The Elementary Solution of Triple Integral Equations and The Solution of Triple Series Equations Involving Associated Legendre Polynomials and their Application

A method is developed for the formal solution of an important class of triple integral equations involving Bessel functions. The solution of the triple integral equations is reduced to two simultaneous Fredholm integral equations and the results obtained are simpler than those of other authors and also superior for the purposes of solution by iteration. In the same manner the formal solution of triple series equations involving associated Legendre polynomials is presented. The solution of the problem is reduced to that of solving a Fredholm integral equation of the first kind. Finally to illustrate the application of the results an electrostatic problem is discussed.     

A New Integral Involving the Product of Bessel Functions and Associated Laguerre Polynomials

A new result for integrals involving the product of Bessel functions and Associated Laguerre polynomials is obtained in terms of the hypergeometric function. Some special cases of the general integral lead to interesting finite and infinite series representations of hypergeometric functions.     

Inequalities for formal power series and entire functions

We present several integral and exponential inequalities for formal power series and for both arbitrary entire functions of exponential type and generalized Borel transforms. They are obtained through certain limit procedures which involve the multiparameter binomial inequalities, integral inequalities for continuous functions, and weighted norm inequalities for analytic functions. Some applications to the confluent hypergeometric functions, Bessel functions, Laguerre polynomials, and trigonometric functions are discussed. Also some generalizations are given.     

Integral inequalities for some special functions

Several integral inequalities for the classical hypergeometric, confluent hypergeometric, and confluent hypergeometric limit functions are given. The related results for Bessel and Whittaker functions as well as for Laguerre, Hermite, and Jacobi polynomials are discussed.     

Properties of the zeros of confluent hypergeometric functions

Several infinite systems of nonlinear algebraic equations satisfied by the zeros of confluent hypergeometric functions are derived. Certain sum rules and other related properties for the zeros follow from these equations. A large class of special functions, which are special cases of confluent hypergeometric functions, is included. This is illustrated in the case of the zeros of Bessel functions and Laguerre polynomials.     

Polynomial sequences associated with the classical linear functionals

This work in mainly devoted to the study of polynomial sequences, not necessarily orthogonal, defined by integral powers of certain first order differential operators in deep connection to the classical polynomials of Hermite, Laguerre, Bessel and Jacobi. This connection is streamed from the canonical element of their dual sequences. Meanwhile new Rodrigues-type formulas for the Hermite and Bessel polynomials are achieved.     

Trace formula for a perturbed operator of Jacobi type

There is a broad class of problems of mathematical physics that lead to the solution of second-order differential equations of some special form. In particular, systems of solutions of such equations are given by classical polynomials (Jacobi, Laguerre, and Hermite polynomials). Such equations are naturally related to second-order differential operators in appropriate Hilbert spaces and the corresponding spectral problems. We consider a Jacobi operator and its perturbation by the operator of multiplication by a function. We derive a trace formula for the perturbed operator and a closed-form expression for the first correction.     

Laguerre functions and representations of su(1,1)

Spectral analysis of a certain doubly infinite Jacobi operator leads to orthogonality relations for confluent hypergeometric functions, which are called Laguerre functions. This doubly infinite Jacobi operator corresponds to the action of a parabolic element of the Lie algebra su(l, 1). The Clebsch-Gordan coefficients for the tensor product representation of a positive and a negative discrete series representation of su(l,l) are determined for the parabolic bases. They turn out to be multiples of Jacobi functions. From the interpretation of Laguerre polynomials and functions as overlap coefficients, we obtain a product formula for the Laguerre polynomials, given by an integral over Laguerre functions, Jacobi functions and continuous dual Hahn polynomials.     

A generalization of classical orthogonal polynomials and the convergence of simultaneous Pade approximants

The concept of generalized classical polyorthogonal polynomials and, in particular, that of generalized Laguerre polynomials, corresponding to a collection of measures with supports on infinite rays in the complex plane, is introduced. The asymptotic behavior of these polynomials and of their corresponding functions of the second kind is investigated. Moreover, generalizations of the Bessel functions and of the Euler integral of the second kind are defined and investigated. The convergence of the simultaneous Pade approximants to certain Stieltjes type functions is proved.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 11, pp. 125–165, 1986.     

A linear homogeneous partial differential equation with entire solutions represented by Bessel polynomials

We have continued our earlier studies on entire solutions of some special type linear homogeneous partial differential equations. Specifically, we deal with entire solutions of the equations that are represented in convergent series of Bessel polynomials, and determine orders and types of the solutions, in terms of their Taylor coefficients, by establishing an analogue of Lindelöf-Pringsheim theorem as well as Wiman-Valiron type theory for such functions. Finally, by using value distribution theory of holomorphic functions, we are able to exhibit some uniqueness theorems of the entire (or meromorphic) solutions.     

Some general theorems on generating functions and their applications

Several general classes of generating functions are established for a certain sequence of functions defined by Equation (1) below. By suitably specializing the various parameters involved, each of these main results can be applied to yield known as well as new generating functions for such familiar orthogonal polynomials as Jacobi, Laguerre, Hermite, and Bessel polynomials, and also for numerous interesting generalizations of these polynomials studied in the literature.     

Interlacing of the zeros of contiguous hypergeometric functions

It is well-known that hypergeometric functions satisfy first order difference-differential equations (DDEs) with rational coefficients, relating the first derivative of hypergeometric functions with functions of contiguous parameters (with parameters differing by integer numbers). However, maybe it is not so well known that the continuity of the coefficients of these DDEs implies that the real zeros of such contiguous functions are interlaced. Using this property, we explore interlacing properties of hypergeometric and confluent hypergeometric functions (Bessel functions and Hermite, Laguerre and Jacobi polynomials as particular cases).     

On expansion problems involving addition theorems and Kapteyn series

The paper deals with general expansions which give as special cases new results involving the Bessel functions, Jacobi, ultraspherical, and Laguerre polynomials, where the degree of the function is incorporated in the argument. In fact, the theorems unify and extend the Neumann-Gegenbauer expansion and its generalization by Fields and Wimp, Cohen, and others, the Kapteyn expansion theory, and the Kapteyn expansion of the second kind. New expressions are given for the Neumann-type degenerate form of a Gegenbauer addition theorem, the Feldheim expansions for the Jacobi and ultraspherical polynomials, and other expressions. Also of interest is the new method of proof, involving differential and integral operators.     

Some families of hypergeometric polynomials and associated integral representations

The main object of this paper is to investigate several general families of hypergeometric polynomials and their associated single-, double-, and triple-integral representations. Some known or new consequences of the general results presented here, involving such classical orthogonal polynomials as the Jacobi, Laguerre, Hermite, and Bessel polynomials, and various other relatively less familiar hypergeometric polynomials, are also considered. Each of the integral representations, which are derived in this paper, may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials.     

The nonlinear diffusion equation in cylindrical coordinates

Nonlinear corrections to some classical solutions of the linear diffusion equation in cylindrical coordinates are studied within quadratic approximation. When cylindrical coordinates are used, we try to find a nonlinear correction using quadratic polynomials of Bessel functions whose coefficients are Laurent polynomials of radius. This usual perturbation technique inevitably leads to a series of overdetermined systems of linear algebraic equations for the unknown coefficients (in contrast with the Cartesian coordinates). Using a computer algebra system, we show that all these overdetermined systems become compatible if we formally add one function on radius W(r). Solutions can be constructed as linear combinations of these quadratic polynomials of the Bessel functions and the functions W(r) and W′(r). This gives a series of solutions to the nonlinear diffusion equation; these are found with the same accuracy as the equation is derived. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 1, pp. 235–245, 2007.     

Luigi Gatteschi’s work on asymptotics of special functions and their zeros

A good portion of Gatteschi’s research publications—about 65%—is devoted to asymptotics of special functions and their zeros. Most prominently among the special functions studied figure classical orthogonal polynomials, notably Jacobi polynomials and their special cases, Laguerre polynomials, and Hermite polynomials by implication. Other important classes of special functions dealt with are Bessel functions of the first and second kind, Airy functions, and confluent hypergeometric functions, both in Tricomi’s and Whittaker’s form. This work is reviewed here, and organized along methodological lines.     

Some families of generating functions for the extended Srivastava polynomials

Almost four decades ago, H.M. Srivastava considered a general family of univariate polynomials, the Srivastava polynomials, and initiated a systematic investigation for this family [10]. In 2001, B. González, J. Matera and H.M. Srivastava extended the Srivastava polynomials by inserting one more parameter [4]. In this study we obtain a family of linear generating functions for these extended polynomials. Some illustrative results including Jacobi, Laguerre and Bessel polynomials are also presented. Furthermore, mixed multilateral and multilinear generating functions are derived for these polynomials.     

The Fourth-order Bessel–type Differential Equation

The Bessel-type functions, structured as extensions of the classical Bessel functions, were defined by Everitt and Markett in 1994. These special functions are derived by linear combinations and limit processes from the classical orthogonal polynomials, classical Bessel functions and the Krall Jacobi-type and Laguerre-type orthogonal polynomials. These Bessel-type functions are solutions of higher-order linear differential equations, with a regular singularity at the origin and an irregular singularity at the point of infinity of the complex plane.

There is a Bessel-type differential equation for each even-order integer; the equation of order two is the classical Bessel differential equation. These even-order Bessel-type equations are not formal powers of the classical Bessel equation.

When the independent variable of these equations is restricted to the positive real axis of the plane they can be written in the Lagrange symmetric (formally self-adjoint) form of the Glazman–Naimark type, with real coefficients. Embedded in this form of the equation is a spectral parameter; this combination leads to the generation of self-adjoint operators in a weighted Hilbert function space. In the second-order case one of these associated operators has an eigenfunction expansion that leads to the Hankel integral transform.

This article is devoted to a study of the spectral theory of the Bessel-type differential equation of order four; considered on the positive real axis this equation has singularities at both end-points. In the associated Hilbert function space these singular end-points are classified, the minimal and maximal operators are defined and all associated self-adjoint operators are determined, including the Friedrichs self-adjoint operator. The spectral properties of these self-adjoint operators are given in explicit form.

From the properties of the domain of the maximal operator, in the associated Hilbert function space, it is possible to obtain a virial theorem for the fourth-order Bessel-type differential equation.

There are two solutions of this fourth-order equation that can be expressed in terms of classical Bessel functions of order zero and order one. However it appears that additional, independent solutions essentially involve new special functions not yet defined. The spectral properties of the self-adjoint operators suggest that there is an eigenfunction expansion similar to the Hankel transform, but details await a further study of the solutions of the differential equation.     

The evaluation of Bessel functions via exp-arc integrals

A standard method for computing values of Bessel functions has been to use the well-known ascending series for small argument, and to use an asymptotic series for large argument; with the choice of the series changing at some appropriate argument magnitude, depending on the number of digits required. In a recent paper, D. Borwein, J. Borwein, and R. Crandall [D. Borwein, J.M. Borwein, R. Crandall, Effective Laguerre asymptotics, preprint at http://locutus.cs.dal.ca:8088/archive/00000334/] derived a series for an “exp-arc” integral which gave rise to an absolutely convergent series for the J and I Bessel functions with integral order. Such series can be rapidly evaluated via recursion and elementary operations, and provide a viable alternative to the conventional ascending-asymptotic switching. In the present work, we extend the method to deal with Bessel functions of general (non-integral) order, as well as to deal with the Y and K Bessel functions.     

New generating functions for Jacobi and related polynomials

In this paper the authors prove a generalization of certain generating functions for Jacobi and related polynomials, given recently by H. M. Srivastava. The method used is due to Pólya and Szegö, and it is based on Rodrigues' formula for the Jacobi polynomials and Lagrange's expansion theorem. A number of special and limiting cases of the main result will give rise to a class of generating functions for ultraspherical, Laguerre and Bessel polynomials.