Bernoulli Identities Research Articles

Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers

The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, is generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of double zeta values. A set of identities for the Ramanujan and generalized Ramanujan polynomials is presented. An alternative proof of Lettington's identity is provided, together with its generalizations to the Hurwitz and Lerch zeta functions, hence to Dirichlet L series, to Eisenstein series, and to general Mellin transforms.The Hurwitz numbers H˜n occur in the Laurent expansion about the origin of a certain Weierstrass ℘ function for a square lattice, and are highly analogous to the Bernoulli numbers. An integral representation of the Laurent coefficients about the origin for general ℘ functions, and for these numbers in particular, is presented. As a Corollary, the asymptotic form of the Hurwitz numbers is determined. In addition, a series representation of the Hurwitz numbers is given, as well as a new recurrence. Other results concern the Matter numbers of the equianharmonic case of the ℘ function.

Some symmetric identities for the generalized Bernoulli, Euler and Genocchi polynomials associated with Hermite polynomials.

In 2008, Liu and Wang established various symmetric identities for Bernoulli, Euler and Genocchi polynomials. In this paper, we extend these identities in a unified and generalized form to families of Hermite–Bernoulli, Euler and Genocchi polynomials. The procedure followed is that of generating functions. Some relevant connections of the general theory developed here with the results obtained earlier by Pathan and Khan are also pointed out.

A symbolic approach to some identities for Bernoulli–Barnes polynomials

The Bernoulli–Barnes polynomials are defined as a natural multidimensional extension of the classical Bernoulli polynomials. Many of the properties of the Bernoulli polynomials admit extensions to this new family. A specific expression involving the Bernoulli–Barnes polynomials has recently appeared in the context of self-dual sequences. The work presented here provides a proof of this self-duality using the symbolic calculus attached to Bernoulli numbers and polynomials. Several properties of the Bernoulli–Barnes polynomials are established by this procedure.

General convolution identities for Bernoulli and Euler polynomials

Using general identities for difference operators, as well as a technique of symbolic computation and tools from probability theory, we derive very general kth order (k≥2) convolution identities for Bernoulli and Euler polynomials. This is achieved by use of an elementary result on uniformly distributed random variables. These identities depend on k positive real parameters, and as special cases we obtain numerous known and new identities for these polynomials. In particular we show that the well-known identities of Miki and Matiyasevich for Bernoulli numbers are special cases of the same general formula.

Convolution Identities for Bernoulli and Genocchi Polynomials

The main purpose of this paper is to derive various Matiyasevich-Miki-Gessel type convolution identities for Bernoulli and Genocchi polynomials and numbers by applying some Euler type identities with two parameters.

Bernoulli Identities and Combinatoric Convolution Sums with Odd Divisor Functions

We study the combinatoric convolution sums involving odd divisor functions, their relations to Bernoulli numbers, and some interesting applications.

Semi-invariants of binary forms and identities for Bernoulli, Euler and Hermite polynomials

Semi-invariants of binary forms and identities for Bernoulli, Euler and Hermite polynomials

A unified presentation of identities involving Weierstrass-type functions and their applications

In this work, we investigate and discuss Weierstrass-type functions ℘ 2 k to the power of m , and then derive a general formula for identities among them. Applications of this general formula include not only producing many of the familiar identities among Weierstrass-type functions, but also infinitely many new ones. Moreover, it is observed that new Bernoulli identities could be constructed systematically by means of the new identities among Weierstrass-type functions.

Symmetry p-adic invariant integral on ℤ p for Bernoulli and Euler polynomials

The main purpose of this paper is to investigate several further interesting properties of symmetry for the p-adic invariant integrals on ℤ p . From these symmetry, we can derive many interesting recurrence identities for Bernoulli and Euler polynomials. Finally we introduce the new concept of symmetry of fermionic p-adic invariant integral on ℤ p . By using this symmetry of fermionic p-adic invariant integral on ℤ p , we will give some relations of symmetry between the power sum polynomials and Euler numbers. The relation between the q-Bernoulli polynomials and q-Dedekind type sums which discussed in Y. Simsek (q-Dedekind type sums related to q-zeta function and basic L-series, J. Math. Anal. Appl. 318 (2006), pp. 333–351) can be also derived by using the properties of symmetry of fermionic p-adic integral on ℤ p .

Some families of Weierstrass-type functions and their applications

In this paper, we introduce the Weierstrass-type functions ℘2k and investigate some of their elementary properties in complex analysis. We focus upon the relations among the classical Weierstrass ℘-function and the Weierstrass-type functions ℘2k . By means of these relations, Rademacher's Bernoulli identity and several other identities can be deduced naturally and straightforwardly, except possibly the properties of the derivatives of the Weierstrass ℘-function. Our presentation provides a new direction to calculate a sub-class of Bernoulli identities without B 2.

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.

A note on Bernoulli identities associated with the Weierstrass ℘-function

Various types of Bernoulli identities have been discussed by Euler, Ramanujan, Rademacher, Eie, and others. In particular, by using certain identities involving zeta functions, Eie constructed several Bernoulli identities systematically. There are many expressions for the Bernoulli number B 2k , which express B 2k as a sum of the products of lower-order Bernoulli numbers in two terms or in three terms. In this article, we show that there is no unique method to calculate the Bernoulli number B 2k as a sum of the products of lower-order Bernoulli numbers in three terms. By employing a technique which is substantially different from the method of Eie’s proof, we derive several Bernoulli identities through the properties of the Weierstrass ℘-function. We also discuss the relationship between these and other known Bernoulli identities.

Identities concerning Bernoulli and Euler polynomials

We establish two general identities for Bernoulli and Euler polynomials, which are of a new type and have many consequences. The most striking result in this paper is as follows: If $n$ is a positive integer, $r+s+t=n$ and $x+y+z=1$, then we have $$rF(s,t;x,y)+sF(t,r;y,z)+tF(r,s;z,x)=0$$ where $$F(s,t;x,y):=\sum_{k=0}^n(-1)^k\binom{s}{k}\binom{t}{n-k}B_{n-k}(x)B_k(y).$$ This symmetric relation implies the curious identities of Miki and Matiyasevich as well as some new identities for Bernoulli polynomials such as $$\sum_{k=0}^n\binom{n}{k}^2B_k(x)B_{n-k}(x)=2\sum^n\Sb k=0 k\not=n-1\endSb\binom{n}{k}\binom{n+k-1}{k}B_k(x)B_{n-k}.$$

New identities involving Bernoulli and Euler polynomials

Using the finite difference calculus and differentiation, we obtain several new identities for Bernoulli and Euler polynomials; some extend Miki's and Matiyasevich's identities, while others generalize a symmetric relation observed by Woodcock and some results due to Sun.

ON EVALUATION OF GENERALIZED EULER SUMS OF EVEN WEIGHT

The classical Euler sum [Formula: see text] cannot be evaluated when the weight p + q is even unless p = 1 or p = q or (p, q) = (2, 4) or (p, q) = (4, 2) [7]. However it is a different story if instead we consider the alternating sums [Formula: see text] and [Formula: see text] They can be evaluated for even weight p + q. In this paper, we shall evaluate a family of generalized Euler sums containing [Formula: see text] when the weight p + q is even via integral transforms of Bernoulli identities.

Applications of the classical umbral calculus

We describe applications of the classical umbral calculus to bilinear generating functions for polynomial sequences, identities for Bernoulli and related numbers, and Kummer congruences.

On sums of certain trigonometric series

Let N be a fixed positive integer. In this paper, we apply a newly developed method of Eie and Lai to the evaluation of certain infinite trigonometric series. Indeed, we are able to express the sums of these series in terms of Bernoulli polynomials. That is, we obtain several new Bernoulli identities. A similar method leads to Ramanujan's identities.

The combinatorial structure of trigonometry

The native mathematical language of trigonometry is combinatorial. Two interrelated combinatorial symmetric functions underlie trigonometry. We use their characteristics to derive identities for the trigonometric functions of multiple distinct angles. When applied to the sum of an infinite number of infinitesimal angles, these identities lead to the power series expansions of the trigonometric functions. When applied to the interior angles of a polygon, they lead to two general constraints satisfied by the corresponding tangents. In the case of multiple equal angles, they reduce to the Bernoulli identities. For the case of two distinct angles, they reduce to the Ptolemy identity. They can also be used to derive the De Moivre‐Cotes identity. The above results combined provide an appropriate mathematical combinatorial language and formalism for trigonometry and more generally polygonometry. This latter is the structural language of molecular organization, and is omnipresent in the natural phenomena of molecular physics, chemistry, and biology. Polygonometry is as important in the study of moderately complex structures, as trigonometry has historically been in the study of simple structures.

On Bernoulli identities and applications

Bernoulli numbers appear as special values of zeta functions at integers and identities relating the Bernoulli numbers follow as a consequence of properties of the corresponding zeta functions. The most famous example is that of the special values of the Riemann zeta function and the Bernoulli identities due to Euler. In this paper we introduce a general principle for producing Bernoulli identities and apply it to zeta functions considered by Shintani, Zagier and Eie. Our results include some of the classical results of Euler and Ramanujan. Kummers congruences play important roles in the investigation of p -adic interpolation of the classical Riemann zeta function.