scholarly journals Some curious results related to a conjecture of Strohmer and Beaver

2020 ◽  
pp. 1-29
Author(s):  
Markus Faulhuber
Keyword(s):  
Special Functions ◽  
Theta Functions ◽  
Zeta Functions ◽  
Analytic Number Theory ◽  
Square Lattice ◽  
Study Results ◽  
Carry Over ◽  
Lower Frame ◽  
Frame Set ◽  
Frame Bound

We study results related to a conjecture formulated by Strohmer and Beaver about optimal Gaussian Gabor frame set-ups. Our attention will be restricted to the case of Gabor systems with standard Gaussian window and rectangular lattices of density 2. Although this case has been fully treated by Faulhuber and Steinerberger, the results in this work are new and quite curious. Indeed, the optimality of the square lattice for the Tolimieri and Orr bound already implies the optimality of the square lattice for the sharp lower frame bound. Our main tools include determinants of Laplace–Beltrami operators on tori as well as special functions from analytic number theory, in particular Eisenstein series, zeta functions, theta functions and Kronecker’s limit formula. We note that our results also carry over to energy minimization problems over lattices and a heat distribution problem over flat tori.

scholarly journals A Survey of Some Recent Developments on Higher Transcendental Functions of Analytic Number Theory and Applied Mathematics

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2294
Author(s):  
Hari Mohan Srivastava
Keyword(s):  
Number Theory ◽  
Special Functions ◽  
Applied Mathematics ◽  
Analytic Number Theory ◽  
Review Article ◽  
Mathematical Functions ◽  
Analytic Number ◽  
Recent Developments ◽  
Transcendental Functions ◽  
Mathematical Sciences

Often referred to as special functions or mathematical functions, the origin of many members of the remarkably vast family of higher transcendental functions can be traced back to such widespread areas as (for example) mathematical physics, analytic number theory and applied mathematical sciences. Here, in this survey-cum-expository review article, we aim at presenting a brief introductory overview and survey of some of the recent developments in the theory of several extensively studied higher transcendental functions and their potential applications. For further reading and researching by those who are interested in pursuing this subject, we have chosen to provide references to various useful monographs and textbooks on the theory and applications of higher transcendental functions. Some operators of fractional calculus, which are associated with higher transcendental functions, together with their applications, have also been considered. Many of the higher transcendental functions, especially those of the hypergeometric type, which we have investigated in this survey-cum-expository review article, are known to display a kind of symmetry in the sense that they remain invariant when the order of the numerator parameters or when the order of the denominator parameters is arbitrarily changed.

scholarly journals The Laplace equation in the exterior of the Hankel contour and novel identities for hypergeometric functions

2013 ◽  
Vol 469 (2157) ◽  
pp. 20130081 ◽  
Author(s):  
A. S. Fokas ◽  
M. L. Glasser
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Laplace Equation ◽  
Special Functions ◽  
Boundary Value ◽  
Zeta Functions ◽  
Neumann Boundary ◽  
Neumann Boundary Value Problem ◽  
Wide Range ◽  
Definition Of

By using conformal mappings, it is possible to express the solution of certain boundary-value problems for the Laplace equation in terms of a single integral involving the given boundary data. We show that such explicit formulae can be used to obtain novel identity for special functions. A convenient tool for deriving this type of identity is the so-called global relation , which has appeared recently in a wide range of boundary-value problems. As a concrete application, we analyse the Neumann boundary-value problem for the Laplace equation in the exterior of the Hankel contour, which appears in the definition of both the gamma and the Riemann zeta functions. By using the explicit solution of this problem, we derive a number of novel identities involving the hypergeometric function. Also, we point out an interesting connection between the solution of the above Neumann boundary-value problem for a particular set of Neumann data and the Riemann hypothesis.

scholarly journals Transport of Solid Particles with Fluid Flow in Rotating Pipes

2019 ◽  
Vol 9 (2) ◽  
pp. 201-206
Author(s):  
F.Sh. Zabirov ◽  
B.M. Latypov ◽  
R.G. Sharafiev ◽  
R.A. Gilmanshin
Keyword(s):  
Fluid Flow ◽  
Experimental Studies ◽  
Equation System ◽  
Solid Particles ◽  
Sand Transport ◽  
Operating Life ◽  
Surface Type ◽  
Horizontal Pipe ◽  
Study Results ◽  
Carry Over

Abstract The article addresses the recent problem of borehole lifting of oil containing sand solids. The presence of sand in oil produced results in a reduced operating life of downhole equipment. The problem of preventing sanding up and sand formation in pumping equipment may be solved and stable sand production may be ensured by producing oil using borehole screw pumps with a surface-type drive, in which the screw is rotated by rotating hollow rods. Rotating hollow rods improve carry-over of sand particles to the surface with rotational oil flow by imparting additional momentum to these particles. Rotational variables of the pipe (cylinder) that enables transport of solids are set only for the air flow moving in a horizontal pipe (cylinder). The purpose of the study is to establish pipe rotational variables in directional wells that enable stable sand transport with fluid flow. Work results have been obtained from numerical studies using the differential equation system and rules of theoretical solid movement, computer simulation and experimental results processing at a laboratory facility. Theoretical study results have been acknowledged by experimental studies. The work establishes the criteria that allow defining the speed range of directional hollow rods that enables carry-over of solids to the surface with fluid flow. Study results may be used to produce oil with submersible screw pumps with a surface-type drive that use hollow sucker rods for pump down.

scholarly journals HYPERGEOMETRIC ZETA FUNCTIONS

2010 ◽  
Vol 06 (01) ◽  
pp. 99-126 ◽  
Author(s):  
ABDUL HASSEN ◽  
HIEU D. NGUYEN
Keyword(s):  
Integral Representation ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Special Functions ◽  
Zeta Functions ◽  
Functional Inequality ◽  
Bernoulli Numbers ◽  
Classical Counterpart ◽  
New Family ◽  
Riemann Zeta

This paper investigates a new family of special functions referred to as hypergeometric zeta functions. Derived from the integral representation of the classical Riemann zeta function, hypergeometric zeta functions exhibit many properties analogous to their classical counterpart, including the intimate connection to Bernoulli numbers. These new properties are treated in detail and are used to demonstrate a functional inequality satisfied by second-order hypergeometric zeta functions.

scholarly journals Indefinite zeta functions

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Gene S. Kopp
Keyword(s):  
Quadratic Forms ◽  
Theta Functions ◽  
Zeta Functions ◽  
Real Quadratic Fields ◽  
Taylor Coefficients ◽  
Dimension 2 ◽  
Complex Symmetric ◽  
Complex Symmetric Matrices ◽  
Indefinite Quadratic ◽  
Real Quadratic

AbstractWe define generalised zeta functions associated with indefinite quadratic forms of signature $$(g-1,1)$$ ( g - 1 , 1 ) —and more generally, to complex symmetric matrices whose imaginary part has signature $$(g-1,1)$$ ( g - 1 , 1 ) —and we investigate their properties. These indefinite zeta functions are defined as Mellin transforms of indefinite theta functions in the sense of Zwegers, which are in turn generalised to the Siegel modular setting. We prove an analytic continuation and functional equation for indefinite zeta functions. We also show that indefinite zeta functions in dimension 2 specialise to differences of ray class zeta functions of real quadratic fields, whose leading Taylor coefficients at $$s=0$$ s = 0 are predicted to be logarithms of algebraic units by the Stark conjectures.

scholarly journals A Quadruple Integral Involving Product of the Struve Hv(βt) and Parabolic Cylinder Du(αx) Functions

Symmetry ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 9
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Zeta Function ◽  
Special Functions ◽  
Zeta Functions ◽  
Parabolic Cylinder ◽  
Fundamental Constants ◽  
Lerch Zeta Function ◽  
Parabolic Cylinder Functions ◽  
Infinite Integral ◽  
Almost All

The objective of the present paper is to obtain a quadruple infinite integral. This integral involves the product of the Struve and parabolic cylinder functions and expresses it in terms of the Hurwitz–Lerch Zeta function. Almost all Hurwitz-Lerch Zeta functions have an asymmetrical zero distributionSpecial cases in terms fundamental constants and other special functions are produced. All the results in the work are new.

scholarly journals Definite Integrals Involving Combinations of Powers and Logarithmic Functions of Complicated Arguments Expressed in Terms of the Hurwitz Zeta Function

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Zeta Function ◽  
Special Functions ◽  
Zeta Functions ◽  
Hurwitz Zeta Function ◽  
Fundamental Constants ◽  
Closed Formula ◽  
Definite Integrals ◽  
Logarithmic Functions

In this manuscript, the authors derive closed formula for definite integrals of combinations of powers and logarithmic functions of complicated arguments and express these integrals in terms of the Hurwitz zeta functions. These derivations are then expressed in terms of fundamental constants, elementary, and special functions. A summary of the results is produced in the form of a table of definite integrals for easy referencing by readers.

scholarly journals The Ihara Zeta Function of the Infinite Grid

10.37236/3561 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Bryan Clair
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Cayley Graph ◽  
Theta Functions ◽  
Zeta Functions ◽  
Multivalued Function ◽  
Ihara Zeta Function ◽  
Grid Graphs ◽  
Infinite Grid ◽  
Limiting Value

The infinite grid is the Cayley graph of $\mathbb{Z} \times \mathbb{Z}$ with the usual generators. In this paper, the Ihara zeta function for the infinite grid is computed using elliptic integrals and theta functions. The zeta function of the grid extends to an analytic, multivalued function which satisfies a functional equation. The set of singularities in its domain is finite.The grid zeta function is the first computed example which is non-elementary, and which takes infinitely many values at each point of its domain. It is also the limiting value of the normalized sequence of Ihara zeta functions for square grid graphs and torus graphs.

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