Special Functions and the Complex Euclidean Group in 3-Space. I

It is known that many of the Special Functions of mathematical physics appear as matrix elements of Lie group representations. This paper is concerned with a beginning attack on the converse problem, i.e., finding conditions that a given function be a matrix element. The methods used are based on a combination of ideas from system theory, functional analysis, Lie theory, differential algebra, and linear ordinary differential equation theory. A key idea is to attach a symbol as an element of a commutative algebra. In favorable cases, this symbol defines a Riemann surface, and a meromorphic differential form on that surface. The topological and analytical invariants attached to this form play a key role in system theory. The Lie algebras of the groups appear as linear differential operators on this Riemann surface. Finally, it is shown how the Picard-Vessiot-Infeld-Hull theory of factorization of linear differential operators leads to realization of many Special Functions as matrix representations of group representations.

We describe a knowledge-based computer project on the special functions of mathematical physics, which uses a unified structural approach to their classification. A prototype program package SFTools is presented. Special functions addressed by the system include hypergeometric-type functions, Heun functions, Painlevé transcendents and orthogonal polynomials. For these functions, basic differential equations are given with emphasis on singularities, integral relations, series expansions, Riemann P-symbols (Riemann scheme) and other important characteristics. In order to structure the information, we use the processes of confluence and reduction of equations widely.

Most of the special functions of mathematical physics are connected with the representation of Lie groups. The action of elements D of the associated Lie algebras as linear differential operators gives relations among the functions in a class, for example, their differential recurrence relations. In this paper, we define fractional generalizations Dμ of these operators in the context of Lie theory, determine their formal properties, and illustrate their use in obtaining interesting relations among the functions. We restrict our attention here to the Euclidean group E(2) and the Bessel functions. We show that the two-variable fractional operator relations lead directly to integral representations for the Bessel functions, reproduce known fractional integrals for those functions when reduced to one variable, and contribute to a coherent understanding of the connection of many properties of the functions to the underlying group structure. We extend the analysis to the associated Legendre functions in a following paper.

Chapter 4 - Special functions

Most of the special functions of mathematical physics are connected with the representation of Lie groups. The action of elements D of the associated Lie algebras as linear differential operators gives relations among the functions in a class, for example, their differential recurrence relations. In this paper, we apply the fractional generalizations Dμ of these operators developed in an earlier paper in the context of Lie theory to the group SO(2,1) and its conformal extension. The fractional relations give a variety of interesting relations for the associated Legendre functions. We show that the two-variable fractional operator relations lead directly to integral relations among the Legendre functions and to one- and two-variable integral representations for those functions. Some of the relations reduce to known fractional integrals for the Legendre functions when reduced to one variable. The results enlarge the understanding of many properties of the associated Legendre functions on the basis of the underlying group structure.

Analysis of function spaces and special functions are closely related to the representation theory of Lie groups. We explain here the connection between the Laguerre functions, the Laguerre polynomials, and the Meixner–Pollacyck polynomials on the one side, and highest weight representations of Hermitian Lie groups on the other side. The representation theory is used to derive differential equations and recursion relations satisfied by those special functions.

We discuss the Bessel differential equation of the first kind of integral order and the associated functions from a Lie-group-theoretical background. All the familiar properties of the Bessel equation and functions are obtained. The analytic methodology developed in the study can easily be adapted to the study of some other special functions of mathematical physics.

On Lie algebras and some special functions of mathematical physics

Gravitational wave signal templates, pattern recognition, and reciprocal Eulerian gamma functions

Almost all the known orthonormal wavelets, except for the Haar and Shannon wavelets, cannot be expressed in closed form. By a closed-form expression we mean a representation of the function in terms of elementary functions, such as trigonometric, exponential, or rational functions, or in terms of special functions of mathematical physics, such as gamma, Bessel, or hypergeometric functions. We outline a procedure for obtaining orthonormal wavelets, as well as nonorthogonal, but interpolating, wavelets. We then apply this procedure to specific cases and obtain a number of orthonormal wavelets and nonorthogonal, but interpolating, wavelets in closed-form expressions.

On p. 307, in formula ( 22) of Section 13.5, for 1 -it2 sin20, read (1 -fc2sin20)1/2.On p. 317, in formula ( 12) of Section 13.7, for sin 0 cos 0, read k sin 0 cos 0.

A theorem is proved which enables one to obtain monotonic properties of the real part, imaginary part, modules and phase of an arbitrary analytic function in the complex plane. The monotonic properties are established from the behavior of the analytic function and its derivative on the boundary of the domain in which the monotonic properties are required. As an applicationsome monotonic results are derived for the Bessel functions $J_\nu (z)$ and $H_\nu ^{(1)} (z)$, where z is complex and $\nu$ is real and positive. The theorem and a corollary can be used to obtain monotonic properties of many other special functions of mathematical physics.

The authors derive a set of unified representations of the Voigt functions in terms of familiar special functions of Mathematical Physics. Some deductions from these representations are also considered.

Two aspects of the application of the asymptotic decomposition method to the decomposition and investigation of nonlinear systems are considered. The first one is the extension of the class of functions in terms of which to form the asymptotic decomposition by applying special functions of mathematical physics. The other aspect is the extension of the asymptotic decomposition method to a new class of differential systems, namely, Pfaffian systems in involution.

It must be difficult to write a good collection of formulas and theorems; it surely is difficult to review one. This book is not meant for reading: It quotes theorem upon theorem without proof and hardly can hold the interest of any reader. It is not meant to be used that way. Rather, I believe, it is meant to remind one of a formula with which one is already vaguely familiar and give its exact form, including the constants which one is likely to forget.