共查询到20条相似文献,搜索用时 15 毫秒
1.
Makoto Ishibashi 《The Ramanujan Journal》2000,4(2):221-229
In this paper we shall construct multiple analogue of the cotangent functions by using the multiple Hurwitz zeta functions and study their properties and special values. In particular, we express the double cotangent functions in terms of generalized eta functions of Berndt and Lewittes. 相似文献
2.
In this paper, we establish an approximate functional equation for the Lerch zeta function, which is a generalization of the Riemann zeta function and the Hurwitz zeta function. 相似文献
3.
H. Mishou 《Lithuanian Mathematical Journal》2007,47(1):32-47
In this paper, we investigate the joint value-distribution for the Riemann zeta function and Hurwitz zeta function attached
with a transcendental real parameter. Especially, we establish the joint universality theorem for these two zeta functions.
Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 39–57, January–March, 2007. 相似文献
4.
Text
We give series expansions for the Barnes multiple zeta functions in terms of rational functions whose numerators are complex-order Bernoulli polynomials, and whose denominators are linear. We also derive corresponding rational expansions for Dirichlet L-functions and multiple log gamma functions in terms of higher order Bernoulli polynomials. These expansions naturally express many of the well-known properties of these functions. As corollaries many special values of these transcendental functions are expressed as series of higher order Bernoulli numbers.Video
For a video summary of this paper, please click here or visit http://youtu.be/2i5PQiueW_8. 相似文献5.
Mohammed Mesk 《Integral Transforms and Special Functions》2018,29(5):402-416
It is shown that the main inequality for several special functions derived in [Masjed-Jamei M. A main inequality for several special functions. Comput Math Appl. 2010;60:1280–1289] can be put in a concise form, and that the main inequalities of the first kind Bessel function, Laplace and Fourier transforms are not valid as presented in the aforementioned paper. To provide alternative inequalities, we give a generalization, being in some cases an improvement, of the Cauchy–Bunyakovsky–Schwarz inequality which can be applied to real functions not necessarily of constant sign. The corresponding discrete inequality is also obtained, which we use to improve the inequalities of the Riemann zeta and the generalized Hurwitz–Lerch zeta functions. Finally, from the main concise inequality, we derive a Turán-type inequality. 相似文献
6.
Min-Soo Kim 《Applied Mathematics Letters》2011,24(1):55-61
In this work we obtain a new approach to closed expressions for sums of products of Bernoulli numbers by using the relation of values at non-positive integers of the important representation of the multiple Hurwitz zeta function in terms of the Hurwitz zeta function. 相似文献
7.
By expressing the sums of products of the Apostol?CBernoulli polynomials in terms of the special values of multiple Hurwitz?CLerch zeta functions at non-positive integers, we obtain the sums of products identity for the Apostol?CBernoulli numbers which is an analogue of the classical sums of products identity for Bernoulli numbers dating back to Euler. 相似文献
8.
Yasushi Komori 《The Ramanujan Journal》2008,17(2):163-183
A surface integral representation of the Mordell-Tornheim double zeta function is given, which is a direct analogue of a well-known
integral representation of the Riemann zeta function of Hankel’s type. As an application, we investigate its values and residues
at integers, where generalizations of a generating function of Bernoulli numbers naturally appear.
相似文献
9.
Miomir S. Stanković Mirjana V. Vidanović Slobodan B. Tricković 《Journal of Computational Analysis and Applications》2003,5(3):313-331
In this paper we consider trigonometric series in terms of the Riemann zeta function and related functions of reciprocal powers. The obtained closed form formulas we apply to the evaluation of the Riemann zeta function and related functions of reciprocal powers. One can establish recursive relations for them and relations between any two of those functions. These closed formulas enable us also to find sums of some Schlömilch series. We give an example which shows how the convergence of a trigonometric series can be accelerated by applying Krylov's method and our formula (7). 相似文献
10.
Djurdje Cvijovi? 《Applied mathematics and computation》2012,218(12):6744-6747
It is demonstrated that the alternating Lipschitz-Lerch zeta function and the alternating Hurwitz zeta function constitute a discrete Fourier transform pair. This discrete transform pair makes it possible to deduce, as special cases and consequences, many (mainly new) transformation relations involving the values at rational arguments of alternating variants of various zeta functions, such as the Lerch and Hurwitz zeta functions and Legendre chi function. 相似文献
11.
Hidehiko Mishou 《Archiv der Mathematik》2008,90(3):230-238
In the previous paper [9] the author proved the joint limit theorem for the Riemann zeta function and the Hurwitz zeta function
attached with a transcendental real number. As a corollary, the author obtained the joint functional independence for these
two zeta functions. In this paper, we study the joint value distribution for the Riemann zeta function and the Hurwitz zeta
function attached with an algebraic irrational number. Especially we establish the weak joint functional independence for
these two zeta functions.
Received: 17 Apri1 2007 相似文献
12.
We discuss zeta extensions in the sense of Kurokawa and Wakayama, Proc. Japan Acad. 2002, for constructing new zeta functions from a given zeta function. This notion appeared when we introduced higher zeta functions such as higher Riemann zeta functions in Kurokawa et al., Kyushu Univ. Preprint, 2003, and a higher Selberg zeta functions in Kurokawa and Wakayama, Comm. Math. Phys., 2004. In this article, we first recall some explicit examples of such zeta extensions and give a conjecture about functional equations satisfied by higher zeta functions. We devote the second part to making a detailed study of the double sine functions which are treated in a framework of the zeta extensions.Mathematics Subject Classification (2000) 11M36.Partially supported by Grant-in-Aid for Scientific Research (B) No. 15340012, and by Grant-in-Aid for Exploratory Research No. 13874004. This is based on the talk at The 2002 Twente Conference on Lie Groups 16–18 Dec. University of Twente, Enschede, The Netherlands. 相似文献
13.
Necdet Batir 《Proceedings Mathematical Sciences》2008,118(4):495-503
We establish various new inequalities for the Hurwitz zeta function. Our results generalize some known results for the polygamma
functions to the Hurwitz zeta function. 相似文献
14.
We show that the sum of the series formed by the so-called hyperharmonic numbers can be expressed in terms of the Riemann zeta function. These results enable us to reformulate Euler's formula involving the Hurwitz zeta function. In additon, we improve Conway and Guy's formula for hyperharmonic numbers. 相似文献
15.
Sergei K. Suslov 《Journal of Approximation Theory》2002,115(2):289-353
We consider explicit expansions of some elementary and q-functions in basic Fourier series introduced recently by Bustoz and Suslov. Natural q-extensions of the Bernoulli and Euler polynomials, numbers, and the Riemann zeta function are discussed as a by-product. 相似文献
16.
Linas Vepštas 《Numerical Algorithms》2008,47(3):211-252
This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies
this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension
of the techniques given by Borwein’s “An efficient algorithm for computing the Riemann zeta function” by Borwein for computing
the Riemann zeta function, to more general series. The algorithm provides a rapid means of evaluating Li
s
(z) for general values of complex s and a kidney-shaped region of complex z values given by ∣z
2/(z–1)∣<4. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be
extended to the entire complex z-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex s and z values. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler–Maclaurin series. The polylogarithm
and the Hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other; thus, either
algorithm can be used to evaluate either function. The Euler–Maclaurin series is a clear performance winner for the Hurwitz
zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms
are superior to the simple Taylor’s series or direct summation. The primary, concrete result of this paper is an algorithm
allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion
of the monodromy group of the polylogarithm is included.
相似文献
17.
Slobodan B. Tričković Miomir S. Stanković Mirjana V. Vidanović 《Integral Transforms and Special Functions》2020,31(5):339-367
ABSTRACTSchlömilch's series is named after the German mathematician Oscar Xavier Schlömilch, who derived it in 1857 as a Fourier series type expansion in terms of the Bessel function of the first kind. However, except for Bessel functions, here we consider an expansion in terms of Struve functions or Bessel and Struve integrals as well. The method for obtaining a sum of Schlömilch's series in terms of the Bessel or Struve functions is based on the summation of trigonometric series, which can be represented in terms of the Riemann zeta and related functions of reciprocal powers and in certain cases can be brought in the closed form, meaning that the infinite series are represented by finite sums. By using Krylov's method we obtain the convergence acceleration of the trigonometric series. 相似文献
18.
Yilmaz Simsek 《Journal of Mathematical Analysis and Applications》2006,318(1):333-351
The main purpose of this paper is to define new generating functions. By applying the Mellin transformation formula to these generating functions, we define q-analogue of Riemann zeta function, q-analogue Hurwitz zeta function, q-analogue Dirichlet L-function and two-variable q-L-function. In particular, by using these generating functions, we will construct new generating functions which produce q-Dedekind type sums and q-Dedekind type sums attached to Dirichlet character. We also give the relations between these sums and Dedekind sums. Furthermore, by using *-product which is given in this paper, we will give the relation between Dedekind sums and q-L function as well. 相似文献
19.
《Indagationes Mathematicae》2017,28(1):84-90
In a recent paper, Byrnes et al. (2014) have developed some recurrence relations for the hypergeometric zeta functions. Moreover, the authors made two conjectures for arithmetical properties of the denominators of the reduced fraction of the hypergeometric Bernoulli numbers. In this paper, we prove these conjectures using some recurrence relations. Furthermore, we assert that the above properties hold for both Carlitz and Howard numbers. 相似文献
20.
Piyush Kumar Bhandari Sushil Kumar Bissu 《Integral Transforms and Special Functions》2016,27(12):956-964
The aim of this paper is to establish the Turán-type inequalities for Struve functions, modified Struve functions, Anger–Weber functions and Hurwitz zeta function, by using a new form of the Cauchy–Bunyakovsky–Schwarz inequality. 相似文献