Comptes Rendus
Théorie des fonctions
Appell and Sheffer sequences: on their characterizations through functionals and examples
Sergio A. Carrillo1 ; Miguel Hurtado1
1 Programa de matemáticas, Universidad Sergio Arboleda, Calle 74 # 14-14, Bogotá, Colombia.
Comptes Rendus. Mathématique, Volume 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.

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

1 Programa de matemáticas, Universidad Sergio Arboleda, Calle 74 # 14-14, Bogotá, Colombia.
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{CRMATH_2021__359_2_205_0,
     author = {Sergio A. Carrillo and Miguel Hurtado},
     title = {Appell and {Sheffer} sequences: on their characterizations through functionals and examples},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {205--217},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {2},
     year = {2021},
     doi = {10.5802/crmath.172},
     language = {en},
}
TY  - JOUR
AU  - Sergio A. Carrillo
AU  - Miguel Hurtado
TI  - Appell and Sheffer sequences: on their characterizations through functionals and examples
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 205
EP  - 217
VL  - 359
IS  - 2
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.172
LA  - en
ID  - CRMATH_2021__359_2_205_0
ER  - 
%0 Journal Article
%A Sergio A. Carrillo
%A Miguel Hurtado
%T Appell and Sheffer sequences: on their characterizations through functionals and examples
%J Comptes Rendus. Mathématique
%D 2021
%P 205-217
%V 359
%N 2
%I Académie des sciences, Paris
%R 10.5802/crmath.172
%G en
%F CRMATH_2021__359_2_205_0
Sergio A. Carrillo; Miguel Hurtado. Appell and Sheffer sequences: on their characterizations through functionals and examples. Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 205-217. doi : 10.5802/crmath.172. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.172/

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[20] Luis M. Navas; Francisco J. Ruiz; Juan L. Varona Appell polynomials as values of special functions, J. Math. Anal. Appl., Volume 459 (2018) no. 1, pp. 419-436 | DOI | MR | Zbl

[21] Luis M. Navas; Francisco J. Ruiz; Juan L. Varona Existence and reduction of generalized Apostol–Bernoulli, Apostol–Euler and Apostol–Genocchi polynomials, Arch. Math., Volume 55 (2019) no. 1, pp. 157-165 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Characterization of Kummer hypergeometric Bernoulli polynomials and applications

Driss Drissi

C. R. Math (2019)

Polynomials with real zeros via special polynomials

Miloud Mihoubi; Said Taharbouchet

C. R. Math (2021)

Formal groups, Bernoulli-type polynomials and L-series

Piergiulio Tempesta

C. R. Math (2007)