Series Of Special Functions Research Articles

On some series of special functions

Series of the form ∑ k = 0 ∞ ∏ i = 1 p ( a i ) k ∏ j = 1 q ( b j ) k w k F k ( z ) are considered, where F k ( z ) are special functions of hypergeometric type. Various types of series involving Bessel, Struve, incomplete gamma functions, and Laguerre, Hermite, Jacobi, Legendre, and Chebyshev polynomials are represented in terms of the Kampé de Fériet and generalized Horn hypergeometric functions of two variables.

Asymptotical Distributions of Eigenvalues of Periodic and Antiperiodic Boundary Value Problems for Second-Order Differential Equations

Abstract—We consider asymptotical distributions of characteristic constants in periodic and antiperiodic boundary value problems for a second-order linear equation with periodic coefficients. This allows one to obtain asymptotical properties of stability and instability zones of solutions. We show that if there are no turning points, i.e., if $$r(t) > 0$$ , then the lengths of instability zones converge to zero as their number increases, while the lengths of stability zones converge to a positive number. If $$r(t) \geqslant 0$$ and function $$r(t)$$ has zeroes, then the lengths of stability and instability zones have finite nonzero limits as the numbers of the corresponding zones infinitely increase. If function $$r(t)$$ is alternating, then the lengths of all stability zones converge to zero and the lengths of all instability zones converge to finite numbers. This yields various stability and instability criteria for solutions of second-order equations with periodic coefficients. The presented results are illustrated by a substantial example. The investigation methods are based on a detailed study of so-called special standard equations and the reduction of original equations to standard equations. Here, asymptotical methods of the theory of singular perturbations and properties of series of special functions are used.

Sector magnets or transverse electromagnetic fields in cylindrical coordinates

The Laplace's equations for the scalar and vector potentials describing electric or magnetic fields in cylindrical coordinates with translational invariance along azimuthal coordinate are considered. The series of special functions which, when expanded in power series in radial and vertical coordinates, in lowest order replicate the harmonic homogeneous polynomials of two variables are found. These functions are based on radial harmonics found by Edwin~M.~McMillan in his more-than-40-years "forgotten" article, which will be discussed. In addition to McMillan's harmonics, second family of adjoint radial harmonics is introduced, in order to provide symmetric description between electric and magnetic fields and to describe fields and potentials in terms of same special functions. Formulas to relate any transverse fields specified by the coefficients in the power series expansion in radial or vertical planes in cylindrical coordinates with the set of new functions are provided. This result is no doubt important for potential theory while also critical for theoretical studies, design and proper modeling of sector dipoles, combined function dipoles and any general sector element for accelerator physics. All results are presented in connection with these problems.

On One Minkowski–Radon Problem and Its Generalizations

We study functions on a sphere with zero weighted means over the circles of fixed radius. A description of these functions is obtained in the form of series in special functions.

Integral relations for solutions of the confluent Heun equation

Firstly, we construct kernels for integral relations among solutions of the confluent Heun equation (CHE). Additional kernels are systematically generated by applying substitutions of variables. Secondly, we establish integral relations between known solutions of the CHE that are power series and solutions that are series of special functions. Thirdly, by using one of the integral relations as an integral transformation we obtain a new series solution of the ordinary spheroidal wave equation (a particular CHE). From this solution we construct new series solutions of the general CHE, and show that these are suitable for solving the radial part of the two-center problem in quantum mechanics.

Analytical modeling of pressure transient behavior for coalbed methane transport in anisotropic media

Resulting from the nature of anisotropy of coal media, it is a meaningful work to evaluate pressure transient behavior and flow characteristics within coals. In this article, a complete analytical model called the elliptical flow model is established by combining the theory of elliptical flow in anisotropic media and Fick's laws about the diffusion of coalbed methane. To investigate pressure transient behavior, analytical solutions were first obtained through introducing a series of special functions (Mathieu functions), which are extremely complex and are hard to calculate. Thus, a computer program was developed to establish type curves, on which the effects of the parameters, including anisotropy coefficient, storage coefficient, transfer coefficient and rate constant, were analyzed in detail. Calculative results show that the existence of anisotropy would cause great pressure depletion. To validate new analytical solutions, previous results were used to compare with the new results. It is found that a better agreement between the solutions obtained in this work and the literature was achieved. Finally, a case study is used to explain the effects of the parameters, including rock total compressibility coefficient, coal medium porosity and anisotropic permeability, sorption time constant, Langmuir volume and fluid viscosity, on bottom-hole pressure behavior. It is necessary to coordinate these parameters so as to reduce the pressure depletion.

On the Kernel of a Hemispherical Funk Transformation and its Local Analogs

Functions on a sphere with zero weighted means over all geodesic balls of a fixed radius are studied. We obtain a description of such functions in the form of series in special functions.

A note on some new series of special functions

Recently, Brychkov [On some new series of special functions. Appl Math Comput. 2007;187:101–104] obtained several finite and infinite series identities involving special functions by making use of an operator related to the Riemann–Liouville fractional calculus operator. In this paper, we present several identities involving the Jacobi polynomials, the generalized Laguerre function and the first Appell's function. These relations are obtained by using a fractional calculus operator related to the Riemann–Liouville operator and a new transformation formula [Tremblay R, Gaboury S, Fugère B-J. A new transformation formula for fractional derivatives with applications. Integral Transforms Spec Funct. 2013;24(3):172–186].

On E-functions of semisimple Lie groups

We develop and describe continuous and discrete transforms of class functions on a compact semisimple, but not simple, Lie group G as their expansions into series of special functions that are invariant under the action of the even subgroup of the Weyl group of G. We distinguish two cases of even Weyl groups—one is the direct product of even Weyl groups of simple components of G and the second is the full even Weyl group of G. The problem is rather simple in two dimensions. It is much richer in dimensions greater than two—we describe in detail E-transforms of semisimple Lie groups of rank 3.

A definitive version of the local two-radii theorem on hyperbolic spaces

The paper deals with various classes of functions that have zero integrals over all balls of a fixed radius in hyperbolic spaces. We describe these classes in terms of series in special functions and prove a uniqueness theorem. These results enabled us to obtain a definitive version of the local two-radii theorem.

Solution of the support problem for several function classes

Several classes of functions with zero integrals over all balls of a fixed radius are studied. A description of functions in such classes as series in special functions is obtained and a theorem on a removable singularity is proved. These results make it possible to solve completely the support problem for several classes of functions with zero ball averages. As a consequence, a far-reaching generalization of the well-known Zalcman's two-radii theorem is obtained.

Uniqueness theorems for some classes of functions with zero spherical means

We study various classes of functions that have zero integrals over all spheres of given radius. For these functions, we obtain a description in the form of a series in special functions; the problem of the existence of a nonzero function of prescribed growth with this property is completely solved.

A definitive version of the local two-radii theorem

Various classes of functions with zero integrals over all balls of a fixed radius are studied. For functions in such classes a description in the form of a series in special functions is obtained and a uniqueness theorem is proved. These results make it possible to solve completely the problem of existence of a non-trivial function with zero integrals over all balls of radius assuming either of two given values.

Stress-strain state of shells with rectangular holes

The stress-strain state of shallow shells with large rectangular holes was investigated. The use of the apparatus of special discontinuous functions reduces the problem to the solution of a fourth-order differential equation for a complex deflection-force function with variable coefficients in the form of delta functions, their derivatives, and column functions. The solution is presented in the form of systems of basis functions composed of series of special functions reflecting the discontinuous character of the forces and moments and having rapid convergence both in the local zone near the hole and in continuous zones. This made it possible to construct effective calculation algorithms and to take into account stress concentration near the corners of the hole.

Integrals and Series of Special Functions.

Integrals and Series of Special Functions.

A spherical wave in a viscoelastic half-space

The propagation of spherical waves in an isotropie elastic medium has been studied sufficiently completely (see, e.g., [1–4]). it is proved [5, 6] that in imperfect solid media, the formation and propagation of waves similar to waves in elastic media are possible. With the use of asymptotic transform inversion methods in [7] a problem of an internal point source in a viscoelastic medium was investigated. The problem of an explosion in rocks in a half-space was considered in [8]. A numerical Laplace transform inversion, proposed by Bellman, is presented in [9] for the study of the action of an explosive pulse on the surface of a spherical cavity in a viscoelastic medium of Voigt type. In the present study we investigate the propagation of a spherical wave formed from the action of a pulsed load on the internal surface of a spherical cavity in a viscoelastic half-space. The potentials of the waves propagating in the medium are constructed in the form of series in special functions. In order to realize viscoelasticity we use a correspondence method [10]. The transform inversion is carried out by means of a representation of the potentials in integral form and subsequent use of asymptotic methods for their calculation. Thus, it becomes possible to investigate the behavior of a medium near the wave fronts. The radial stress is calculated on the surface of the cavity.

Notes on the History of the Uses of Analyticity in Operator Theory

provided that d(X) #0. Heref and g are members of the function class C[[a, b]. The function d(X) is an entire analytic function of the complex parameter X and D(s, t; X), which is a continuous function of (s, t), is also an entire analytic function of X. The display of analyticity in the solution of the Fredholm equation of the second kind is an early signal of the important role which analyticity was destined to play in spectral theory. Anticipations of spectral theory itself can be traced at least as far back as 1836, which was the date of publication of the work of Sturm [38], [39] and Liouville [28], [29] on second-order ordinary differential equations and the associated eigenvalue and expansion theorems. In 1904 Hilbert [20] showed how to transform a differential boundary value problem into a problem of solving an integral equation. Moreover, the Green's function of the differential problem turned out, under certain conditions, to be a kernel of Hilbert-Schmidt type, from which followed a great many things of interest. The utility of analytic functions and the calculus of residues in connection with expansions in series of special functions has been known since the work of Cauchy. In 1894 Poincare [34] dealt with the problem of the vibrating membrane and related the eigenvalues and eigenfunction expansions to the meromorphic character of the Green's function's dependence on the parameter. In 1908 Birkhoff made explicit the use of the meromorphic behavior of the Green's