Averaged Integral Transforms

The use of differintegral operators $D^\nu $ with $D = {d / {dz}}$ and $\nu \in \mathbb{C}$ and hyperdifferential operators, together with some operational calculus manipulations leads to new formulae of representation of usual special functions. These formulae are useful for obtaining generating functions and differential recurrence relations of orthogonal polynomials.

6.-7 - DEFINITE INTEGRALS OF SPECIAL FUNCTIONS

ABSTRACTAn earlier method for obtaining indefinite integrals of special function from the second-order linear equations which define them has been reformulated in terms of Riccati equations, which are nonlinear and first-order. For this application the nonlinearity is not a problem and the first-order property is a great advantage. Integrals can be derived using fragments of these Riccati equations and here only two specific fragment types are examined in detail. These fragments allow general integration formulas to be derived using quadrature. Other results will be presented separately. Results are presented here for Airy functions, Bessel functions, complete elliptic integrals, associated Legendre functions and Gauss hypergeometric functions. All results have been checked by differentiation using Mathematica.

A theory of special functions should derive as many of the numerous formulas and properties as possible from some guiding principles. One of these principles is the F-equation, which Truesdell has put forward, and which is satisfied by all simple special functions. Many results about addition theorems, generating functions, integral representations etc. follow immediately from the F-equation. One disadvantage, however, is that the F-equation does not constitute a specific property of the simple special functions, generating instead functions of a much more general character.

Fourier series are derived for generalizations of the three canonical Legendre incomplete elliptic integrals using a hypergeometric series approach. The Fourier series for the incomplete Epstein–Hubbell integrals are obtained as special cases of the generalization of the Legendre integrals of the first and second kinds. The Fourier series for the integrals of the first and second kinds, and those for the Epstein–Hubbell integrals, were obtained recently using a different approach, but the series obtained for the generalization of the incomplete integral of the third kind is new. All cases of the integral of the third kind are given, with the modulus and the parameter being complex variables, and the Fourier coefficients are given in terms of the Kampé de Feriet function for all cases.

6–7 - Definite Integrals of Special Functions

The main aim of this paper is to give the definitions of Caputo fractional derivative operators and show their use in the special function theory. For this purpose, we introduce new types of incomplete hypergeometric functions and obtain their integral representations. Furthermore, we define incomplete Caputo fractional derivative operators and show that the images of some elementary functions under the action of incomplete Caputo fractional operators give a new type of incomplete hypergeometric functions. This definition helps us to obtain linear and bilinear generating relations for the new type incomplete Gauss hypergeometric functions.

Sign problems in elliptic integral solution of planar elastica theory

We give a brief account of the key properties of elliptic hypergeometric integrals—a relatively recently discovered top class of transcendental special functions of hypergeometric type. In particular, we describe an elliptic generalization of Euler’s and Selberg’s beta integrals, elliptic analogue of the Euler–Gauss hypergeometric function and some multivariable elliptic hypergeometric functions on root systems. The elliptic Fourier transformation and corresponding integral Bailey lemma technique is outlined together with a connection to the star-triangle relation and Coxeter relations for a permutation group. We review also the interpretation of elliptic hypergeometric integrals as superconformal indices of four dimensional supersymmetric quantum field theories and corresponding applications to Seiberg type dualities.

The present paper deals with the definition and study of new kinds of hypergeometric matrix functions within complex analysis. We also get the radius of regularity and matrix recurrence relations are then developed for l(m,n)-hypergeometric matrix function of two complex variables. We give a different approach to prove the effect of the differential operator on this function. Finally, another hypergeometric matrix function, namely, p l(m,n)-hypergeometric matrix function of two complex variables are defined, its components when the positive integer p is greater than one, provide a matrix partial differential equation satisfied by these function and some of their properties are investigated.

This paper explores a novel application of the Laplace transform in cryptographic schemes by employing a linear combination of Bessel functions and exponential functions. By leveraging the transform’s properties, we develop an encryption-decryption framework that is computationally efficient and highly secure against certain types of cryptographic attacks. The exponential decay observed in Laplace-transformed domains and the complex behaviour of Bessel functions are powerful tools for data encoding and concealment. Analytical and numerical results validate the method’s effectiveness, showcasing its potential for modern cryptographic systems. Cryptography has long been a cornerstone of secure communication, with its methods evolving in tandem with technological advancements. While traditional schemes rely on number theory or algebraic structures, this paper explores an alternative approach that combines Laplace transforms with special functions. Specifically, we focus on employing Bessel functions in conjunction with exponential functions to construct an encryption algorithm. The Laplace transform’s unique properties, such as linearity, time-shifting, and frequency domain representation, offer a fertile ground for developing innovative cryptographic methods. Then the algorithm described in this study generates a dynamic cipher by processing plaintext using polynomial, exponential, and factorial-based operations. The four fundamental steps comprising the computational process are text preprocessing, dynamic coefficient calculation, encryption packaging, and reverse analysis. The computational cost of each step varies depending on the number of characters processed. The symbol for this character quantity is n. The cryptographic analysis of the problem is discussed in detail.

The life and career of Academician A. D. Kovalenko, a prominent Ukrainian mechanician, are briefly described. His achievements in thermomechanics are outlined. Among them are the thermomechanical fundamentals of thermoelasticity theory; exact solutions to various stress-strain problems for circular plates and shells of revolution of varying thickness found using the theory of special functions (in particular, hypergeometric functions of the second kind) as the basic mathematical tool; methods widely used to design turbomachine parts; and linear and nonlinear viscoelasticity theory developed considering the temperature dependence of material properties

A certain double integral transformation

The hypergeometric functions are one of the most important and special functions in mathematics. They are the generalization of the exponential functions. Particularly the ordinary hypergeometric function 2F1(a, b; c; z) is represented by hypergeometric series and is a solution to a second order differential equation. Similarly, Laplace transform is a form of integral transform that converts linear differential equations to algebraic equations. This paper aims to study the convergence of hypergeometric function and Laplace transform of some hypergeometric functions. Moreover, some relationships between Laplace transformation and hypergeometric functions is established in the concluding section of this paper.

We extend a cell growth model with a deterministic cell growth rate and asymmetric division to include dispersion. This extension yields a second order functional differential equation, the solution of which is required to be a probability density function for the size of a cell (as measured by DNA content). We study the equation for certain choice of non constant coeffcients that correspond to dispersion, growth and splitting rates. This choice of coeffcients leads to a Bessel type operator, and it is shown that there is a unique probability distribution function that solves the equation. The solution is constructed using the Mellin transform and is given in terms of an infinite series of modified Bessel functions.