Bell-Based Bernoulli Polynomials with Applications (original) (raw)

A Note on Bell-Based Bernoulli and Euler Polynomials of Complex Variable

Computer Modeling in Engineering & Sciences

In this article, we construct the generating functions for new families of special polynomials including two parametric kinds of Bell-based Bernoulli and Euler polynomials. Some fundamental properties of these functions are given. By using these generating functions and some identities, relations among trigonometric functions and two parametric kinds of Bell-based Bernoulli and Euler polynomials, Stirling numbers are presented. Computational formulae for these polynomials are obtained. Applying a partial derivative operator to these generating functions, some derivative formulae and finite combinatorial sums involving the aforementioned polynomials and numbers are also obtained. In addition, some remarks and observations on these polynomials are given.

Various applications of the (exponential) complete Bell polynomials

2010

In a rather straightforward manner, we develop the well-known formula for the Stirling numbers of the first kind in terms of the (exponential) complete Bell polynomials where the arguments include the generalised harmonic numbers. We also show how the (exponential) complete Bell polynomials feature in a number of other areas of mathematical interest.

Properties and applications of Bell polynomials of two variables

In this article, we introduce Bell polynomials of two variables within the framework of generating functions and explore various properties associated with them. Specifically, we delve into explicit representations, summation formulae, recurrence relations, and addition formulas. Additionally, we present the matrix form and product formula for these polynomials. Finally, we introduce the two-variable Bell-based Stirling polynomials of the second kind and outline their corresponding results. This study contributes to a deeper understanding of the properties and applications of Bell polynomials in mathematical analysis.

The generalized Stirling and Bell numbers revisited

Journal of Integer Sequences, 2012

The generalized Stirling numbers S s;h (n, k) introduced recently by the authors are shown to be a special case of the three parameter family of generalized Stirling numbers S(n, k; α, β, r) considered by Hsu and Shiue. From this relation, several properties of S s;h (n, k) and the associated Bell numbers B s;h (n) and Bell polynomials B s;h|n (x) are derived. The particular case s = 2 and h = −1 corresponding to the meromorphic Weyl algebra is treated explicitly and its connection to Bessel numbers and Bessel polynomials is shown. The dual case s = −1 and h = 1 is connected to Hermite polynomials. For the general case, a close connection to the Touchard polynomials of higher order recently introduced by Dattoli et al. is established, and Touchard polynomials of negative order are introduced and studied. Finally, a q-analogue S s;h (n, k|q) is introduced and first properties are established, e.g., the recursion relation and an explicit expression. It is shown that the q-deformed numbers S s;h (n, k|q) are special cases of the type-II p, qanalogue of generalized Stirling numbers introduced by Remmel and Wachs, providing the analogue to the undeformed case (q = 1). Furthermore, several special cases are discussed explicitly, in particular, the case s = 2 and h = −1 corresponding to the q-meromorphic Weyl algebra considered by Diaz and Pariguan.

On new identities for Bell's polynomials

Discrete Mathematics, 2005

In this work, we propose two new methods for the determination of new identities for Bell's polynomials. The first method is based on the Lagrange inversion formula, and the second is based on the binomial sequences. These methods allow the easy recovery of known identities and deduction of some new identities of these polynomials.

Derivation of Bell Polynomials of the Second Kind

Arxiv preprint arXiv:1104.5065, 2011

New methods for derivation of Bell polynomials of the second kind are presented. The methods are based on an ordinary generating function and its composita. The relation between a composita and a Bell polynomial is demonstrated. Main theorems are written and examples of Bell polynomials for trigonometric functions, polynomials, radicals, and Bernoulli functions are given.

The role of binomial type sequences in determination identities for Bell polynomials

2008

Our paper deals about identities involving Bell polynomials. Some identities on Bell polynomials derived using generating function and successive derivatives of binomial type sequences. We give some relations between Bell polynomials and binomial type sequences in first part, and, we generalize the results obtained in [4] in second part.