The (Exponential) Bipartitional Polynomials and Polynomial Sequences of Trinomial Type: Part I (original) (raw)

#A11 Integers 13 (2013) Generalized Binomial Expansions and Bernoulli Polynomials

2014

We investigate generalized binomial expansions that arise from two-dimensional sequences satisfying a broad generalization of the triangular recurrence for binomial coefficients. In particular, we present a new combinatorial formula for such sequences in terms of a ‘shift by rank’ quasi-expansion based on ordered set partitions. As an application, we give a new proof of Dilcher’s formula for expressing generalized Bernoulli polynomials in terms of classical Bernoulli polynomials.

Some identities involving polynomial coefficients

arXiv: Number Theory, 2015

By polynomial (or extended binomial) coefficients, we mean the coefficients in the expansion of integral powers, positive and negative, of the polynomial 1+t+cdots+tm1+t +\cdots +t^{m}1+t+cdots+tm; mgeq1m\geq 1mgeq1 being a fixed integer. We will establish several identities and summation formul\ae\ parallel to those of the usual binomial coefficients.

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

Filomat

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 Kampé de Fériet based Bernoulli polynomials are considered as an member of Legendre-Gould-Hopper based Appell family and certain results for this member are also obtained.

Polynomials of multipartitional type and inverse relations

Discussiones Mathematicae - General Algebra and Applications, 2011

Chou, Hsu and Shiue gave some applications of Faà di Bruno's formula to characterize inverse relations. Our aim is to develop some inverse relations connected to the multipartitional type polynomials involving to binomial type sequences.

Odd and even Lidstone-type polynomial sequences. Part 1: basic topics

Advances in Difference Equations, 2018

Two new general classes of polynomial sequences called respectively odd and even Lidstone-type polynomials are considered. These classes include classic Lidstone polynomials of first and second kind. Some characterizations of the two classes are given, including matrix form, conjugate sequences, generating function, recurrence relations, and determinant forms. Some examples are presented and some applications are sketched.

Extended Forms of Certain Hybrid Special Polynomials Related to Appell Sequences

Bulletin of the Malaysian Mathematical Sciences Society, 2018

The use of integral transforms and operational methods is a fairly useful tool to deal with new families of special polynomials. In this article, the extended Laguerre-Gould-Hopper-Appell polynomials are introduced by means of generating function and determinant definition. Their quasi-monomial properties are also established. Examples of some members belonging to the extended Laguerre-Gould-Hopper-Appell polynomials are considered, and their contour and 3-D plots are drawn.

Generalized Binomial Expansions and Bernoulli Polynomials

Integers, 2014

In this paper we investigate generalized binomial expansions that arise from two-dimensional sequences satisfying a broad generalization of the triangular recurrence for binomial coefficients. In particular, we present a new combinatorial formula for such sequences in terms of a 'shift by rank' quasi-expansion based on ordered set partitions. As an application, we give a new proof of Dilcher's formula for expressing generalized Bernoulli polynomials in terms of classical Bernoulli polynomials.

Odd and Even Lidstone-type polynomial sequences. Part 2: applications

Calcolo, 2020

Two new general classes of polynomial sequences called respectively odd and even Lidstone-type polynomials are considered. These classes include classic Lidstone polynomials of first and second kind. Some characterizations of the two classes are given, including matrix form, conjugate sequences, generating function, recurrence relations, and determinant forms. Some examples are presented and some applications are sketched.

New recurrence relations for several classical families of polynomials

Journal of Difference Equations and Applications, 2021

In this paper we derive new recurrence relations for the following families of polynomials: Nörlund polynomials, generalized Bernoulli polynomials, generalized Euler polynomials, Bernoulli polynomials of the second kind, Buchholz polynomials, generalized Bessel polynomials and generalized Apostol-Euler polynomials. The recurrence relations are derived from a differential equation of first order and a Cauchy integral representation obtained from the generating function of these polynomials.

Identities related to special polynomials and combinatorial numbers

Filomat

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.