scholarly journals On the Application of the Plan Formula to the Study of the Zeta-Function of Zeros of Entire Function

2020 ◽  
pp. 135-140
Author(s):  
Vyacheslav I. Kuzovatov ◽  
Alexander M. Kytmanov ◽  
Azimbai Sadullaev
Keyword(s):  
Entire Function ◽  
Integral Representation ◽  
Zeta Function ◽  
Explicit Expression

We consider an application of the Plan formula to the study of the properties of the zeta- function of zeros of entire function. Based on this formula, we obtained an explicit expression for the kernel of the integral representation of the zeta-function in this case

scholarly journals On the Zeta-Function of Zeros of Entire Function

2021 ◽  
Vol 14 (5) ◽  
pp. 599-603
Author(s):  
Vyacheslav I. Kuzovatov ◽  
◽  
Alexander M. Kytmanov ◽  
Azimbai Sadullaev ◽  
Keyword(s):  
Entire Function ◽  
Integral Representation ◽  
Zeta Function ◽  
Explicit Expression

This article is devoted to the study of the properties of the zeta-function of zeros of entire function. We obtained an explicit expression for the kernel of the integral representation of the zetafunction in one case

scholarly journals An extension ofq-zeta function

2004 ◽  
Vol 2004 (49) ◽  
pp. 2649-2651 ◽  
Author(s):  
T. Kim ◽  
L. C. Jang ◽  
S. H. Rim
Keyword(s):  
Integral Representation ◽  
Zeta Function ◽  
Hurwitz Zeta Function

We will define the extension ofq-Hurwitz zeta function due to Kim and Rim (2000) and study its properties. Finally, we lead to a useful new integral representation for theq-zeta function.

scholarly journals Fourier expansion and integral representation generalized Apostol-type Frobenius–Euler polynomials

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Alejandro Urieles ◽  
William Ramírez ◽  
María José Ortega ◽  
Daniel Bedoya
Keyword(s):  
Fourier Series ◽  
Integral Representation ◽  
Zeta Function ◽  
Fourier Expansion ◽  
Series Representation ◽  
Hurwitz Zeta Function ◽  
Euler Polynomials ◽  
Genocchi Polynomials ◽  
Lerch Zeta Function ◽  
Rational Arguments

Abstract The main purpose of this paper is to investigate the Fourier series representation of the generalized Apostol-type Frobenius–Euler polynomials, and using the above-mentioned series we find its integral representation. At the same time applying the Fourier series representation of the Apostol Frobenius–Genocchi and Apostol Genocchi polynomials, we obtain its integral representation. Furthermore, using the Hurwitz–Lerch zeta function we introduce the formula in rational arguments of the generalized Apostol-type Frobenius–Euler polynomials in terms of the Hurwitz zeta function. Finally, we show the representation of rational arguments of the Apostol Frobenius Euler polynomials and the Apostol Frobenius–Genocchi polynomials.

scholarly journals Spectral Theory from the Second-Orderq-Difference Operator

2007 ◽  
Vol 2007 ◽  
pp. 1-14 ◽  
Author(s):  
Lazhar Dhaouadi
Keyword(s):  
Integral Representation ◽  
Zeta Function ◽  
Spectral Theory ◽  
Spectral Measure ◽  
Resolvent Operator ◽  
Difference Operator ◽  
Poincaré Inequality ◽  
Second Order ◽  
Poincare Inequality ◽  
The Resolvent Operator

Spectral theory from the second-orderq-difference operatorΔqis developed. We give an integral representation of its inverse, and the resolvent operator is obtained. As application, we give an analogue of the Poincare inequality. We introduce the Zeta function for the operatorΔqand we formulate some of its properties. In the end, we obtain the spectral measure.

scholarly journals An estimate for the Mellin transform of powers of Hardy's function.

2010 ◽  
Vol Volume 33 ◽  
Author(s):  
Matti Jutila
Keyword(s):  
Entire Function ◽  
Zeta Function ◽  
Mellin Transform ◽  
Critical Line ◽  
Classical Estimate ◽  
International Audience ◽  
Hardy’S Function

International audience We show that a certain modified Mellin transform $\mathcal M(s)$ of Hardy's function is an entire function. There are reasons to connect $\mathcal M(s)$ with the function $\zeta(2s-1/2)$, and then the orders of $\mathcal M(s)$ and $\zeta(s)$ should be comparable on the critical line. Indeed, an estimate for $\mathcal M(s)$ is proved which in the particular case of the critical line coincides with the classical estimate of the zeta-function.

scholarly journals A new derivation of $\zeta(-r)=-\frac{B_{r+1}}{r+1}$

2019 ◽  
Author(s):  
Sumit Kumar Jha
Keyword(s):  
Integral Representation ◽  
Zeta Function ◽  
Explicit Formula ◽  
Riemann Zeta Function ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Master Theorem

In this note, we give a new derivation for the fact that $\zeta(-r)=-\frac{B_{r+1}}{r+1}$ where $\zeta(s)$ represents the Riemann zeta function, and $B_{r}$ represents the Bernoulli numbers. Our proof uses the well-known explicit formula for the Bernoulli numbers in terms of the Stirling numbers of the second kind, and the Ramanujan's master theorem to obtain an integral representation for the Riemann zeta function.

scholarly journals The Newtonian Potential and the Demagnetizing Factors of the General Ellipsoid

2017 ◽  
Author(s):  
Giovanni Di Fratta
Keyword(s):  
Integral Representation ◽  
Explicit Expression ◽  
Representation Formula ◽  
Newtonian Potential ◽  
Bounded Domains ◽  
Dimensional Problem ◽  
Smooth Functions ◽  
Integral Representation Formula ◽  
Transport Theorem ◽  
Stokes Integral

The objective of this paper is to present a modern and concise new derivation for the explicit expression of the interior and exterior Newtonian potential generated by homogeneous ellipsoidal domains in $\mathbb{R}^N$ (with $N \geqslant 3$). The very short argument is essentially based on the application of Reynolds transport theorem in connection with Green-Stokes integral representation formula for smooth functions on bounded domains of$\mathbb{R}^N$, which permits to reduce the N-dimensional problem to a 1-dimensional one. Due to its physical relevance, a separate section is devoted to the derivation of the demagnetizing factors of the general ellipsoid which are one of the most fundamental quantities in ferromagnetism.

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