Special Functions Of Mathematical Physics Research Articles

Integral equations for special functions of Heun class

Abstract. Integral equations equivalent to boundary problems for the differen-tial equations of Heun class are studied. A new method of their derivation isproposed which is not as general as the known ones but is efficient in creatingintegral equations of a certain type. A number of new integral equations are ob-tained, some related to the double-confluent Heun equation. Formal calculationsat the first stage are followed by rigorous formulations. 1. IntroductionThe well-known special functions of mathematical physics, Bessel functions, hyper-geometric functions, etc., are defined as solutions of linear ordinary second-order dif-ferential equations with polynomial coefficients. Below we call any solution of suchequations a special function. The special functions can be expressed by means of in-tegral representations in terms of elementary functions. These representations are ofparticular importance for further study of additional characteristics of these functions:recurrence relations, asymptotic expansions, etc.There is no hope of finding such representations for all special functions in ourmore general sense; however, there exist more sophisticated integral relations andintegral equations which do succeed. Our main goal in this article is to deduce severalsimple integral equations for some special functions, namely of Heun type and thecorresponding confluent cases [4, 10]. These functions usually arise as eigenfunctionsof an eigenvalue problem associated with the initial differential equation.Several of our results are known. The main tools used previously for their derivationare either utilization of some integral transforms [6] or separation of variables in anauxiliary partial differential equation (pde) [1, 2, 5, 7-9]. Our approach is based onthe polynomial structure of the original equation, and it permits us to obtain newrelations as well as to give a new derivation of the old ones in a unique manner.On the other hand, it does not give the large variety of relations depending on anadditional parameter obtained by separation of variables of the mentioned pde. Infact these two methods are complementary to each other.2. Major definitionsBy a Heun equation we mean a second-order linear homogeneous differential equationwith four regular singular points. With the help of a linear transformation of theindependent variable these points can be located at z

On the Kummer Solutions of the Hypergeometric Equation

One of the oldest, and still one of the most interesting, applications of group theory arises in the study of the transformations of an ordinary differential equation. If we know that a given differential equation admits a group of transformations, then we know that the solution set must admit that same group of transformations, and we can deduce properties of all the solutions from the properties of any one of them. A case in point is offered by the celebrated hypergeometric equation (See Eq. (1) below), whose solutions include many of the most interesting special functions of mathematical physics. In his book [3], Einar Hille notes that this equation has a venerable history associated with such names as Gauss, Euler, Riemann, and Kummer. The hypergeometric equation is in fact a prototype: every ordinary differential equation of second order with at most three regular singular points can be brought to the hypergeometric equation by means of suitably chosen changes of variable [S]. In 1836 Kummer published a set of 6 distinct solutions of the hypergeometric equation. These include the hypergeometric function of Gauss, and all of them could be expressed in terms of Gauss's function (See Table 1 below). A useful summary of their basic properties is found in [1, p. 105ff.]. A glance at the list of these Kummer solutions reveals a rather complicated set of relationships which pleads for some simple explanation. We show here that the Kummer solutions are related by a finite group of transformations which serve to explain their relationships and to exemplify the use of transformation groups in the study of differential equations.

Implicit rule formation in symbolic computation

To facilitate symbolic computation in certain areas of applied mathematics, using Mathematica, 1 1 Mathematica is a trademark of Wolfram Research, Inc. we give interrelated names to identities and corresponding transformation rules. This has let us write procedures to form the rules automatically, once names that conform to a particular simple style have been assigned to the identities. We have used the methodology in applications to quantum chemistry, robot kinematics and other topics, and explain it here by reference to examples that use elementary functions and special functions of mathematical physics. We think it is of interest beyond the specifics of Mathematica usage.

An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities

It is shown that every ‘proper-hypergeometric’ multisum/integral identity, orq-identity, with a fixed number of summations and/or integration signs, possesses a short, computer-constructible proof. We give a fast algorithm for finding such proofs. Most of the identities that involve the classical special functions of mathematical physics are readily reducible to the kind of identities treated here. We give many examples of the method, including computer-generated proofs of identities of Mehta-Dyson, Selberg, Hille-Hardy,q-Saalschutz, and others. The prospect of using the method for proving multivariate identities that involve an arbitrary number of summations/integrations is discussed.

Quantum mechanics on the Macintosh

Based on the interactive program Interquanta, Quantum Mechanics on the Macintosh, uses extensive 3-D graphics to guide the student through computer experiments in the quantum mechanics of free particle motion, bound states and scattering, tunneling, two-particle interactions, and more. It also includes a section on special functions of mathematical physics. With more than 200 problems, the text and programs provide students with practical experiences in using such hard-to-visualize concepts as complex amplitudes, eigenvalues, and scattering cross sections. The diskette included with the book provides two versions of the programs, one for use in computers with a mathematical coprocessor, the other optimised for machines without a coprocessor. For this new edition the software has been updated and extended.

Some aspects of the group-theoretic approach in problems of nonlinear mechanics

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.

Analytic stellar structure

This paper continues to elaborate on analytic methods to construct models for the internal structure of solar-type stars. Since a detailed stellar model is desired, a nonlinear analytic density distribution in terms of a two-parameter family of models has been assumed. Hydrostatic equilibrium and energy conservation determine the conditions in the gravitationally stabilized stellar fusion reactor. The results show once more that methods of differential and integral calculus provide a laboratory for the application of special functions of mathematical physics in stellar structure.

Lie group theory of the bessel equation of the first kind of integral order

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.

A new approach to the theory of hypergeometric series and special functions of mathematical physics

A new approach to the theory of hypergeometric series and special functions of mathematical physics

Introduction to a covariant theory of special functions of mathematical physics

Using harmonic analysis techniquies, the covariant expression of Jacobi and Hermite polynomials on an n-dimensional space endowed with a metric g of signature (p+, q−) is given. The properties of these polynomials are studied and their relations with the hypergeometric function are given.

Transition probabilities for some ‘special' diffusions

We study the transition probabilities of the diffusions dXt = (1 – exp(Xt ))dt + dWt and dXt = – tanh Xtdt + dWt , in terms of special functions of mathematical physics (confluent hypergeometric and Legendre functions, respectively).

Transition probabilities for some ‘special' diffusions

We study the transition probabilities of the diffusions dXt = (1 – exp(Xt))dt + dWt and dXt = – tanh Xtdt + dWt, in terms of special functions of mathematical physics (confluent hypergeometric and Legendre functions, respectively).

Table errata: Formulas and theorems for the special functions of mathematical physics [second edition, Springer, Berlin, 1948; MR 10, 38] by W. Magnus and F. Oberhettinger

Table errata: Formulas and theorems for the special functions of mathematical physics [second edition, Springer, Berlin, 1948; MR 10, 38] by W. Magnus and F. Oberhettinger

Table errata: Formulas and theorems for the special functions of mathematical physics [Springer, New York, 1966; MR 38 #1291] by W. Magnus, F. Oberhettinger and R. P. Soni

Table errata: Formulas and theorems for the special functions of mathematical physics [Springer, New York, 1966; MR 38 #1291] by W. Magnus, F. Oberhettinger and R. P. Soni

Some algebraic, geometric, and system-theoretic properties of the Special Functions of mathematical physics

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.

Special Functions of Mathematical Physics and Chemistry

Special Functions of Mathematical Physics and Chemistry

Table errata: Formulas and theorems for the special functions of mathematical physics [third edition, Springer, New York, 1966; MR 38 #1291] by W. Magnus, F. Oberhettinger and R. P. Soni

Table errata: Formulas and theorems for the special functions of mathematical physics [third edition, Springer, New York, 1966; MR 38 #1291] by W. Magnus, F. Oberhettinger and R. P. Soni

Table errata: Formulas and theorems for the special functions of mathematical physics [third edition, Springer, New York, 1966; MR 38 #1291] by W. Magnus, F. Oberhettinger and R. P. Soni

Table errata: Formulas and theorems for the special functions of mathematical physics [third edition, Springer, New York, 1966; MR 38 #1291] by W. Magnus, F. Oberhettinger and R. P. Soni

The Connection Problem for General Linear Ordinary Differential Equations at Two Regular Singular Points with Applications in the Theory of Special Functions

A two point connection problem for local solutions at two regular singularities of a general linear ordinary differential equation is studied. Explicit formulas for the connection coefficients are obtained which have a wide field of applications especially in the theory of special functions of mathematical physics. Applications to the ellipsoidal wave equation and to Heun’s equation are considered.

A new equation for quantum dissipative systems

In this paper, we propose a new approach to the description of quantum dissipative systems. It is based on a finite-difference retarded equation governing the time evolution of the state vector. We exhibit the fundamental features of the equation, as well as the properties of its solutions, with particular emphasis to the case of the free particle. Moreover, a detailed analysis shows that our equation provides a natural way to understand the origin of important special functions of mathematical physics. Finally, we discuss the physical interpretation of our results and we examine with care the decay of the state vector belonging to excited energy levels.