Let p be a fixed prime number. Throughout this paper, the symbols Z,Zp,Qp,C, and Cp will denote the ring of rational integers, the ring of p-adic integers, the field of p-adic rational numbers, the complex number field, and the completion of the algebraic closure of Qp, respectively. Let N be the set of natural numbers and Z N ∪ {0}. Let vp be the normalized exponential valuation of Cp with |p|...

Abstract Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying A ′ n +1 ( x ) = + 1) with 0 constant polynomial. This allows us to obtain in simple way some known relations involving Apostol-Bernoulli polynomials, Apostol-Euler and generalized Bernoulli attached primitive Dirichlet character.

We develop polynomials in z ∈ C for which some generalized harmonic numbers are special cases at z= 0. By using the Riordan array method, we explore interesting relationships between these polynomials, the generalized Stirling polynomials, the Bernoulli polynomials, the Cauchy polynomials and the Nörlund polynomials. © 2007 Elsevier B.V. All rights reserved.

graham higman has defined coset diagrams for psl(2,ℤ). these diagrams are composed of fragments, and the fragments are further composed of two or more circuits. q. mushtaq has proved in 1983 that existence of a certain fragment γ of a coset diagram in a coset diagram is a polynomial f in ℤ[z]. higman has conjectured that, the polynomials related to the fragments are monic and for a fixed degree...

Recently, several Bell based polynomials such as Bernoulli, Euler, Genocchi and Apostol versions were defined investigated. The main aim of this paper is to introduce the general family Appell polynomials, which includes many new members in addition existing ones, investigate their properties including determinantal representation, recurrence relation, derivative formula, shift operators differ...

We introduce and investigate the Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials by means of a suitable theirs generating polynomials. We establish several interesting properties of these polynomials. Also, we gave some propositions two theorems and one corollary.

The generalized mixed type Bernoulli-Gegenbauer polynomials of order α >−1/2 are special obtained by use the generating function method. These represent an interesting mixture between two classes functions, namely Bernoulli and Gegenbauer polynomials. main purpose this paper is to discuss some their algebraic analytic properties.

We consider the modified q-analogue of Riemann zeta function which is defined by ζq(s)= ∑∞ n=1(qn(s−1)/[n]s), 0< q < 1, s ∈ C. In this paper, we give q-Bernoulli numbers which can be viewed as interpolation of the above q-analogue of Riemann zeta function at negative integers in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers. Also, we will treat some...

is a well-known formula from classical analysis giving a relation between the finite sum of values of a function f , whose first m derivatives are absolutely integrable on [a, n], and its integral, for a,m, n ∈ N, a < n. This elementary formula appears often in introductory texts [2, 17], usually in reference to a particular application—Stirling’s asymptotic formula. However, general approaches...