scholarly journals Special functions and Gauss–Thakur sums in higher rank and dimension

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Quentin Gazda ◽  
Andreas Maurischat
Keyword(s):  
Generating Functions ◽  
Function Field ◽  
Special Function ◽  
Special Functions ◽  
General Construction ◽  
Period Lattice ◽  
Higher Rank ◽  
Algebraic Elements ◽  
Anderson Generating Functions

AbstractAnderson generating functions have received a growing attention in function field arithmetic in the last years. Despite their introduction by Anderson in the 1980s where they were at the heart of comparison isomorphisms, further important applications, e.g., to transcendence theory have only been discovered recently. The Anderson–Thakur special function interpolates L-values via Pellarin-type identities, and its values at algebraic elements recover Gauss–Thakur sums, as shown by Anglès and Pellarin. For Drinfeld–Hayes modules, generalizations of Anderson generating functions have been introduced by Green–Papanikolas and – under the name of “special functions” – by Anglès, Ngo Dac and Tavares Ribeiro. In this article, we provide a general construction of special functions attached to any Anderson A-module. We show direct links of the space of special functions to the period lattice, and to the Betti cohomology of the A-motive. We also undertake the study of Gauss–Thakur sums for Anderson A-modules, and show that the result of Anglès–Pellarin relating values of the special functions to Gauss–Thakur sums holds in this generality.

scholarly journals MODELING OF DISTURBANCES IN STUDY OF CONTROL SYSTEMS

2020 ◽  
Vol 2020 (4) ◽  
pp. 70-79
Author(s):  
Mikhail Vasilevich Lyakhovets ◽  
Georgiy Valentinovich Makarov ◽  
Alexandr Sergeevich Salamatin
Keyword(s):  
Control Systems ◽  
Generating Functions ◽  
Special Functions ◽  
Computer Training ◽  
Full Scale ◽  
Analytical Models ◽  
Scale Model ◽  
Data Series ◽  
Dynamic Databases ◽  
Natural Data

The article is devoted to questions of synthesis of full-scale - model realizations of data series on the basis of natural data for modeling of controllable and uncontrollable influences at research of operating and projected control systems, and also in training systems of computer training. The possibility of formation of model effects on the basis of joint use of multivariate dynamic databases and natural data simulator is shown. Dynamic databases store information that characterizes the typical representative situations of systems in the form of special functions - generating functions. Multiple variability of dynamic databases is determined by the type of the selected generating function, the methods of obtaining parameters (coefficients) of this function, as well as the selected accuracy of approximation. The situation models recovered by generating functions are used as basic components (trends) in the formation of the resulting full-scale - model implementations and are input into the natural data simulator. The data simulator allows for each variant of initial natural data to form an implementation of the perturbation signal with given statistical properties on a given simulation interval limited by the initial natural implementation. This is achieved with the help of a two-circuit structure, where the first circuit is responsible for evaluation and cor-rection of initial properties of the natural signal, and the second - for iterative correction of deviations of properties of the final implementation from the specified ones. The resulting realizations reflect the properties of their full-scale components, which are difficult to describe by analytical models, and are supplemented by model values, allowing in increments to correct the properties to the specified ones. The given approach allows to form set of variants of course of processes on the basis of one situation with different set degree of uncertainty and conditions of functioning.

The Special Function Field Sieve

2002 ◽  
Vol 16 (1) ◽  
pp. 81-98 ◽  
Author(s):  
Oliver Schirokauer
Keyword(s):  
Function Field ◽  
Special Function

scholarly journals Definite Integral of Algebraic, Exponential and Hyperbolic Functions Expressed in Terms of Special Functions

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1425
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Special Function ◽  
Special Functions ◽  
Complex Numbers ◽  
Definite Integral ◽  
Hyperbolic Functions ◽  
Closed Form Solutions ◽  
Definite Integrals ◽  
Arbitrary Complex ◽  
Famous Book

While browsing through the famous book of Bierens de Haan, we came across a table with some very interesting integrals. These integrals also appeared in the book of Gradshteyn and Ryzhik. Derivation of these integrals are not listed in the current literature to best of our knowledge. The derivation of such integrals in the book of Gradshteyn and Ryzhik in terms of closed form solutions is pertinent. We evaluate several of these definite integrals of the form ∫0∞(a+y)k−(a−y)keby−1dy, ∫0∞(a+y)k−(a−y)keby+1dy, ∫0∞(a+y)k−(a−y)ksinh(by)dy and ∫0∞(a+y)k+(a−y)kcosh(by)dy in terms of a special function where k, a and b are arbitrary complex numbers.

A general construction of mutually orthogonal latin squares of prime order

2018 ◽  
Vol 7 (1-2) ◽  
pp. 77-93
Author(s):  
J. A. Saka ◽  
O. O. Oyadare
Keyword(s):  
Structural Properties ◽  
Generating Function ◽  
Generating Functions ◽  
Prime Order ◽  
Latin Squares ◽  
General Construction ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Complete Set ◽  
General Method

This paper presents a general method of constructing a complete set of Mutually Orthogonal Latin Squares (MOLS) of the order of any prime, via the use of generating functions dened on the nite eld of this order. Apart from using the generating function to get a complete set of Mutually Orthogonal Latin Squares, the studies of the generating functions opens up the possibility of getting at the deep structural properties of MOLS. Copious examples were given for detailed illustrations.

scholarly journals Investigation of the k-Analogue of Gauss Hypergeometric Functions Constructed by the Hadamard Product

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 714
Author(s):  
Mohamed Abdalla ◽  
Muajebah Hidan
Keyword(s):  
Hypergeometric Functions ◽  
Hadamard Product ◽  
Special Function ◽  
Function Theory ◽  
Special Functions ◽  
Integral Transforms ◽  
Integral Representations ◽  
Convergence Properties ◽  
Pochhammer Symbol ◽  
Definition Of

Traditionally, the special function theory has many applications in various areas of mathematical physics, economics, statistics, engineering, and many other branches of science. Inspired by certain recent extensions of the k-analogue of gamma, the Pochhammer symbol, and hypergeometric functions, this work is devoted to the study of the k-analogue of Gauss hypergeometric functions by the Hadamard product. We give a definition of the Hadamard product of k-Gauss hypergeometric functions (HPkGHF) associated with the fourth numerator and two denominator parameters. In addition, convergence properties are derived from this function. We also discuss interesting properties such as derivative formulae, integral representations, and integral transforms including beta transform and Laplace transform. Furthermore, we investigate some contiguous function relations and differential equations connecting the HPkGHF. The current results are more general than previous ones. Moreover, the proposed results are useful in the theory of k-special functions where the hypergeometric function naturally occurs.

scholarly journals Lie theoretic origin of some generating functions of Fox’s H-Function

2012 ◽  
Vol 43 (2) ◽  
pp. 179-185
Author(s):  
D.K. Jain ◽  
Renu Jain
Keyword(s):  
Generating Functions ◽  
Special Functions ◽  
Addition Theorems ◽  
Matrix Elements ◽  
Powerful Technique ◽  
Group Theoretic Method ◽  
In Series ◽  
Special Case ◽  
Fox’S H Function ◽  
Basis Vectors

The group theoretic method for achieving unification of diverse mass of literature of special functions is most recent of such efforts and is definitely the most elegant one. In this method the special functions emerge as basis vectors and matrix elements of local multiplier representation of some well known groups. This dual role played by special functions affords a powerful technique for derivation of several generating functions and addition theorems for them. The present paper aims at harnessing this technique to generate, derive and interpret certain expansion of Fox's H-function in series of H-function. In the special case these expansions reduce to corresponding results for G-function.

scholarly journals Double and dual numbers. SU(2) groups, two-component spinors and generating functions

2020 ◽  
Vol 66 (4 Jul-Aug) ◽  
pp. 418
Author(s):  
G. F. Torres del Castillo ◽  
K. C. Gutiérrez-Herrera
Keyword(s):  
Laplace Equation ◽  
Generating Functions ◽  
Special Functions ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Dual Numbers ◽  
Unitary Matrices ◽  
Double Numbers ◽  
Two Component ◽  
Toroidal Coordinates

We explicitly show that the groups of $2 \times 2$ unitary matrices with determinant equal to 1 whose entries are double or dual numbers are homomorphic to ${\rm SO}(2,1)$ or to the group of rigid motions of the Euclidean plane, respectively, and we introduce the corresponding two-component spinors. We show that with the aid of the double numbers we can find generating functions for separable solutions of the Laplace equation in the $(2 + 1)$ Minkowski space, which contain special functions that also appear in the solution of the Laplace equation in the three-dimensional Euclidean space, in spheroidal and toroidal coordinates.

scholarly journals A Tutorial on the Basic Special Functions of Fractional Calculus

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Probability Theory ◽  
Integral Equation ◽  
Fractional Calculus ◽  
Diffusion Equation ◽  
Special Function ◽  
Special Functions ◽  
Fractional Diffusion ◽  
Basic Properties ◽  
Fractional Relaxation ◽  
Time Fractional Diffusion Equation

In this tutorial survey we recall the basic properties of the special function of the Mittag-Leffler and Wright type that are known to be relevant in processes dealt with the fractional calculus. We outline the major applications of these functions. For the Mittag-Leffler functions we analyze the Abel integral equation of the second kind and the fractional relaxation and oscillation phenomena. For the Wright functions we distinguish them in two kinds. We mainly stress the relevance of the Wright functions of the second kind in probability theory with particular regard to the so-called M-Wright functions that generalizes the Gaussian and is related with the time-fractional diffusion equation.

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