Dmitry V. Dolgy, Dae San Kim, taekyun Kim
In this paper, we give some new identities of Carlitz q-Bernoulli polynomials under symmetry group S 3 . The derivatives of identities are based on the q-Volkenborn integral expression of the generating function for the Carlitz q-Bernoulli polynomials and the q-Volkenborn integral equations that can be expressed as the exponential generating functions for the q-power sums.
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New convolution identities of hypergeometric Bernoulli polynomials are presented. Two different approaches to proving these identities are discussed, corresponding to the two equivalent definitions of hypergeometric Bernoulli polynomials as Appell sequences.
In the present paper, we obtain new interesting relations and identities of the Apostol-Bernoulli polynomials of higher order, which are derived using a Bernoulli polynomial basis. Finally, by utilizing our method, we also derive formulas for the convolutions of Bernoulli and Euler polynomials, expressed via Apostol-Bernoulli polynomials of higher order.
In this paper we give some relation involving values of q-Bernoulli, q-Euler and Bernstein polynomials. From these relations, we obtain some interesting identities on the q-Bernoulli, q-Euler and Bernstein polynomials.
The purpose of this paper is to give symmetric identities for higher-order degenerate q- Bernoulli polynomials arising from the p-adic q-integral on Zp.
In this paper, we study the Carlitz's degenerate Bernoulli numbers and polynomials and give some formulae and identities related to those numbers and polynomials.
In this paper, we prove new identities for Bernoulli polynomials that extend Alzer and Kwong's results. The key idea is to use the Volkenborn integral over $\mathbb Z_p$ of the Bernoulli polynomials to establish recurrence relations on the integrands. Also, some known identities are obtained by our approach.
In this paper, we give some interesting and new identities of q-Bernoulli numbers which are derived from convolutions on the ring of p-adic integers.
In this paper, we give some new identities of symmetry for q-Bernoulli polynomials under the symmetric group of degree n arising from p-adic q-integrals on Zp.
In this paper, we derive some new and interesting idebtities for Bernoulli, Euler and Hermite polynomials associated with Chebyshev polynomials.
The purpose of this paper is to give some new identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials.
In this paper, by the properties of p-adic invariant integral on Zp, we establish various identities concerning the generalized Bernoulli numbers and polynomials. From the symmetric properties of p-adic invariant integral on Zp, we give some interesting relationship between the power sums and the generalized Bernoulli polynomials.
The purpose of this paper is to derive some identities of the higher order generalized twisted Bernoulli numbers and polynomials attached to $χ$ from the properties of the p-adic invariant integrals.
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 \ge 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 …
A number of identities are proved by using Stirling transforms. These identities involve Stirling numbers of the first and second kinds, hyperharmonic and derangement numbers, Bernoulli and Euler numbers and polynomials, powers, power sums, and factorials.
In this paper, we derive eight basic identities of symmetry in three variables related to generalized Bernoulli polynomials and generalized power sums. All of these are new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the $p$-adic integral expression of the generating function for the generalized Bernoulli polynomials and the quotient of $p$-adic integrals that can be expressed as the exponential generating function for the generalized power sums.
In this paper we establish plenty of number theoretic and combinatoric identities involving generalized Bernoulli and Stirling numbers of both kinds. These formulas are deduced from Pascal type matrix representations of Bernoulli and Stirling numbers. For this we define and factorize a modified Pascal matrix corresponding to Bernoulli and Stirling cases.
In this paper, we consstruct a new extended q-Bernoulli numbers and poly nomials. From these numbers, we derive the multiple zeta functions and give some relations between multiple Bernoulli numbers and multiple zeta functions.
Recently, the higher-order Carlitz's q-Bernoulli polynomials are represented as q-Volkenborn integral on Zp by Kim. A question was asked in [13] as to finding the extended formulaeof symmetries for Bernoulli polynomials which are related to Carlitz q-Bernoulli polynomials. In this paper, we give some new identities of symmetry for the higher-order Carlitz q-Bernoulli polynomials whch are derived from multivariate q-Volkenborn integrals on Zp. We note that they are a partial answer to that question. .
Recently, Kim-Kim Introduced some interesting identities of symmetry for q-Bernoulli polynomials under symmetry group of degree n. In this paper, we study the degenerate q-Euler polynomials and derive some identities of symmetry for these polynomials arising from the p-adic q-integral on Zp.
We investigate some interesting properties of Bernstein polynomials associated with boson p-adic integrals on Zp.
In this paper, we study umbral calculus to have alternative ways of obtaining our results. That is, we derive some interesting identities of the higher-order Bernoulli, Euler and Hermite polynomials arising from umbral calculus to have alternative ways.
In this paper, we derive eight basic identities of symmetry in three variables related to $q$-Bernoulli polynomials and the $q$-analogue of power sums. These and most of their corollaries are new, since there have been results only about identities of symmetry in two variables. These abundance of symmetries shed new light even on the existing identities so as to yield some further interesting ones. The derivations of identities are based on the $p$-adic integral expression of the generating function for the $q$-Bernoulli polynomials and the quotient of integrals that can be expressed as the ex…
We derive eight identities of symmetry in three variables related to generalized twisted Bernoulli polynomials and generalized twisted power sums, both of which are twisted by ramified roots of unity. All of these are new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the $p$-adic integral expression of the generating function for the generalized twisted Bernoulli polynomials and the quotient of $p$-adic integrals that can be expressed as the exponential generating function for the generalized twisted power sums.
In this paper, we derive eight basic identities of symmetry in three variables related to Bernoulli polynomials and power sums. These and most of their corollaries are new, since there have been results only about identities of symmetry in two variables. These abundance of symmetries shed new light even on the existing identities so as to yield some further interesting ones. The derivations of identities are based on the $p$-adic integral expression of the generating function for the Bernoulli polynomials and the quotient of integrals that can be expressed as the exponential generating functio…
Recently, Kim-Jang-Yi have introduced q-Bernstein polynomials. From these q-Berstein polynomials, we investigte some properties related to q-Stirling numbes and q-Bernoulli numbes.
We prove formulas for the Bernoulli numbers by using the Newton-Girard identities to evaluate the Riemann zeta function at positive even integers. To do this, we define a sequence of positive integers, a sequence of polynomials, and a sequence of linear operators on the space of functions. We prove properties of these polynomials, such as the positivity of their coefficients, and present a combinatorial formula for the Bernoulli numbers as a positive sum over plane trees which can be generalized as a transform of sequences. We also combinatorially prove the Newton-Girard identities using the s…
We derive three identities of symmetry in two variables and eight in three variables related to generalized twisted Bernoulli polynomials and generalized twisted power sums, both of which are twisted by unramified roots of unity. The case of ramified roots of unity was treated previously. The derivations of identities are based on the $p$-adic integral expression, with respect to a measure introduced by Koblitz, of the generating function for the generalized twisted Bernoulli polynomials and the quotient of $p$-adic integrals that can be expressed as the exponential generating function for the…
Recently, $λ$-Bernoulli and $λ$-Euler numbers are studied in [5, 10]. The purpose of this paper is to present a systematic study of some families of the $q$-extensions of the $λ$-Bernoulli and the $λ$-Euler numbers by using the bosonic $p$-adic $q$-integral and the fermionic $p$-adic $q$-integral. The investigation of these $λ$-$q$-Bernoulli and $λ$-$q$-Euler numbers leads to interesting identities related to these objects. The results of the present paper cover earlier results concerning $q$-Bernoulli and $q$-Euler numbers. By using derivative operator to the generating functions of $λ$-$q$-B…
We introduce the degenerate Bernoulli numbers of the second kind as a degenerate version of the Bernoulli numbers of the second kind. We derive a family of nonlinear differential equations satisfied by a function closely related to the generating function for those numbers. We obtain explicit expressions for the coefficients appearing in those differential equations and the degenerate Bernoulli numbers of the second kind. In addition, as an application and from those differential equations we have an identity expressing the degenerate Bernoulli numbers of the second kind in terms of those numb…
I recent years, many mathematicians studied various degenerate version of some spcial polynomials of which quite a few interesting results were discovered. In this paper, we introduce the type 2 degenerate Bernoulli polynomials of the second kind and their higher-order analogues, and study some identities and expressions for these polynomials. Specially, we obtain a relation between the type 2 degenerate Bernoulli polynomials of the second kind and degenerate Bernoulli polynomials of the second kind, and identity involving hihger-order analogues of those polynomials and the degenerate stirling…
In this paper, we will deal with some new formulae for two product Genocchi polynomials together with both Euler polynomials and Bernoulli polynomials. We get some applications for Genocchi polynomials. Our applications possess a number of interesting properties to study in Theory of Analytic numbers which we express in the present paper.
In this paper, we obtain a generalization of an identity due to Carlitz on Bernoulli polynomials. Then we use this generalized formula to derive two symmetric identities which reduce to some known identities on Bernoulli polynomials and Bernoulli numbers, including the Miki identity.
In this paper, we give some interesting identities of higher-order Bernoulli, Frobenius-Euler and Euler polynomials arising from umbral calculus. From our method of this paper, we can derive many interesting identities of special polynomials
In recent years, studying degenerate versions of various special polynomials and numbers have attracted many mathematicians. Here we introduce degenerate type 2 Bernoulli polynomials, fully degenerate type 2 Bernoulli polynomials and degenerate type 2 Euler polynomials, and their corresponding numbers, as degenerate and type 2 versions of Bernoulli and Euler numbers. Regarding to those polynomials and numbers, we derive some identities, distribution relations, Witt type formulas and analogues for the Bernoulli's interpretation of powers of the first $m$ positive integers in terms of Bernoulli …
The joy of multisection: elementary approaches to Ramanujan's lacunary identities for Bernoulli numbers
本文用几何对称性与生成函数两种多重分段方法重新推导并推广 Ramanujan 的 Bernoulli 数缺项恒等式。
In this paper we obtain several new identities for Bernoulli and Euler polynomials; some of them extend Miki's and Matiyasevich's identities. Our new method involves differences and derivatives of polynomials.
We derive several symmetric identities for Bernoulli and Euler polynomials which imply some known identities. Our proofs depend on the new technique developed in part I and some identities obtained in [European J. Combin. 24(2003), 709--718].
The relations between the Bernoulli and Eulerian polynomials of higher order and the complete Bell polynomials are found that lead to new identities for the Bernoulli and Eulerian polynomials and numbers of higher order. General form of these identities is considered and generating function for polynomials satisfying this general identity is found.
Dedekind sums occur in the transformation behaviour of the logarithm of the Dedekind eta-function under substitutions from the modular group. In 1892, Dedekind showed a reciprocity relation for the Dedekind sums. Apostol generalized Dedekind sums by replacing the first Bernoulli function appearing in them by any Bernoulli functions and derived a reciprocity relation for the generalized Dedekind sums. In this paper, we consider poly-Dedekind sums which are obtained from the Dedekind sums by replacing the first Bernoulli function by any type 2 poly-Bernoulli functions of arbitrary indices and pr…
In this paper, we consider higher-order Bernoulli and poly-Bernoulli mixed type polynomials and we give some interesting identities of those polynomials arising from umbral calculus.
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.
In this paper, we consider Barnes' multiple Bernoulli and poly-Bernoulli mixed-type polynomials. From the properties of Sheffer sequences of these polynomials arising from umbrral calculus, we derive new and interesting identities.
We present a new method for the derivation of convolution identities for finite sums of products of Bernoulli numbers. Our approach is motivated by the role of these identities in quantum field theory and string theory. We first show that the Miki identity and the Faber-Pandharipande-Zagier (FPZ) identity are closely related, and give simple unified proofs which naturally yield a new Bernoulli number convolution identity. We then generalize each of these three identities into new families of convolution identities depending on a continuous parameter. We rederive a cubic generalization of Miki'…
In this paper we prove a family of identities for Bernoulli numbers parameterized by triples of integers $(a,b,c)$ with $a+b+c=n-1$, $n\ge 4$. These identities are deduced while translating into homotopical terms the gauge action on the Maurer Cartan Set which can be seen an abstraction of the behaviour of gauge infinitesimal transformations in classical gauge theory. We show that Euler and Miki's identities, well known and apparently non related formulas, are linear combinations of our family and they satisfy a particular symmetry relation.
Using the methods of classical invariant theory a general approach to finding of identities for Bernulli, Euler and Hermite polynomials is proposed.
In this paper, we derive some identities involving special numbers and moments of random variables by using the generating functions of the moments of certain random variables. Here the related special numbers are Stirling numbers of the first and second kinds, degenerate Stirling numbers of the first and second kinds, derangement numbers, higher-order Bernoulli numbers and Bernoulli numbers of the second kind.
In this paper we study q-Bernoulli numbers and polynomials related to q-Stirling numbers. From thsese studying we investigate some interesting q-stirling numbers' identities related to q-Bernoulli numbers.
In this paper, we investigate some properties of q-Bernoulli polynomi- als arising from q-umbral calculus. Finally, we derive some interesting identities of q-Bernoulli polynomials from our investigation.
In this paper, we give some recurrence formula and new and interesting identities for the poly-Bernoulli numbers and polynomials which are derived from umbral calculus.
We consider the $1$-dimensional reflected Brownian motion and $3$-dimensional Bessel process and the general models. By decomposing the hitting times of consecutive sites into loops, we obtain identities, called loop identities, for the generating functions of the hitting times. After proving this decomposition both combinatorially and inductively, we consider the case that sites are equally distributed. Then, from loop identities, we derive expressions of Bernoulli and Euler polynomials, in terms of Euler polynomials of higher-orders.
In this paper we consider the weighted q-Bernoulli numbers and polynomials which are differnt type of Carlitz's q-Bernoulli numbers and polynomials. From these numbers and polynomials, we derive some interesting formulaes and identities.
We present a relationship between the generalized hyperharmonic numbers and the poly-Bernoulli polynomials, motivated from the connections between harmonic and Bernoulli numbers. This relationship yields numerous identities for the hyper-sums and several congruences.
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.
With this paper we introduce a new series representation of $ζ(3)$, which is based on the Clausen representation of odd integer zeta values. Although, relatively fast converging series based on the Clausen representation exist for $ζ(3)$, their convergence behavior is very slow compared to BBP-type formulas, and as a consequence they are not used for explicit numerical computations. The reason is found in the fact that the corresponding Clausen function can be calculated analytically for a few rational arguments only, where $x=\frac{1}{6}$ is the smallest one. Using polylogarithmic identities …
In this paper, we give new identities involving Phillips q-Bernstein polynomials and we derive some interesting properties of q-Berstein polynomials associated with q-Stirling numbers and q-Bernoulli polynomials.
In this paper, we investigate some identities of Laguerre polynomials involving Bernoulli and Euler polynomials which are derived from umbral calculus.
In this paper, we introduce the new fully degenerate poly-Bernoulli numbers and polynomials and investigate some properties of these polynomials and numbers. From our properties, we derive some identities for the fully degenerate poly-Bernoulli numbers and polynomials.
The main aim of this paper is to investigate and introduce relations between the numbers of k-ary Lyndon words and unified zeta-type functions which was defined by Ozden et al [15, p. 2785]. Finally, we give some identities on generating functions for the numbers of k-ary Lyndon words and some special numbers and polynomials such as the Apostol-Bernoulli numbers and polynomials, Frobenius-Euler numbers, Euler numbers and Bernoulli numbers.
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{…
In this paper, we consider Hermite and poly-Bernoulli mixed-type polynomials and investigate the properties of those polynomials which are derived from umbral calculus. Finally, we give various identities associated with Stirling numbers, Bernoulli and Frobenius-Euler polynomials of higher order.
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