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.     

Extended Zeilberger's algorithm for identities on Bernoulli and Euler polynomials

We present a computer algebra approach to proving identities on Bernoulli polynomials and Euler polynomials by using the extended Zeilberger's algorithm given by Chen, Hou and Mu. The key idea is to use the contour integral definitions of the Bernoulli and Euler numbers to establish recurrence relations on the integrands. Such recurrence relations have certain parameter free properties which lead to the required identities without computing the integrals. Furthermore two new identities on Bernoulli numbers are derived.     

The Haruki-Rassias and related integral representations of the Bernoulli and Euler polynomials

Haruki and Rassias [H. Haruki, T.M. Rassias, New integral representations for Bernoulli and Euler polynomials, J. Math. Anal. Appl. 175 (1993) 81-90] found the integral representations of the classical Bernoulli and Euler polynomials and proved them by making use of the properties of certain functional equation. In this sequel, we rederive, in a completely different way, the results of Haruki and Rassias and deduce related and new integral representations. Our proofs are quite simple and remarkably elementary.     

Bernoulli多项式和Euler多项式的关系

本文给出了 Bernoulli- Euler数之间的关系和 Bernoulli- Euler多项式之间的关系 ,从而深化和补充了有关文献中的相关结果 .     

Genocchi polynomials associated with the Umbral algebra

The aim of this paper is to study on the Genocchi polynomials of higher order on P, the algebra of polynomials in the single variable x over the field C of characteristic zero and P^∗, the vector spaces of all linear functional on P. By using the action of a linear functional L on a polynomial p(x) Sheffer sequences and Appell sequences, we obtain some fundamental properties of the Genocchi polynomials. Furthermore, we give relations between, the first and second kind Stirling numbers, Euler polynomials of higher order and Genocchi polynomials of higher order.     

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.     

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.     

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 Appell sequences of polynomials of Bernoulli and Euler type

A construction of new sequences of generalized Bernoulli polynomials of first and second kind is proposed. These sequences share with the classical Bernoulli polynomials many algebraic and number theoretical properties. A class of Euler-type polynomials is also presented.     

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.     

关于Lucas多项式平方和的恒等式

利用广义Lucas多项式L n(x,y)的性质,通过构造组合和式T n(x,y;tx2),结合Bernoulli多项式的生成函数和Euler多项式的生成函数,采用分析学中的方法,得到两个有关L2n(x,y)的恒等式.并从这一结果出发,得到了两个推论,推广了相关文献的一些结果.     

Some identities and congruences concerning Euler numbers and polynomials

In this paper, using the properties of the moments of p-adic measures, we establish some identities and Kummer likewise congruences concerning Euler numbers and polynomials. In the preliminaries, we introduce the Laplace transform which is an important tool for the determination of the moments of p-adic measures. We also give a sequence n(dn) linked to Euler numbers and which satisfies the same type of congruences and identities as the Euler numbers. At the end, for p=2, we give congruences on Euler numbers involving the sequence n(dn).     

On the q-extension of Euler and Genocchi numbers

Carlitz has introduced an interesting q-analogue of Frobenius-Euler numbers in [L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. 15 (1948) 987-1000; L. Carlitz, q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc. 76 (1954) 332-350]. He has indicated a corresponding Stadudt-Clausen theorem and also some interesting congruence properties of the q-Euler numbers. A recent author's study of more general q-Euler and Genocchi numbers can be found in previous publication [T. Kim, L.C. Jang, H.K. Pak, A note on q-Euler and Genocchi numbers, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001) 139-141]. In this paper we give a new construction of q-Euler numbers, which are different from Carlitz's q-extension and author's q-extension in previous publication (see [T. Kim, L.C. Jang, H.K. Pak, A note on q-Euler and Genocchi numbers, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001) 139-141]). By using our q-extension of Euler numbers, we can also consider a new q-extension of Genocchi numbers and obtain some interesting relations between q-extension of Euler numbers and q-extension of Genocchi numbers.     

Summation formulae for finite tangent and secant sums

In a series of papers [6], [7], [8], [9] and [10] it has been shown that nine remarkably general families of the finite trigonometric sums could be summed in closed form by making use of the calculus of residues and choosing a particularly convenient integration contour. In this sequel, new summation formulae for three general families of finite tangent and secant sums have been deduced by the same approach.     

Some results on sums of products of Bernoulli polynomials and Euler polynomials

In this paper, by the methods of partial fraction decomposition and generating functions, we establish an explicit expression for sums of products of l Bernoulli polynomials and n?l Euler polynomials, i.e., for sums $$S_n^{(k)}(y;l,k-l):= \sum_{\substack{j_1+\cdots+j_k=n\\j_1,\dots,j_k\geq0}} \binom {n}{j_1,\dots,j_k} B_{j_1}(x_1)\cdots B_{j_l}(x_l)E_{j_{l+1}}(x_{l+1}) \cdots E_{j_k}(x_k). $$ This result is then used to deal with various other types of sums of products of Bernoulli polynomials and Euler polynomials. Some of them are expressed in terms of $S_{n}^{(k)}(y;l,k-l)$ and can be computed directly, while the others satisfy certain recurrences and can be determined recursively. As a consequence, many known results are special cases of ours.     

Diophantine Equations and Bernoulli Polynomials

Given m, n 2, we prove that, for sufficiently large y, the sum 1 ⁿ +···+ y ⁿ is not a product of m consecutive integers. We also prove that for m n we have 1 ^m +···+ x ^m 1 ⁿ +···+ y ⁿ , provided x, y are sufficiently large. Among other auxiliary facts, we show that Bernoulli polynomials of odd index are indecomposable, and those of even index are almost indecomposable, a result of independent interest.