Higher Order Bernoulli Numbers Research Articles

New results on Bernoulli numbers of higher order

Let r be any positive integer. In this paper, we revisit explicit and recurrence formulas satisfied by the Bernoulli numbers Bn(r) of higher order r. By using the unsigned Stirling numbers, we give them new forms and a clearer description. The second part of this study consists of providing and proving analogues of Kummer congruences and a von Staudt–Clausen theorem for the numbers Bn(r)∕(rnr), as well as investigating their numerators. Moreover, we study the p-integrality of these numbers. In the third part of this paper, we construct a new family of Eisenstein series Ek(r)(z) whose constant terms are the numbers Bn(r)∕(rnr). We prove a congruence for these new Eisenstein series. This generalizes the classical von Staudt–Clausen’s and Kummer’s congruences of Eisenstein series. Our study leads to several applications in algebraic number theory.

The 2-adic analysis of Stirling numbers of the second kind via higher order Bernoulli numbers and polynomials

Several new estimates for the [Formula: see text]-adic valuations of Stirling numbers of the second kind are proved. These estimates, together with criteria for when they are sharp, lead to improvements in several known theorems and their proofs, as well as to new theorems, including a long-standing open conjecture by Lengyel. The estimates and criteria all depend on our previous analysis of powers of [Formula: see text] in the denominators of coefficients of higher order Bernoulli polynomials. The corresponding estimates for Stirling numbers of the first kind are also proved. Some attention is given to asymptotic cases, which will be further explored in subsequent publications.

Poly-Cauchy numbers with level 2

ABSTRACT We introduce poly-Cauchy numbers with level 2. Poly-Cauchy numbers may be interpreted as a kind of generalizations of the classical Cauchy numbers by using the inverse relation of exponentials and logarithms. On the contrary, poly-Bernoulli numbers can be from the inverse relation of logarithms and exponentials. In this similar stream, poly-Cauchy numbers with level 2 may be yielded from the inverse relation about the hyperbolic sine function, which is a 2-step function of the exponential function. In this article, we show several expressions, relations, and properties about poly-Cauchy numbers with level 2. Poly-Cauchy numbers with level 2 can be expressed in terms of multinomial coefficients with combinatorial summation, Stirling numbers of the first kind, or iterated integrals. We also give some recurrence relations for poly-Cauchy numbers with level 2. When the index is negative, the double summation may be formulated as a closed form. A simple case of Cauchy numbers with level 2 has some more relations with D numbers from higher-order Bernoulli numbers or complementary Euler numbers. We prove some more expressions in determinants, continued fractions or by Trudi's formula.

Some identities involving special numbers and moments of random variables

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.

On the twisted Daehee polynomials with q-parameter

The n th twisted Daehee numbers with q-parameter are closely related to higher-order Bernoulli numbers and Bernoulli numbers of the second kind. In this paper, we give a p-adic integral representation of the twisted Daehee polynomials with q-parameter, and we derive some interesting properties related to the n th twisted Daehee polynomials with q-parameter.

Sums of Products Involving Power Sums of φ(n) Integers

A sequence of rational numbers as a generalization of the sequence of Bernoulli numbers is introduced. Sums of products involving the terms of this generalized sequence are then obtained using an application of Faà di Bruno's formula. These sums of products are analogous to the higher order Bernoulli numbers and are used to develop the closed form expressions for the sums of products involving the power sums Ψk(x,n):=∑d|n‍μ(d)dkSkx/d, n∈ℤ+ which are defined via the Möbius function μ and the usual power sum Sk(x) of a real or complex variable x. The power sum Sk(x) is expressible in terms of the well-known Bernoulli polynomials by Sk(x):=(Bk+1(x+1)-Bk+1(1))/(k+1).

Identities on products of Genocchi numbers

In this paper, we discuss the properties of a class of Genocchi numbers by generating functions and Riordan arrays, we establish some identities involving Genocchi numbers, the Stirling numbers, the generalized Stirling numbers, the higher order Bernoulli numbers and Cauchy numbers. Further, we get asymptotic value of some sums relating the Genocchi numbers.

On the Modified q‐Bernoulli Numbers of Higher Order with Weight

The purpose of this paper is to give some properties of the modified q‐Bernoulli numbers and polynomials of higher order with weight. In particular, by using the bosonic p‐adic q‐integral on ℤp, we derive new identities of q‐Bernoulli numbers and polynomials with weight.

Identities on the Weighted q‐Bernoulli Numbers of Higher Order

We give a new construction of the weighted q‐Bernoulli numbers and polynomials of higher order by using multivariate p‐adic q‐integral on ℤp.

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.

Closed-form summation of the Dowker and related sums

Finite sums of powers of cosecants appear in a wide range of physical problems. We, through a unified approach which uses contour integrals and residues, establish the summation formulas for two general families of such sums. One of them is the family which was first studied and summed in closed form by Dowker [Phys. Rev. D 36, 3095 (1987)], while the other is related to it and has not been studied before. Our summation formulas of the Dowker sums involve only the Stirling numbers of the first kind and the (ordinary) Bernoulli polynomials and numbers, unlike the earlier summation formulas in which either the higher-order Bernoulli numbers and polynomials or the multiple sums involving the Bernoulli numbers and their products, were used. A great deal of other (known or presumably new) closed-form summations follows as straightforward corollaries to these formulas. Among them are two special cases of the celebrated Verlinde’s formula and numerous sums encountered in various physical problems by McCoy and Orrick [J. Stat. Phys. 83, 839 (1996)], Gervois and Mehta [J. Math. Phys. 36, 5098 (1995)], and Henkel and Lacki [Phys. Lett. A 138, 105 (1989)].

Kummer congruence for the Bernoulli numbers of higher order

The authors studied the properties of Bernoulli numbers of higher order [Appl. Math. Comput., in press; Bull. Aust. Math. 65 (2002) 59]. For q=1, we can also find their results [Proc. Jangjeon Math. Soc. 1 (2000) 97; Arch. Math. 76 (2001) 190; Proc. Jangjeon Math. Soc. 1 (2000) 161; Adv. Stud. Contemp. Math. 2 (2000) 9; Proc. Jangjeon Math. Soc. 2 (2001) 23; J. Math. Phys. A 34 (2001) L643; Proc. Jangjeon Math. Soc. 2 (2001) 19; Proc. Jangjeon Math. Soc. 2 (2001) 9; Proc. Jangjeon Math. Soc. 3 (2001) 63]. The authors suggested the question to inquire the proof of Kummer congruence for Bernoulli numbers of higher order [Appl. Math. Comput., in press]. In this paper we give a proof of Kummer type congruence for the Bernoulli numbers of higher order, which is an answer to a part of the question in [Appl. Math. Comput., in press].

Arithmetic properties of Bernoulli numbers of higher order

Arithmetic properties of Bernoulli numbers of higher order

Some theorems on Bernoulli numbers of higher order

Some theorems on Bernoulli numbers of higher order