Mutually inverse series relating Ferrers and associated Legendre functions and generating functions pertaining to them

This article deals with three types of mutually inverse series relating Ferrers and associated Legendre functions of arbitrary complex indexes and orders established on the base of integral representations by using a number of generating functions (some of them are novel) for polynomials expressed in terms of hypergeometric or generalized hypergeometric polynomials. High-order asymptotics of above-mentioned polynomials are indicated. In particular, a uniform asymptotics for Gegenbauer polynomials of degree n and index a-n is obtained for large n. In special cases, the inverse series turn into novel mutually inverse finite sums relating Gegenbauer or associated Legendre polynomials of different arguments as well as into novel connection formulas for Gegenbauer and associated Legendre polynomials.Comment: 27 page

Avoided crossings in mesoscopic systems: electron propagation on a non-uniform magnetic cylinder

We consider an electron constrained to move on a surface with revolution symmetry in the presence of a constant magnetic field $B$ parallel to the surface axis. Depending on $B$ and the surface geometry the transverse part of the spectrum typically exhibits many crossings which change to avoided crossings if a weak symmetry breaking interaction is introduced. We study the effect of such perturbations on the quantum propagation. This problem admits a natural reformulation to which tools from molecular dynamics can be applied. In turn, this leads to the study of a perturbation theory for the time dependent Born-Oppenheimer approximation

Ferrers functions of arbitrary degree and order and related functions

Numerous novel integral and series representations for Ferrers functions of the first kind (associated Legendre functions on the cut) of arbitrary degree and order, various integral, series and differential relations connecting Ferrers functions of different orders and degrees as well as a uniform asymptotic expansion are derived in this article. Simple proofs of four generating functions for Ferrers functions are given. Addition theorems for Ferrers functions of the argument tanh(a+b) are proved by basing on generation functions for three families of hypergeometric polynomials. Relations for Gegenbauer polynomials and Ferrers associated Legendre functions (associated Legendre polynomials) are obtained as special cases.Comment: 39 pages. 1) Inessential details are omitted in the proof of Theorem 13. 2) The statement of Theorem 26 is slightly changed. This change has enabled me to write a new generating function for Ferrers functions as a corollary. 3) The statement of Theorem 29 (Theorem 28 in the previous version) is corrected. 4) Typos are correcte

On certain Heun functions, associated functions of discrete variables, and applications

Two sets of the Heun functions are introduced via integrals. Theorems about expanding functions with respect to these sets are proven. A number of integral and series representations as well as integral equations and asymptotic formulas are obtained for these functions. Some of the coefficients of the series are orthogonal (J-orthogonal) functions of discrete variables and may be interpreted as orthogonal polynomials. Other sets of the coefficients are biorthonormal. Expanding infinite vectors to series with respect to the coefficients is discussed. Certain Legendre functions of complex degree are limiting cases of the studied functions. This leads to new relations for Legendre functions and associated integral transforms. The treated Heun functions find a use for solving dual Fourier series equations which are reduced to the Fredholm integral equations of the second kind. Explicit solutions are obtained in a special case

Dual and triple Fourier-Bessel series equations

AbstractA method for solving dual and triple Fourier-Bessel series equations is proposed. It is based on a novel operator transforming Bessel functions into the sine function and on an inversion formula analogous to one for Bessel series. As a result, the dual and triple equations are transformed into the Fredholm integral equations of the second kind or into the singular integral equations of a well-known type. The suggested approach differs from the existing and provides new possibilities for applications. This is demonstrated by the torsional problem for an annular punch in contact with an inhomogeneous elastic cylinder. The asymptotic solution is derived as a distance between the punch edge and the lateral cylinder surface is short provided that the punch hole is small.A number of novel results for series containing Bessel functions is obtained as well