Degenerate Bernoulli polynomials, generalized factorial sums, and their applications

We prove a general symmetric identity involving the degenerate Bernoulli polynomials and sums of generalized falling factorials, which unifies several known identities for Bernoulli and degenerate Bernoulli numbers and polynomials. We use this identity to describe some combinatorial relations between these polynomials and generalized factorial sums. As further applications we derive several identities, recurrences, and congruences involving the Bernoulli numbers, degenerate Bernoulli numbers, generalized factorial sums, Stirling numbers of the first kind, Bernoulli numbers of higher order, and Bernoulli numbers of the second kind.     

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.     

Peters type polynomials and numbers and their generating functions: Approach with p‐adic integral method

The aim of this paper is to introduce and investigate some of the primary generalizations and unifications of the Peters polynomials and numbers by means of convenient generating functions and p‐adic integrals method. Various fundamental properties of these polynomials and numbers involving some explicit series and integral representations in terms of the generalized Stirling numbers, generalized harmonic sums, and some well‐known special numbers and polynomials are presented. By using p‐adic integrals, we construct generating functions for Peters type polynomials and numbers (Apostol‐type Peters numbers and polynomials). By using these functions with their partial derivative eqautions and functional equations, we derive many properties, relations, explicit formulas, and identities including the Apostol‐Bernoulli polynomials, the Apostol‐Euler polynomials, the Boole polynomials, the Bernoulli polynomials, and numbers of the second kind, generalized harmonic sums. A brief revealing and historical information for the Peters type polynomials are given. Some of the formulas given in this article are given critiques and comments between previously well‐known formulas. Finally, two open problems for interpolation functions for Apostol‐type Peters numbers and polynomials are revealed.     

Notes on generalization of the Bernoulli type polynomials

Recently, Srivastava et al. introduced a new generalization of the Bernoulli, Euler and Genocchi polynomials (see [H.M. Srivastava, M. Garg, S. Choudhary, Russian J. Math. Phys. 17 (2010) 251-261] and [H.M. Srivastava, M. Garg, S. Choudhary, Taiwanese J. Math. 15 (2011) 283-305]). They established several interesting properties of these general polynomials, the generalized Hurwitz-Lerch zeta functions and also in series involving the familiar Gaussian hypergeometric function. By the same motivation of Srivastava’s et al. [11] and [12], we introduce and derive multiplication formula and some identities related to the generalized Bernoulli type polynomials of higher order associated with positive real parameters a, b and c. We also establish multiple alternating sums in terms of these polynomials. Moreover, by differentiating the generating function of these polynomials, we give a interpolation function of these polynomials.     

On the generalization of the Euler polynomials with the real parameters

In this study, we give multiplication formula for generalized Euler polynomials of order α and obtain some explicit recursive formulas. The multiple alternating sums with positive real parameters a and b are evaluated in terms of both generalized Euler and generalized Bernoulli polynomials of order α. Finally we obtained some interesting special cases.     

Analysis of the Bernstein basis functions: an approach to combinatorial sums involving binomial coefficients and Catalan numbers

We give some alternative forms of the generating functions for the Bernstein basis functions. Using these forms,we derive a collection of functional equations for the generating functions. By applying these equations, we prove some identities for the Bernstein basis functions. Integrating these identities, we derive a variety of identities and formulas, some old and some new, for combinatorial sums involving binomial coefficients, Pascal's rule, Vandermonde's type of convolution, the Bernoulli polynomials, and the Catalan numbers. Copyright © 2014 John Wiley & Sons, Ltd.     

On the Character Sum of Polynomials and the Two-term Exponential Sums

The main purpose of this paper is using the analytic method and the properties of the character sums to study the computational problem of one kind hybrid power mean involving the character sums of polynomials and the two-term exponential sums,and give several interesting identities and asymptotic formulae for them.     

Higher-order convolutions for Bernoulli and Euler polynomials

We prove convolution identities of arbitrary orders for Bernoulli and Euler polynomials, i.e., sums of products of a fixed but arbitrary number of these polynomials. They differ from the more usual convolutions found in the literature by not having multinomial coefficients as factors. This generalizes a special type of convolution identity for Bernoulli numbers which was first discovered by Yu. Matiyasevich.     

高阶Bernoulli多项式和高阶Euler多项式的关系

利用发生函数的方法，讨论了高阶Bernoulli数和高阶Euler数，高阶Bernoulli多项式和高阶Euler多项式之间的关系，得到了经典Bernoulli数和Euler数，经典Bernoulli多项式和Euler多项式之间的新型关系。     

Positivity of certain sums over Jacobi kernel polynomials

We present a computer-assisted proof of positivity of sums over kernel polynomials for ultraspherical Jacobi polynomials.     

An explicit formula for generalized potential polynomials and its applications

We define the generalized potential polynomials associated to an independent variable, and prove an explicit formula involving the generalized potential polynomials and the exponential Bell polynomials. We use this formula to describe closed type formulas for the higher order Bernoulli, Eulerian, Euler, Genocchi, Apostol-Bernoulli, Apostol-Euler polynomials and the polynomials involving the Stirling numbers of the second kind. As further applications, we derive several known identities involving the Bernoulli numbers and polynomials and Euler polynomials, and new relations for the higher order tangent numbers, the higher order Bernoulli numbers of the second kind, the numbers , the higher order Bernoulli numbers and polynomials and the higher order Euler polynomials and their coefficients.     

Computation methods for combinatorial sums and Euler‐type numbers related to new families of numbers

The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well‐known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we give not only a computational algorithm for these numbers but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers. Copyright © 2016 John Wiley & Sons, Ltd.     

退化高阶伯努利多项式与广义等幂和多项式的对称等式

研究了退化伯努利多项式与广义等幂和多项式的对称关系,获得了关于多个退化高阶伯努利多项式与广义等幂和多项式的若干对称关系.     

An identity of symmetry for the Bernoulli polynomials

The main purpose of this paper is to prove an identity of symmetry for the higher order Bernoulli polynomials. It turns out that the recurrence relation and multiplication theorem for the Bernoulli polynomials which discussed in [F.T. Howard, Application of a recurrence for the Bernoulli numbers, J. Number Theory 52 (1995) 157-172], as well as a relation of symmetry between the power sum polynomials and the Bernoulli numbers developed in [H.J.H. Tuenter, A symmetry of power sum polynomials and Bernoulli numbers, Amer. Math. Monthly 108 (2001) 258-261], are all special cases of our results.     

Euler多项式的若干对称恒等式

Using the generating functions, we prove some symmetry identities for the Euler polynomials and higher order Euler polynomials, which generalize the multiplication theorem for the Euler polynomials. Also we obtain some relations between the Bernoulli polynomials, Euler polynomials, power sum, alternating sum and Genocchi numbers.     

Faulhaber’s theorem on power sums

We observe that the classical Faulhaber’s theorem on sums of odd powers also holds for an arbitrary arithmetic progression, namely, the odd power sums of any arithmetic progression a+b,a+2b,…,a+nb is a polynomial in na+n(n+1)b/2. While this assertion can be deduced from the original Fauhalber’s theorem, we give an alternative formula in terms of the Bernoulli polynomials. Moreover, by utilizing the central factorial numbers as in the approach of Knuth, we derive formulas for r-fold sums of powers without resorting to the notion of r-reflective functions. We also provide formulas for the r-fold alternating sums of powers in terms of Euler polynomials.     

An extension of Vietoris's inequalities

We establish a best possible extension of a famous Theorem of Vietoris about the positivity of a general class of cosine sums. Our result refines and sharpens several earlier generalizations of this Theorem, and settles some open questions regarding the possibility of further improvement of it. Some new inequalities for trigonometric sums are given. We show that our results have applications within the context of positive sums of Gegenbauer polynomials and quadrature methods. We also obtain some existing estimates for the location of zeros of certain trigonometric polynomials under a weakened condition on their coefficients. 2000 Mathematics Subject ClassificationPrimary—42A05; Secondary—42A32,26D05, 26D15, 33B15, 33C45 Dedicated to Richard Askey on the occasion of his 70th birthday     

The convergence of partial sums of interpolating polynomials

For functions of ΛBV, we study the convergence of the partial sums of interpolating polynomials. An estimate is found for the Fourier-Lagrange coefficients of these functions. For functions in BV, convergence is shown at points of discontinuity if the order of the polynomial increases sufficiently rapidly compared to the order of the partial sum. A Dirichlet-Jordan type theorem is shown for functions of harmonic bounded variation, and this result is shown to be best possible.     

Divisibility properties of a class of binomial sums

We study congruence and divisibility properties of a class of combinatorial sums that involve products of powers of two binomial coefficients, and show that there is a close relationship between these sums and the theorem of Wolstenholme. We also establish congruences involving Bernoulli numbers, and finally we prove that under certain conditions the sums are divisible by all primes in specific intervals.     

多项式的特征和与二项指数和的混合均值

本文主要利用解析方法以及二项指数和与Dirichlet特征的性质，研究多项式的特征和与二项三次指数和的混合均值的计算问题，并得到一个较强的渐近公式.