q-Hardy-Berndt type sums associated with q-Genocchi type zeta and q-l-functions

The aim of this paper is to define new generating functions. By applying a derivative operator and the Mellin transformation to these generating functions, we define q-analogue of the Genocchi zeta function, q-analogue Hurwitz type Genocchi zeta function, and q-Genocchi type l-function. We define partial zeta function. By using this function, we construct p-adic interpolation functions which interpolate generalized q-Genocchi numbers at negative integers. We also define p-adic meromorphic functions on C_p. Furthermore, we construct new generating functions of q-Hardy-Berndt type sums and q-Hardy-Berndt type sums attached to Dirichlet character. We also give some new relations, related to these sums.     

Markov measures and extended zeta functions

In this paper we study a family of representations of the Cuntz algebras O _p where p is a prime. These algebras are built on generators and relations. They are C ^?-algebras and their representations are a part of non-commutative harmonic analysis. Starting with specific generators and relations we pass to an ambient C ^?-algebra, for example in one of the Cuntz-algebras. Our representations are motivated by the study of frequency bands in signal processing: We construct induced measures attached to those representations which turned out to be related to a class of zeta functions. For a particular case those measures give rise to a class of Markov measures and q-Bernoulli polynomials. Our approach is amenable to applications in problems from dynamics and mathematical physics: We introduce a deformation parameter q, and an associated family of q-relations where the number q is a ??quantum-deformation,?? and also a parameter in a scale of (Riemann-Ruelle) zeta functions. Our representations are used in turn in a derivation of formulas for this q-zeta function.     

Multiple Dedekind zeta functions and evaluations of extended multiple zeta values

We define the number field analog of the zeta function of d-complex variables studied by Zagier in (First European Congress of Mathematics, vol. II (Paris, 1992), Progress in Mathematics, vol. 120, Birkhauser, Basel, 1994, pp. 497-512). We prove that in certain cases this function has a meromorphic continuation to C^d, and we identify the linear subvarieties comprising its singularities. We use our approach to meromorphic continuation to prove that there exist infinitely many values of these functions at regular points in their extended domains which can be expressed as a rational linear combination of values of the Dedekind zeta function.     

On asymptotics of the q-exponential and q-gamma functions

In this work we present a derivation for the complete asymptotic expansions of Euler?s q-exponential function and Jackson?s q-gamma function via Mellin transform. These formulas are valid everywhere, uniformly on any compact subset of the complex plane.     

Multiple zeta values for classical special functions

The Ramanujan Journal - For primes $$p>3$$ we give a formula for the traces of pth Hecke operators on space of cusp forms of weight k and level 1 involving p-adic hypergeometric functions....     

q-Functions and extreme topological measures

q-Functions provide a method for constructing topological measures. We give necessary and sufficient conditions for a composition of a q-function and a topological measure to be a topological measure. Regular and extreme step q-functions are characterized by certain regions in Rn. Then extreme q-functions are used to study extreme topological measures. For example, we prove (under some assumptions on the underlying set) that given n, there are different types of extreme topological measures with values 0,1/n,…,1. In contrast, in the case of measures the only extreme points are {0,1}-valued, i.e., point masses.     

On asymptotics of q-Gamma functions

In this paper we derive some asymptotic formulas for the q-Gamma function Γq(z) for q tending to 1.     

Note on the Euler q-zeta functions

We consider the q-analogue of the Euler zeta function which is defined by

     

Partial zeta functions

In the present paper, we study analytic properties of the zeta functions defined by the Euler products over elements in subsets of the set of prime elements.     

Multiple zeta values

The definition of multiple zeta values is extended in the paper. The preservation of the main properties known for multiple zeta values in the sense of their classic definition is proved.     

On the qth convergence exponent of entire functions

In this paper we obtain a growth relation for entire functions of qth order with respect to the distribution of its zeros. We also derive certain relations between the qth convergence exponents of two or more entire functions. The most striking result of the paper is: If f(z) has at least one zero, then

$\text{lim sup}\text{r→∞}\text{log n}\text{(r)}\text{log}{}^{\mathrm{[q+1]}}{}_{\mathrm{r}}\text{=?(q)}$

, where n(r) is the number of zeros of f(z) in $\text{¦}\text{z}\text{¦ ? r}$ and

$\text{?(q)=g.l.b.}\text{α:α>0 and}\text{∑}\text{∞}\text{n=1}\text{(log}{}^{\mathrm{[q]}}\text{r}{}_{\mathrm{n}}\text{)}{}^{\mathrm{-\alpha }}\text{<∞}$

.     

Multiple zeta functions and asymptotic structure of free Abelian groups of finite rank

We define the multiple zeta function of the free Abelian group Z^d as

ζ_Z^d(s₁,…,s_d)=∑_|Z^d:H|<∞α₁(H)^−s₁?α_d(H)^−s_d,     

Shintani-Barnes zeta and gamma functions

We show that Shintani's work on multiple zeta and gamma functions can be simplified and extended by exploiting difference equations. We re-prove many of Shintani's formulas and prove several new ones. Among the latter is a generalization to the Shintani-Barnes gamma functions of Raabe's 1843 formula , and a further generalization to the Shintani zeta functions. These explicit formulas can be interpreted as “vanishing period integral” side conditions for the ladder of difference equations obeyed by the multiple gamma and zeta functions. We also relate Barnes’ triple gamma function to the elliptic gamma function appearing in connection with certain integrable systems.     

Multiple zeta values and ideles

The Ramanujan Journal - In this paper we give two idelic representations of the multiple zeta values—one using iterated integrals over the finite ideles and the other using iterated integrals...