This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we give a new symmetry identities for the generalized higher-order (h, q)-Bernoulli polynomials which are derive from the properties of symmetry of Volkenborn...

The Bernoulli polynomials Bk restricted to 0, 1 and extended by periodicity have nth sine and cosine Fourier coefficients of the formCk/n . In general, the Fourier coefficients of any polynomial restricted to 0, 1 are linear combinations of terms of the form 1/n . If we can make this linear combination explicit for a specific family of polynomials, then by uniqueness of Fourier series, we get a...

In the paper, the authors recall some known determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, find alternative determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, and present several new recurrence relations for the Bernoulli numbers and polynomials.

Komori, Matsumoto and Tsumura introduced a zeta function ?r (s, ?) associated with root system ?. In this paper, we introduce q-analogue of function, denoted by a, ?; q), investigate its properties. We show that ‘Weyl group symmetric' linear combination q) can be written as multiple integral over torus involving functions ?s. For positive integers k, ?k regarded q-analogues the periodic Bernoul...

As a generalization of 2D Bernoulli polynomials, neo-Bernoulli polynomials are introduced from a point of view involving the use of nonexponential generating functions. Their relevant recurrence relations, the differential equations satisfied by them and some other properties are obtained. Especially, we obtain the relationships between them and neo-Hermite polynomials. We also study some other...

In this paper, by using the Bernoulli numbers and exponential complete Bell polynomials, we establish four general asymptotic expansions for hyperfactorial functions ∏ k=1 n k q , which have only odd power terms or even terms. We derive recurrences parameter sequences in these give some special recurrences.

Six approaches to the theory of Bernoulli polynomials are known; these are associated with the names of J. Bernoulli [2], L. Euler [4], E. Lucas [8], P. E. Appell [1], A. Hürwitz [6] and D. H. Lehmer [7]. In this note we deal with a new determinantal definition for Bernoulli polynomials recently proposed by F. Costabile [3]; in particular, we emphasize some consequent procedures for automatic c...

Let q be regarded as either a complex number q ∈ C or a p-adic number q ∈ Cp. If q ∈ C, then we always assume |q| < 1. If q ∈ Cp, we normally assume |1− q|p < p − 1 p−1 , which implies that q = exp(x log q) for |x|p ≤ 1. Here, | · |p is the p-adic absolute value in Cp with |p|p = 1 p . The q-basic natural number are defined by [n]q = 1−q 1−q = 1 + q + · · · + q , ( n ∈ N), and q-factorial are a...