Generalized Associated Legendre Functions and their Applications

Abstract

ForewordIn many areas of applied mathematics and in particular, in applied analysis and differential equations various types of special functions become essential tools for scientists and engineers. They appear in the solutions of initial-value or boundary-value problems of mathematical physics and usually satisfy certain classes of ordinary differential equations. One of the important classes of special functions is of hypergeometric type. It includes all classical hypergeometric functions as the well-known Gaussian hypergeometric function, Bessel, Macdonald, Neumann, Legendre, Whittaker, Lommel, Thompson, Kummer, Tricomi, Appel, Wright etc. and the generalized hypergeometric functions p F g , JS-function of Mac-Robert, Meijer's G-function and Fox's if-function.As is known the hypergeometric functions can be defined through the contour integrals of the Mellin-Barnes type and some of them properties and asymptotic representations can be derived using the theory of the Mellin transform. This approach enables also to study integral transformations whose kernels depend upon variables and parameters of the special functions which are involved in them.This book is devoted to the theory and applications of the generalized associated Legendre functions of the first and the second kind P™' n {z) and Q™' n (z), which are important representatives of the hypergeometric functions. They occur as generalizations of classical Legendre functions P£{z) and Q"(z) of the first and the second kind respectively. The authors use various methods of contour integration, consider solutions of the corresponding differential equations in order to obtain important properties of the generalized associated Legendre functions as them series representations, asymptotic formulas in a neighborhood of singular points, zeros properties, connection with the Jacobi functions, relations with Bessel functions, elliptic integrals and incomplete beta functions.Some of the interesting aspects, which are considered here involve classes of dual and triple integral equations being associated with the function P™j'? 2+lV (cosha). This function represents the kernel of the generalized Mehler-Fock transform of order (m,n),which, in turn, is one of the basic integral transformations (the so-called index transforms) depending upon a parameter of the Legendre function. It should be remarked that the authors consider fractional integro-differential properties of the generalized associated Legendre functions and give various generalizations of Buschman-Erdelyi's type integral operators. This book also presents the theory of factorization and composition structure of integral operators associated with generalized Legendre functions, which has important applications in finding solutions of the corresponding integral equations. The method of the considered integral equations gives series of examples of integrals over variables and parameters of the generalized Legendre functions. One of such sources of formulas is served by the pair of the following Mehler-Fock trans...

Scalar and tensor power spectra in a universe with positive curvature index and maximally symmetric spatial section

“…where Ξ = τ − cot −1 n and Υ = η − cot −1 n. which is greater than 0.94 (because n ≥ 3), so we can use the asymptotic formulas of the associated Legendre functions [43], thus…”

Generalized hybrid Mehler-Fock-type integral transformation of the first kind and its applications

“…For j ≥ 2, we take the two real functions A (µ) j −1/2+ib j (cosh r) and B (µ) j −1/2+ib j (cosh r) as a fundamental system of solutions [8].…”

“…A natural generalization of the differential operator (1) is the operator [7,8] Λ (µ) = d 2 dr 2 + coth r d dr…”

Fundamental Solutions of the Cauchy Problem for Invariant Λ(µ)-Hyperbolic Operators on Riemannian Manifolds

“…where P (μ) −1/2+iβ (cosh y) is the generalized associated Legendre function of the first kind [4], as a solution of the generalized Legendre equation…”