Order Bernoulli Numbers Research Articles

A note on degenerate poly-Genocchi numbers and polynomials

Recently, some mathematicians have been studying a lot of degenerate versions of special polynomials and numbers in some arithmetic and combinatorial aspects. Our research is also interested in this field. In this paper, we introduce a new type of the degenerate poly-Genocchi polynomials and numbers, based on Kim and Kim’s (J. Math. Anal. Appl. 487(2):124017, 2020) modified polyexponential function. The paper is divided into two parts. In Sect. 2, we consider a new type of the degenerate poly-Genocchi polynomials and numbers constructed from the modified polyexponential function. We also show several combinatorial identities related to the degenerate poly-Genocchi polynomials and numbers. Some of them include the degenerate and other special polynomials and numbers such as the Stirling numbers of the first kind, the degenerate Stirling numbers of the second kind, degenerate Euler polynomials, degenerate Bernoulli polynomials and Bernoulli numbers of order α, etc. In Sect. 3, we also introduce the degenerate unipoly Genocchi polynomials attached to an arithmetic function by using the degenerate polylogarithm function. We give some new explicit expressions and identities related to degenerate unipoly Genocchi polynomials and special numbers and polynomials.

Identities and Computation Formulas for Combinatorial Numbers Including Negative Order Changhee Polynomials

The purpose of this paper is to construct generating functions for negative order Changhee numbers and polynomials. Using these generating functions with their functional equation, we prove computation formulas for combinatorial numbers and polynomials. These formulas include Euler numbers and polynomials of higher order, Stirling numbers, and negative order Changhee numbers and polynomials. We also give some properties of these numbers and polynomials with their generating functions. Moreover, we give relations among Changhee numbers and polynomials of negative order, combinatorial numbers and polynomials and Bernoulli numbers of the second kind. Finally, we give a partial derivative of an equation for generating functions for Changhee numbers and polynomials of negative order. Using these differential equations, we derive recurrence relations, differential and integral formulas for these numbers and polynomials. We also give p-adic integrals representations for negative order Changhee polynomials including Changhee numbers and Deahee numbers.

Multiparameter Higher Order Daehee and Bernoulli Numbers and Polynomials

This paper gives a new generalization of higher order Daehee and Bernoulli numbers and polynomials. We define the multiparameter higher order Daehee numbers and polynomials of the first and second kind. Moreover, we derive some new results for these numbers and polynomials. The relations between these numbers and Stirling and Bernoulli numbers are obtained. Furthermore, some interesting special cases of the generalized higher order Daehee and Bernoulli numbers and polynomials are deduced.

Contour Integration Underlies Fundamental Bernoulli Number Recurrence

One solution to a relatively recent American Mathematical Monthly problem, requesting the evaluation of a real definite integral, could be couched in terms of a contour integral which vanishes a priori. While the required real integral emerged on setting to zero the real part of the contour quadrature, the obligatory, simultaneous vanishing of the imaginary part alluded to still another pair of real integrals forming the first two entries in the infinite log-sine sequence, known in its entirety. It turns out that identical reasoning, utilizing the same contour but a slightly different analytic function thereon, sufficed not only to evaluate that sequence anew, on the basis of a vanishing real part, but also, in setting to zero its conjugate imaginary part, to recover the fundamental Bernoulli number recurrence. The even order Bernoulli numbers $B_{2k}$ entering therein were revealed on the basis of their celebrated connection to Riemann,’s zeta function $\zeta(2k)$. Conversely, by permitting the related Bernoulli polynomials to participate as integrand factors, Euler’s connection itself received an independent demonstration, accompanied anew by an elegant log-sine evaluation, alternative to that already given. And, while the Bernoulli recurrence is intended to enjoy here the pride of place, this note ends on a gloss wherein all the motivating real integrals are recovered yet again, and in quite elementary terms, from the Fourier series into which the Taylor development for $\text{Log}(1-z)$ blends when its argument $z$ is restricted to the unit circle.

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 and Euler numbers of higher order

Some theorems on Bernoulli and Euler numbers of higher order

A Note on Bernoulli and Euler Numbers of Order ± p

A Note on Bernoulli and Euler Numbers of Order ± p