Higher-order Bernoulli Polynomials Research Articles

Derivation of computational formulas for certain class of finite sums: Approach to generating functions arising from p$$ p $$‐adic integrals and special functions

The aim of this paper is to construct generating functions for certain families of special finite sums by using the Newton–Mercator series, hypergeometric functions, and ‐adic integral. By using these generating functions with their functional and partial derivative equations, many novel computational formulas involving the special finite sums of (inverse) binomial coefficients, the Bernoulli type polynomials and numbers, Euler polynomials and numbers, the Stirling numbers, the (alternating) harmonic numbers, the Leibnitz polynomials, and others are derived. We also develop a computation algorithm for these finite sums and provide some of their special values. By using these finite sums and combinatorial numbers, we find some formulas involving multiple alternating zeta functions, the Bernoulli polynomials of higher order and the Euler polynomials of higher order. We then obtain a decomposition from these formulas, which are related to the multiple Hurwitz zeta functions.

Two-Variable Type 2 Poly-Fubini Polynomials

In the present work, a new extension of the two-variable Fubini polynomials is introduced by means of the polyexponential function, which is called the two-variable type 2 poly-Fubini polynomials. Then, some useful relations including the Stirling numbers of the second and the first kinds, the usual Fubini polynomials, and the higher-order Bernoulli polynomials are derived. Also, some summation formulas and an integral representation for type 2 poly-Fubini polynomials are investigated. Moreover, two-variable unipoly-Fubini polynomials are introduced utilizing the unipoly function, and diverse properties involving integral and derivative properties are attained. Furthermore, some relationships covering the two-variable unipoly-Fubini polynomials, the Stirling numbers of the second and the first kinds, and the Daehee polynomials are acquired.

Some identities of Lah\u2013Bell polynomials

Recently, the nth Lah–Bell number was defined as the number of ways a set of n elements can be partitioned into nonempty linearly ordered subsets for any nonnegative integer n. Further, as natural extensions of the Lah–Bell numbers, Lah–Bell polynomials are defined. We study Lah–Bell polynomials with and without the help of umbral calculus. Notably, we use three different formulas in order to express various known families of polynomials such as higher-order Bernoulli polynomials and poly-Bernoulli polynomials in terms of the Lah–Bell polynomials. In addition, we obtain several properties of Lah–Bell polynomials.

Some new identities and inequalities for Bernoulli polynomials and numbers of higher order related to the Stirling and Catalan numbers

The aim of this paper is to give some new relations, identities, and inequalities for the Bernoulli polynomials and numbers of higher order, the Stirling numbers of the second kind, the Eulerian numbers, and the Catalan numbers. By applying the Laplace transformation to the generating function of the Bernoulli polynomials of higher order, a novel formula for these polynomials is obtained. Integral and series representations for these polynomials and numbers are given. Moreover, the upper bound and the lower bound for the Bernoulli numbers of negative order are given. Some inequalities including the Bernoulli numbers of negative order and the Stirling numbers of the second kind are also given. Finally, appropriate ligaments of the definitions and results introduced here with those in earlier as well as oncoming investigations will be designated.

Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials

In this paper, we investigate some identities on Bernoulli numbers and polynomials and those on degenerate Bernoulli numbers and polynomials arising from certain p-adic invariant integrals on Z p . In particular, we derive various expressions for the polynomials associated with integer power sums, called integer power sum polynomials and also for their degenerate versions. Further, we compute the expectations of an infinite family of random variables which involve the degenerate Stirling polynomials of the second and some value of higher-order Bernoulli polynomials.

Identities for Bernoulli polynomials related to multiple Tornheim zeta functions

We show that each member of a doubly infinite sequence of highly nonlinear expressions of Bernoulli polynomials, which can be seen as linear combinations of certain higher-order convolutions, is a multiple of a specific product of linear factors. The special case of Bernoulli numbers has important applications in the study of multiple Tornheim zeta functions. The proof of the main result relies on properties of Eulerian polynomials and higher-order Bernoulli polynomials.

Some Symmetric Identities Involving the Stirling Polynomials Under the Finite Symmetric Group

In the paper, the authors present some symmetric identities involving the Stirling polynomials and higher order Bernoulli polynomials under all permutations in the finite symmetric group of degree n. These identities extend and generalize some known results.

On the integral of products of higher-order Bernoulli and Euler polynomials

In this paper, we derive a formula on the integral of products of the higher-order Euler polynomials. By the same way, similar relations are obtained for $l$ higher-order Bernoulli polynomials and $r$ higher-order Euler polynomials. Moreover, we establish the connection between the results and the generalized Dedekind sums and Hardy--Berndt sums. Finally, the Laplace transform of Euler polynomials is given.

Fourier series of functions associated with higher-order Bernoulli polynomials

Fourier series of functions associated with higher-order Bernoulli polynomials

An explicit form of the polynomial part of a restricted partition function

We prove an explicit formula for the polynomial part of a restricted partition function, also known as the first Sylvester wave. This is achieved by way of some identities for higher-order Bernoulli polynomials, one of which is analogous to Raabe’s well-known multiplication formula for the ordinary Bernoulli polynomials. As a consequence of our main result we obtain an asymptotic expression of the first Sylvester wave as the coefficients of the restricted partition grow arbitrarily large.

A note on higher-order Bernoulli polynomials

Let be the -dimensional vector space over Q. From the property of the basis for the space , we derive some interesting identities of higher-order Bernoulli polynomials.

Construction a new generating function of Bernstein type polynomials

Main purpose of this paper is to reconstruct generating function of the Bernstein type polynomials. Some properties this generating functions are given. By applying this generating function, not only derivative of these polynomials but also recurrence relations of these polynomials are found. Interpolation function of these polynomials is also constructed via Mellin Transformation. This function interpolates these polynomials at negative integers which are given explicitly. Moreover, relations between these polynomials, the generalized Stirling numbers, and Bernoulli polynomials of higher order are given. Furthermore some applications associated with B\'ezier curve are given.

A New Generating Function of (q-) Bernstein-Type Polynomials and Their Interpolation Function

The main object of this paper is to construct a new generating function of the (q-) Bernstein-type polynomials. We establish elementary properties of this function. By using this generating function, we derive recurrence relation and derivative of the (q-) Bernstein-type polynomials. We also give relations between the (q-) Bernstein-type polynomials, Hermite polynomials, Bernoulli polynomials of higher order, and the second-kind Stirling numbers. By applying Mellin transformation to this generating function, we define interpolation of the (q-) Bernstein-type polynomials. Moreover, we give some applications and questions on approximations of (q-) Bernstein-type polynomials, moments of some distributions in Statistics.

Summation formulae for finite cotangent sums

Recently, a half-dozen remarkably general families of the finite trigonometric sums were summed in closed-form by choosing a particularly convenient integration contour and making use of the calculus of residues. In this sequel, we show that this procedure can be further extended and we find the summation formulae, in terms of the higher order Bernoulli polynomials and the ordinary Bernoulli polynomials, for four general families of the finite cotangent sums.

Symmetry Properties of Higher-Order Bernoulli Polynomials

We investigate properties of identities and some interesting identities of symmetry for the Bernoulli polynomials of higher order using the multivariate -adic invariant integral on .

Expression for restricted partition function through Bernoulli polynomials

Explicit expressions for restricted partition function W(s,dm) and its quasiperiodic components Wj(s,dm) (called Sylvester waves) for a set of positive integers dm={d1,d2,…,dm} are derived. The formulas are represented in a form of a finite sum over Bernoulli polynomials of higher order with periodic coefficients.

Summation of a family of finite secant sums

We use contour integrals and the Cauchy residue theorem in order to derive several summation formulas, in terms of the higher-order Bernoulli polynomials and the ordinary Bernoulli and Euler polynomials, for a remarkably general family of secant sums. Numerous (known or new) special cases are shown to follow readily from the summation formulas presented in this paper.

Some relationships between the generalized Apostol–Bernoulli polynomials and Hurwitz–Lerch Zeta functions

The main object of this paper is to further investigate the generalized Apostol–Bernoulli polynomials of higher order, which were introduced and studied recently by Luo and Srivastava [2005, Journal of Mathematical Analysis and Applications, 308, 290–302; 2006, Computers and Mathematics with Applications, 51, 631–642]. Here, we first derive an explicit representation of these generalized Apostol–Bernoulli polynomials of higher order in terms of a generalization of the Hurwitz–Lerch Zeta function and then proceed to establish a functional relationship between the generalized Apostol–Bernoulli polynomials of rational arguments and the Hurwitz (or generalized) Zeta function. Our results would provide extensions of those given earlier by (for example) Apostol [1951, Pacific Journal of Mathematics, 1, 161–167] and Srivastava [2000, Mathematical Proceedings of the Cambridge Philosophical Society, 129, 77–84].

Some polynomials associated with Williams' limit formula for $\zeta (2n)$

An interesting limit formula for the Riemann Zeta function . Each of our closed-form summation formulas, expressing a finite trigonometric sum in terms of higher-order Bernoulli polynomials, is capable of yielding many (new or known) special cases and consequences.

Casimir effect around a cone.

The vacuum average of the energy density of a free, massless scalar field around a conical flux-tube singularity in d+1 space-time dimensions is calculated. A complex contour method is employed and the results involve higher-order Bernoulli polynomials. The formulas might have applications to cosmic strings.