Akad. 4, 157–187 (1877)
Touchard, J.: Nombres exponentiels et nombres de Bernoulli. Can. J. Math. 8, 305–320 (1956)
van der Poorten, A.J.: A proof that Euler missed ...Apéry’s proof of the irrationality of \(\zeta (3)\), Math. Intelligencer 1(4), 195–203 (1978/79)
Viennot, G.: Une théorie combinatoire des nombres d’Euler et Genocchi, Séminaire de Théorie des nombres de l’Université Bordeaux, Exposé no. 11, 1980–1981, Publications de l’Université Bordeaux I
Viennot, G.: Une théorie combinatoire des polynômes orthogonaux généraux. Université du Québec à Montréal, Lecture Notes (1983)
Wilson, J.: Some hypergeometric orthogonal polynomials. SIAM J. Anal. 11, 690–701 (1980)
Wyman, M., Moser, L.: On some polynomials of Touchard. Can. J. Math. 8, 321–322 (1956)
Zeng, J.: Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials. Proc. Lond. Math. Soc. 3(1), 1–22 (1992)
Zeng, J.: Sur quelques propriétés de symétrie des nombres de Genocchi. Discret. Math. 153(1–3), 319–333 (1996)
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Dedicated to the memory of Richard Askey.
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Gessel, I.M., Zeng, J. Moments of orthogonal polynomials and exponential generating functions. Ramanujan J 61, 675–700 (2023). https://doi.org/10.1007/s11139-022-00548-6
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DOI: https://doi.org/10.1007/s11139-022-00548-6
Keywords
- Moments
- Orthogonal polynomials
- Wilson polynomials
- Askey–Wilson polynomials
- Genocchi numbers
- Genocchi median numbers
- Dumont–Foata polynomials