Cangul-Ozden-Simsek[1] constructed the q-Genocchi numbers of high order using a fermionic p-adic integral on Zp, and gave Witt’s formula and the interpolation functions of these numbers. In this paper, we present the generalization of the higher order q-Euler numbers and q-Genocchi numbers of Cangul-Ozden-Simsek. We define q-extensions of w-Euler numbers and polynomials, and w-Genocchi numbers ...

Motivated by Kurt’s work [Filomat 30 (4) 921-927, 2016], we …rst consider a class of a new generating function for (p; q)-analog of Apostol type polynomials of order including Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials of order . By making use of their generating function, we derive some useful identities. We also introduce (p; q)-analog of Stirling numbers of second kind...

Every finite field has many multiplicative generators. However, finding one in polynomial time is an important open problem. In fact, even finding elements of high order has not been solved satisfactorily. In this paper, we present an algorithm that for any positive integer c and prime power q, finding an element of order exp(Ω( √ qc)) in the finite field Fq(qc−1)/(q−1) in deterministic time (q...

This paper presents explicit formulas and algorithms to compute the eigenvalues and eigenvectors of order-one-quasiseparable matrices. Various recursive relations for characteristic polynomials of their principal submatrices are derived. The cost of evaluating the characteristic polynomial of an N × N matrix and its derivative is only O(N). This leads immediately to several versions of a fast q...

Copyright q 2010 Lee-Chae Jang et al. 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. We investigate several arithmetic properties of h, q-Genocchi polynomials and numbers of higher order.

Let N∗(m) be the minimal length of a polynomial with ±1 coefficients divisible by (x − 1)m. Byrnes noted that N∗(m) ≤ 2m for each m, and asked whether in fact N∗(m) = 2m. Boyd showed that N∗(m) = 2m for all m ≤ 5, but N∗(6) = 48. He further showed that N∗(7) = 96, and that N∗(8) is one of the 5 numbers 96, 144, 160, 176, or 192. Here we prove that N∗(8) = 144. Similarly, let m∗(N) be the maxima...

Pólya’s enumeration theorem states that the number of labelings of a finite set up to symmetry is given by a polynomial in the number of labels. We give a new perspective on this theorem by generalizing it to partially ordered sets and order preserving maps. Further we prove a reciprocity statement in terms of strictly order preserving maps generalizing a classical result by Stanley (1970). We ...

Let p be a fixed prime number. Throughout this paper Zp, Qp, and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp. For x ∈ Cp, we use the notation x q 1 − q / 1 − q . Let UD Zp be the space of uniformly differentiable functions on Zp, and let vp be the normalized exponential valuation of Cp wi...

In this ‎article‎‎, ‎an ‎ap‎plied matrix method, which is based on Bernouli Polynomials, has been presented to find approximate solutions of ‎high order ‎Volterra ‎integro-differential‎ equations. Through utilizing this approach, the proposed equations reduce to a system of algebric equations with unknown Bernouli coefficients. A number of numerical ‎illustrations‎ have been ‎solved‎ to ‎assert...