Twisted (h,q)-Bernoulli numbers and polynomials related to twisted (h,q)-zeta function and L-function☆

By Yilmaz Simsek. 2006

157 citations recorded by the source corpus.

Abstract In this paper, by using q-Volkenborn integral, we construct new generating functions of the new twisted ( h , q ) -Bernoulli polynomials and numbers. By applying the Mellin transformation to these generating functions, we obtain integral representations of the new twisted ( h , q ) -zeta function and twisted ( h , q ) -L-function, which interpolate the twisted ( h , q ) -Bernoulli numbers and generalized twisted ( h , q ) -Bernoulli numbers at non-positive integers, respectively. Furthermore, relation between twisted ( h , q ) -zeta function and twisted ( h , q ) -L-function are proved. Some new relations, related to twisted ( h , q ) -Bernoulli polynomials and numbers, are given.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/70947d3bc81a69afe7216a72823b2a74324730c9

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In Algebra and Number Theory

In Applied Mathematics

Citations in the corpus

A New Generating Function of (q-) Bernstein-Type Polynomials and Their Interpolation Function (2010, background): “. These polynomials are related to the many branches of Mathematics. By substituting v = 1 into the above, we have F(t) = tetx et − 1 = X∞ n=1 B n tn n! , where B n is usual Bernoulli polynomials cf. [18]. The usual second kind Stirling numbers with parameters (n,k), denote by S(n,k), that is the nu”
Special functions related to Dedekind-type DC-sums and their applications (2009, methodology): “n=0 Bn(z)t /n!, |t| < 2π, where Bn(0) = Bn stands for the Bernoulli numbers ([1–64]).”
On p-adic twisted q-L-functions related to generalized twisted Bernoulli numbers (2006, methodology): “In [ 25 ], the author defined twisted (h, q)-Bernoulli numbers, zeta functions, and an L-function.”
On the multiple q-Genocchi and Euler numbers (2008, background): “This distribution yields the p-adic bosonic q-integral by the rule Iq(f) = ∫ Zp f(x) dμq(x) = ∫ X f(x) dμq(x) (see [12, 17, 18]), where f ∈ UD(Zp) = the space of uniformly differentiable functions on Zp with values in Cp.”
Complete sum of products of (h, q)-extension of Euler polynomials and numbers (2010, methodology): “Recently, twisted (h,q)-Bernoulli and Euler numbers and polynomials were studied by several authors cf.(see [15], [4], [20], [21], [ 22 ], [14], [13], [16]).”
q-Extensions of Some Relationships Between the Bernoulli and Euler Polynomials (2011, background): “[27], Simsek ([28, 29] and [30]) and Srivastava et al.”
On the Generating Function for Bernstein Polynomials (2010, unknown): “([1], [5], [8], [9]).”
New approach to the complete sum of products of the twisted (h, q)-Bernoulli numbers and polynomials (2007, methodology): “Twisted version of Witt’s formula for B (h) n,w(z, q) is given by the following theorem [25]: Theorem 2.”
Symmetry p-adic invariant integral on ℤ p for Bernoulli and Euler polynomials (2007, methodology): “ n, where B n are the n-th ordinary Bernoulli numbers. In the recent the p-adic invariant integral on Z p are studied by several researchers in the area of number theory and mathematical physics (see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]). In [23”
Euler Numbers and Polynomials Associated with Zeta Functions (2008, methodology): “ i [i]q , cf.[ 13, 17, 31]. In complex plane, the ordinary Bernoulli numbers are a sequence of signed rational numbers that can be deﬁned by the identity t et −1 = X∞ n=0 Bn tn n! , |t| < 2π , cf.[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 2”