scholarly journals Generalizations of the Bernoulli and Appell polynomials

2004 ◽  
Vol 2004 (7) ◽  
pp. 613-623 ◽  
Author(s):  
Gabriella Bretti ◽  
Pierpaolo Natalini ◽  
Paolo E. Ricci
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Bernoulli Polynomials ◽  
Factorization Method ◽  
Bernoulli Numbers ◽  
Appell Polynomials

We first introduce a generalization of the Bernoulli polynomials, and consequently of the Bernoulli numbers, starting from suitable generating functions related to a class of Mittag-Leffler functions. Furthermore, multidimensional extensions of the Bernoulli and Appell polynomials are derived generalizing the relevant generating functions, and using the Hermite-Kampé de Fériet (or Gould-Hopper) polynomials. The main properties of these polynomial sets are shown. In particular, the differential equations can be constructed by means of the factorization method.

scholarly journals A generalization of the Bernoulli polynomials

2003 ◽  
Vol 2003 (3) ◽  
pp. 155-163 ◽  
Author(s):  
Pierpaolo Natalini ◽  
Angela Bernardini
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Bernoulli Polynomials ◽  
Factorization Method ◽  
Bernoulli Numbers

A generalization of the Bernoulli polynomials and, consequently, of the Bernoulli numbers, is defined starting from suitable generating functions. Furthermore, the differential equations of these new classes of polynomials are derived by means of the factorization method introduced by Infeld and Hull (1951).

scholarly journals Some families of differential equations associated with the Hermite-based Appell polynomials and other classes of Hermite-based polynomials

Filomat ◽  
2014 ◽  
Vol 28 (4) ◽  
pp. 695-708 ◽  
Author(s):  
H.M. Srivastava ◽  
M.A. Özarslan ◽  
Banu Yılmaz
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Bernoulli Polynomials ◽  
Factorization Method ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
Partial Differential ◽  
Genocchi Polynomials

Recently, Khan et al. [S. Khan, G. Yasmin, R. Khan and N. A. M. Hassan, Hermite-based Appell polynomials: Properties and Applications, J. Math. Anal. Appl. 351 (2009), 756-764] defined the Hermite-based Appell polynomials by G(x, y, z, t) := A(t)?exp(xt + yt2 + zt3) = ??,n=0 HAn(x, y, z) tn/n! and investigated their many interesting properties and characteristics by using operational techniques combined with the principle of monomiality. Here, in this paper, we find the differential, integro-differential and partial differential equations for the Hermite-based Appell polynomials via the factorization method. Furthermore, we derive the corresponding equations for the Hermite-based Bernoulli polynomials and the Hermite-based Euler polynomials. We also indicate how to deduce the corresponding results for the Hermite-based Genocchi polynomials from those involving the Hermite-based Euler polynomials.

scholarly journals Certain results of hybrid families of special polynomials associated with appell sequences

Filomat ◽  
2019 ◽  
Vol 33 (12) ◽  
pp. 3833-3844 ◽  
Author(s):  
Ghazala Yasmin ◽  
Abdulghani Muhyi
Keyword(s):  
Generating Functions ◽  
Mixed Type ◽  
Bernoulli Polynomials ◽  
Operational Method ◽  
Appell Polynomials ◽  
Special Polynomials ◽  
Appell Sequences

In this article, the Legendre-Gould-Hopper polynomials are combined with Appell sequences to introduce certain mixed type special polynomials by using operational method. The generating functions, determinant definitions and certain other properties of Legendre-Gould-Hopper based Appell polynomials are derived. Operational rules providing connections between these formulae and known special polynomials are established. The 2-variable Hermite Kamp? de F?riet based Bernoulli polynomials are considered as an member of Legendre-Gould-Hopper based Appell family and certain results for this member are also obtained.

scholarly journals Voros coefficients for the hypergeometric differential equations and Eynard–Orantin’s topological recursion: Part II: For confluent family of hypergeometric equations

2019 ◽  
Vol 4 (1) ◽  
Author(s):  
Kohei Iwaki ◽  
Tatsuya Koike ◽  
Yumiko Takei
Keyword(s):  
Free Energy ◽  
Differential Equations ◽  
Classical Limit ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Spectral Curves ◽  
Topological Recursion ◽  
Hypergeometric Equations ◽  
Explicit Expressions

Abstract We show that each member of the confluent family of the Gauss hypergeometric equations is realized as quantum curves for appropriate spectral curves. As an application, relations between the Voros coefficients of those equations and the free energy of their classical limit computed by the topological recursion are established. We will also find explicit expressions of the free energy and the Voros coefficients in terms of the Bernoulli numbers and Bernoulli polynomials. Communicated by: Youjin Zhang

scholarly journals Properties of Partially Degenerate Complex Appell Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1508
Author(s):  
Dojin Kim ◽  
Sangil Kim
Keyword(s):  
Differential Equations ◽  
Exponential Type ◽  
Generating Functions ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
Polynomial Sequences ◽  
General Properties ◽  
Symmetric Identities ◽  
Degenerate Version

Degenerate versions of polynomial sequences have been recently studied to obtain useful properties such as symmetric identities by introducing degenerate exponential-type generating functions. As part of our continued work in degenerate versions of generating functions, we subsequently present our study on degenerate complex Appell polynomials by considering a partially degenerate version of the generating functions of ordinary complex Appell polynomials in this paper. We only consider partially degenerate generating functions to retain the crucial properties of the Appell sequence, and we present useful identities and general properties by splitting complex values into their real and imaginary parts; moreover, we provide several explicit examples. Additionally, the differential equations satisfied by degenerate complex Bernoulli and Euler polynomials are derived by the quasi-monomiality principle using Appell-type polynomials.

scholarly journals Computation of k-ary Lyndon words using generating functions and their differential equations

Filomat ◽  
2018 ◽  
Vol 32 (10) ◽  
pp. 3455-3463
Author(s):  
Irem Kucukoglu ◽  
Yilmaz Simsek
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Prime Numbers ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Combinatorial Sums ◽  
Higher Order Derivative ◽  
Lyndon Words ◽  
Higher Order Differential Equations

By using generating functions technique, we investigate some properties of the k-ary Lyndon words. We give an explicit formula for the generating functions including not only combinatorial sums, but also hypergeometric function. We also derive higher-order differential equations and some formulas related to the k-ary Lyndon words. By applying these equations and formulas, we also derive some novel identities including the Stirling numbers of the second kind, the Apostol-Bernoulli numbers and combinatorial sums. Moreover, in order to compute numerical values of the higher-order derivative for the generating functions enumerating k-ary Lyndon words with prime number length, we construct an efficient algorithm. By applying this algorithm, we give some numerical values for these derivative equations for selected different prime numbers.

scholarly journals The actions on the generating functions for the family of the generalized Bernoulli polynomials

Filomat ◽  
2017 ◽  
Vol 31 (1) ◽  
pp. 35-44
Author(s):  
Mustafa Alkan ◽  
Yilmaz Simsek
Keyword(s):  
Cyclic Group ◽  
Generating Functions ◽  
Periodic Group ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Group Homomorphism ◽  
Complex Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
The Family ◽  
Generalized Bernoulli Polynomials

In this paper, we study the generalization Bernoulli numbers and polynomials attached to a periodic group homomorphism from a finite cyclic group to the set of complex numbers and derive new periodic group homomorphism by using a fixed periodic group homomorphism. Hence, we obtain not only multiplication formulas, but also some new identities for the generalized Bernoulli polynomials attached to a periodic group homomorphism.

On some classes of differential equations and associated integral equations for the Laguerre–Appell polynomials

2018 ◽  
Vol 9 (3) ◽  
pp. 185-194 ◽  
Author(s):  
Subuhi Khan ◽  
Mumtaz Riyasat ◽  
Shahid Ahmad Wani
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Integral Equations ◽  
Recurrence Relations ◽  
Laguerre Polynomials ◽  
Factorization Method ◽  
Volterra Integral Equations ◽  
Appell Polynomials ◽  
Partial Differential ◽  
Genocchi Polynomials

Abstract The article aims to explore some new classes of differential equations and associated integral equations for some hybrid families of Laguerre polynomials. The recurrence relations and differential, integro-differential and partial differential equations for the hybrid Laguerre–Appell polynomials are derived via the factorization method. An analogous study of these results for the hybrid Laguerre–Bernoulli, Euler and Genocchi polynomials is presented. Further, the Volterra integral equations for the hybrid Laguerre–Appell polynomials and for their corresponding members are also explored.

scholarly journals Multiple-Poly-Bernoulli Polynomials of the Second Kind Associated with Hermite Polynomials

2017 ◽  
Vol 58 (1) ◽  
pp. 97-112 ◽  
Author(s):  
Waseem A. Khan ◽  
M. Ghayasuddin ◽  
M. Shadab
Keyword(s):  
Generating Functions ◽  
Hermite Polynomials ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
General Symmetry ◽  
Bernoulli Numbers And Polynomials ◽  
New Class ◽  
Summation Formulae

Abstract In this paper, we introduce a new class of Hermite multiple-poly-Bernoulli numbers and polynomials of the second kind and investigate some properties for these polynomials. We derive some implicit summation formulae and general symmetry identities by using different analytical means and applying generating functions. The results derived here are a generalization of some known summation formulae earlier studied by Pathan and Khan.

Identities related to special polynomials and combinatorial numbers

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4833-4844 ◽  
Author(s):  
Eda Yuluklu ◽  
Yilmaz Simsek ◽  
Takao Komatsu
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Euler Numbers ◽  
Special Polynomials ◽  
Central Factorial Numbers

The aim of this paper is to give some new identities and relations related to the some families of special numbers such as the Bernoulli numbers, the Euler numbers, the Stirling numbers of the first and second kinds, the central factorial numbers and also the numbers y1(n,k,?) and y2(n,k,?) which are given Simsek [31]. Our method is related to the functional equations of the generating functions and the fermionic and bosonic p-adic Volkenborn integral on Zp. Finally, we give remarks and comments on our results.

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