Théorie des fonctions
Appell and Sheffer sequences: on their characterizations through functionals and examples
Comptes Rendus. Mathématique, Tome 359 (2021) no. 2, pp. 205-217.

L’objectif de cet article est de présenter une nouvelle récurrence simple pour les suites d’Appell et de Sheffer en termes de la fonctionnelle linéaire qui les définit, et d’expliquer comment cela équivaut à plusieurs caractérisations bien connues qui apparaissent dans la littérature. Nous donnons aussi plusieurs exemples, y compris des représentations intégrales des opérateurs inverses associés aux polynômes de Bernoulli et d’Euler, et une nouvelle représentation intégrale des polynômes d’Hermite d-orthogonaux remis à l’échelle, qui généralise l’opérateur de Weierstrass associé aux polynômes d’Hermite.

The aim of this paper is to present a new simple recurrence for Appell and Sheffer sequences in terms of the linear functional that defines them, and to explain how this is equivalent to several well-known characterizations appearing in the literature. We also give several examples, including integral representations of the inverse operators associated to Bernoulli and Euler polynomials, and a new integral representation of the re-scaled Hermite d-orthogonal polynomials generalizing the Weierstrass operator related to the Hermite polynomials.

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DOI : 10.5802/crmath.172
Classification : 05A40, 11B83, 11B68
Mots clés : Sheffer and Appell sequences, Bernoulli, Euler and Hermite $d$-orthogonal polynomials
Carrillo, Sergio A. 1 ; Hurtado, Miguel 1

1 Programa de matemáticas, Universidad Sergio Arboleda, Calle 74 # 14-14, Bogotá, Colombia.
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Carrillo, Sergio A.; Hurtado, Miguel. Appell and Sheffer sequences: on their characterizations through functionals and examples. Comptes Rendus. Mathématique, Tome 359 (2021) no. 2, pp. 205-217. doi : 10.5802/crmath.172. http://www.numdam.org/articles/10.5802/crmath.172/

[1] Aceto, Lidia; Cação, Isabel A matrix approach to Sheffer polynomials, J. Math. Anal. Appl., Volume 446 (2017) no. 1, pp. 87-100 | DOI | MR | Zbl

[2] Aceto, Lidia; Malonek, Helmuth R.; Tomaz, Graça A unified matrix approach to the representation of Appell polynomials, Integral Transforms Spec. Funct., Volume 26 (2015) no. 6, pp. 426-441 | DOI | MR | Zbl

[3] Adell, José A.; Lekuona, Alberto Binomial convolution and transformations of Appell polynomials, J. Math. Anal. Appl., Volume 456 (2017) no. 1, pp. 16-33 | DOI | MR | Zbl

[4] Adell, José A.; Lekuona, Alberto Closed form expressions for Appell polynomials, Ramanujan J., Volume 49 (2019), pp. 567-583 | DOI | MR | Zbl

[5] Apostol, Tom M. On the Lerch zeta function, Pac. J. Math., Volume 1 (1951) no. 2, pp. 161-167 | DOI | MR | Zbl

[6] Appell, Paul E. Sur une classe de polynômes, Ann. Sci. Éc. Norm. Supér., Volume 9 (1880), pp. 119-144 | DOI | Numdam | Zbl

[7] Balser, Werner Formal power series and linear systems of meromorphic ordinary differential equations, Universitext, Springer, 2000 | Zbl

[8] Boas, Ralph P. Stieljes moment problem for functions of bounded variation, Bull. Am. Math. Soc., Volume 45 (1939), pp. 399-404 | DOI

[9] Borwein, Jonathan M.; Calkin, Neil J.; Manna, Dante Euler–Boole summation revisited, Am. Math. Mon., Volume 116 (2009) no. 5, pp. 387-412 | DOI | MR | Zbl

[10] Bourbaki, Nicolas Éléments de mathématique. Fonctions d’une variable réelle. Théorie élémentaire, Springer, 2007 | DOI | Zbl

[11] Candelpergher, Bernard Ramanujan summation of divergent series, Lecture Notes in Mathematics, 2185, Springer, 2017 | MR | Zbl

[12] Comtet, Louis Advanced combinatorics. The art of finite and infinite expansions, D. Reidel Publishing Co., 1974 | Zbl

[13] Costabile, Francesco A.; Longo, Elisabetta J. An algebraic approach to Sheffer polynomial sequences, Integral Transforms Spec. Funct., Volume 25 (2010) no. 4, pp. 295-311 | DOI | MR | Zbl

[14] Costabile, Francesco A.; Longo, Elisabetta J. A determinantal approach to Appell polynomials, J. Comput. Appl. Math., Volume 236 (2010), pp. 1528-1542 | DOI | MR | Zbl

[15] Douak, Khalfa The relation of the d-orthogonal polynomials to the Appell polynomials, J. Comput. Appl. Math., Volume 70 (1996), pp. 279-295 | DOI | MR | Zbl

[16] Drissi, Driss Characterization of Kummer hypergeometric Bernoulli polynomials and applications, C. R. Math. Acad. Sci. Paris, Volume 357 (2019), pp. 743-751 | DOI | MR | Zbl

[17] Gould, Henry W.; Hopper, A. T. Operational formulas connected with two generalizations of Hermite polynomials, Duke Math. J., Volume 29 (1962) no. 1, pp. 51-63 | DOI | MR | Zbl

[18] Hassen, Abdul; Nguyen, Hieu Hypergeometric Bernoulli polynomials and Appell sequences, Int. J. Number Theory, Volume 5 (2008) no. 4, pp. 767-774 | DOI | MR | Zbl

[19] Magnus, Wilhelm; Oberhettinger, Fritz; Soni, Raj P. Formulas and theorems for special functions of mathematical physics, Springer, 1966 | DOI

[22] Nørlund, Niels E. Mémoire sur les polynômes de Bernoulli, Acta Math., Volume 43 (1922), pp. 121-196 | DOI

[23] Nørlund, Niels E. Vorlesungen über differenzen-rechnung, Grundlehren der Mathematischen Wissenschaften, 13, Springer, 1924 | DOI

[24] Olver, Frank W. Asymptotics and special functions, A K Peters, 1997 | DOI | MR | Zbl

[25] NIST Handbook of Mathematical Functions (Olver, Frank W.; Lozier, Daniel W.; Boisvert, Ronald F.; Clark, Charles W., eds.), US Department of Commerce, 2010 http://dlmf.nist.gov | Zbl

[26] Roman, Steven Umbral calculus, Pure and Applied Mathematics, 111, Academic Press Inc., 1984 | MR | Zbl

[27] Roman, Steven; Rota, Gian-Carlo Umbral calculus, Adv. Math., Volume 27 (1978) no. 1, pp. 95-128 | DOI | MR | Zbl

[28] Rota, Gian-Carlo Finite operator calculus, Academic Press Inc., 1975 | Zbl

[29] Sheffer, Isador M. Some properties of polynomial sets of type zero, Duke Math. J., Volume 5 (1939) no. 3, pp. 590-622 | DOI | MR

[30] Sheffer, Isador M. Note on Appell polynomials, Bull. Am. Math. Soc., Volume 51 (1945) no. 10, pp. 739-744 | DOI | MR | Zbl

[31] Shohat, James The relation of the classical orthogonal polynomials to the polynomials of Appell, Am. J. Math., Volume 58 (1936) no. 3, pp. 453-464 | DOI | MR | Zbl

[32] Sun, Ping Moment representation of Bernoulli polynomial, Euler polynomial and Gegenbauer polynomials, Stat. Probab. Lett., Volume 77 (2007) no. 7, pp. 748-751 | DOI | MR | Zbl

[33] Ta, Bao Quoc Probabilistic approach to Appell polynomials, Expo. Math., Volume 33 (2015) no. 3, pp. 269-294 | DOI | MR | Zbl

[34] Tempesta, Piergiulio On Appell sequences of polynomials of Bernoulli and Euler type, J. Math. Anal. Appl., Volume 341 (2008) no. 2, pp. 1295-1310 | DOI | MR | Zbl

[35] Thorne, C. J. A property of Appell sets, Am. Math. Mon., Volume 52 (1945) no. 4, pp. 191-193 | DOI | MR | Zbl

[36] Touchard, Jacques Sur les cycles des substitutions, Acta Math., Volume 70 (1939) no. 4, pp. 243-297 | DOI | MR | Zbl

[37] Wang, Weiping A determinantal approach to Sheffer sequences, Linear Algebra Appl., Volume 463 (2014), pp. 228-254 | DOI | MR | Zbl

[38] Yang, Yongzhi Determinant representations of Appell polynomial sequences, Oper. Matrices, Volume 2 (2008) no. 4, pp. 517-524 | DOI | MR | Zbl

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