Comptes Rendus
Number theory/Mathematical analysis
Characterization of Kummer hypergeometric Bernoulli polynomials and applications
Comptes Rendus. Mathématique, Volume 357 (2019) no. 10, pp. 743-751.

In this paper, we present two characterizations of the sequences of Kummer hypergeometric polynomials Ba,b,n(x) and Kummer hypergeometric polynomials of the second kind Ka,b,n(x), which are respectively defined by the exponential generating functions:

extM(a,a+b;t)=n=0Ba,b,n(x)tnn! and extU(a,a+b;t)=n=0Ka,b,n(x)tnn! with M(a,b;t)=n=0(a)n(b)ntnn!,
where U(a,a+b;t) is the Kummer hypergeometric function of the second kind.

First we construct Gauss–Weierstrass-type convolution operators Twa,b with a well-chosen kernel (density) function for each sequence of Kummer hypergeometric polynomials and for Kummer hypergeometric polynomials of the second kind. Then we characterize Kummer hypergeometric polynomials as the only Appell polynomials having a weighted-integral mean equal to zero. Our approach is inspired by the Gauss–Weierstrass convolution transform for Hermite polynomials and the Kummer integral representation for confluent hypergeometric functions.

Dans cet article, nous présentons deux caractérisations des suites Ba,b,n(x) et Ka,b,n(x) de polynômes hypergéométriques de type Kummer définies par leurs fonctions génératrices :

extM(a,a+b;t)=n=0Ba,b,n(x)tnn! et extU(a,a+b;t)=n=0Ka,b,n(x)tnn! avec M(a,b;t)=n=0(a)n(b)ntnn!,
U(a,a+b;t) est la fonction hypergéométrique de Kummer de seconde espèce.

Premièrement, nous construisons des opérateurs de convolution Twa,b du type Gauss–Weierstrass pour chacune des suites de polynômes de Kummer de première et de seconde espèces. Deuxièmement, nous caractérisons les polynômes hypergéométriques de Kummer Ba,b,n(x) comme étant les seuls polynômes ayant une moyenne intégrale pondérée égale à zero. Cette approche nous a été inspirée par la transformation de Gauss–Weierstrass pour les polynômes de Hermite et par la représentation intégrale de type Euler–Kummer pour les fonctions hypergéométriques.

Published online:
DOI: 10.1016/j.crma.2019.10.004

Driss Drissi 1

1 Department of Mathematics, Rowan University, Glassboro, NJ 08028, USA
     author = {Driss Drissi},
     title = {Characterization of {Kummer} hypergeometric {Bernoulli} polynomials and applications},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {743--751},
     publisher = {Elsevier},
     volume = {357},
     number = {10},
     year = {2019},
     doi = {10.1016/j.crma.2019.10.004},
     language = {en},
AU  - Driss Drissi
TI  - Characterization of Kummer hypergeometric Bernoulli polynomials and applications
JO  - Comptes Rendus. Mathématique
PY  - 2019
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EP  - 751
VL  - 357
IS  - 10
PB  - Elsevier
DO  - 10.1016/j.crma.2019.10.004
LA  - en
ID  - CRMATH_2019__357_10_743_0
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%T Characterization of Kummer hypergeometric Bernoulli polynomials and applications
%J Comptes Rendus. Mathématique
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%R 10.1016/j.crma.2019.10.004
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Driss Drissi. Characterization of Kummer hypergeometric Bernoulli polynomials and applications. Comptes Rendus. Mathématique, Volume 357 (2019) no. 10, pp. 743-751. doi : 10.1016/j.crma.2019.10.004.

[1] M. Abramovitz; I. Stegun Handbook of Mathematical Functions with Formulae, Dover, New York, 1972

[2] P.E. Appell Sur une classe de polynômes, Ann. Éc. Norm. Supér. (2), Volume 9 (1882), pp. 119-144

[3] P.E. Appell Sur l'équation δ2zdx2δzδy=0 et la théorie de la chaleur, J. Math. Pures Appl. (4), Volume 8 (1892), pp. 187-216

[4] L.F.A. Arbogast Du Calcul des Dérivations, LeVrault Frères, Strasbourg, 1800

[5] K.L. Bell; N.S. Scott Coulomb functions (Negative Energies), Comput. Phys. Commun., Volume 20 (1980), pp. 447-458

[6] P. Boyle; A. Potapchik Application of high-precision computing for pricing arithmetic Asian options, Genoa, Italy, 9–12 July 2006, ACM, New York (2006), pp. 39-46

[7] L. Carlitz The Staudt–Clausen theorem, Math. Mag., Volume 34 (1960–1961), pp. 131-146

[8] L. Comtet Advanced Combinatorics, Reidel Publishing Co, Boston, MA, USA, 1974

[9] F.A. Costabile; E. Longo A determinantal approach to Appell polynomials, J. Comput. Appl. Math., Volume 234 (2010), pp. 1528-1542

[10] R. Dere; Y. Simsek; H.M. Srivastava A unified presentation of three families of generalized Apostol-type polynomials based upon the theory of umbrel calculus and the umbrel algebra, J. Number Theory, Volume 133 (2013), pp. 3245-3263

[11] K. Dilcher; L. Malloch Arithmetic Properties of Bernoulli-Padé Numbers and Polynomials, J. Number Theory, Volume 92 (2002), pp. 330-347

[12] K. Dilcher Bernoulli numbers and confluent hypergeometric functions, Urbana-Champaign, IL, USA, 2000 (B. Berndt; M.A. Bennett; N. Boston; H.G. Diamond; A.J. Hildebrand; W. Philipp, eds.), A.K. Peters, Natick, MA, USA (2002), pp. 343-363

[13] G.N. Georgiev; M.N. Georgieva-Grosse A new property of complex Kummer function and its application to waveguide propagation, IEEE Antennas Wirel. Propag. Lett., Volume 2 (2003), pp. 306-309

[14] G.N. Georgiev; M.N. Georgieva-Grosse The Kummer confluent hypergeometric function and some of its applications in the theory of azimuthally magnetized circular ferrite waveguides, J. Telecommun. Inf. Technol., Volume 3 (2005), pp. 112-128

[15] A. Hassen; H. Nguyen Hypergeometric Zeta Functions, Int. J. Number Theory, Volume 6 (2010) no. 1, pp. 99-126

[16] E. Hille Notes on linear transformations II Analyticity of semi-groups, Ann. of Math. (2), Volume 40 (1939), pp. 1-47

[17] F.T. Howard Some sequences of rational numbers related to the exponential function, Duke Math. J., Volume 34 (1967), pp. 701-716

[18] F.T. Howard Numbers Generated by the Reciprocal of ex1x, Math. Comput., Volume 31 (1977) no. 138, pp. 581-598

[19] K. Knopp Infinite Sequences and Series, Dover, New York, USA, 1956

[20] H. Lehmer A new approach to Bernoulli polynomials, Amer. Math. Mon., Volume 95 (1988), pp. 905-911

[21] J. Riordan Derivatives of composite functions, Bull. Amer. Math. Soc., Volume 52 (1946), pp. 664-667

[22] K. Weierstrass, Sitzungsberichte der Königlich-Preussischen Akademie der Wissenschaften zu Berlin (1885), pp. 633-639 (789–805)

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