Generalized Associated Legendre Functions and Their Applications

Book excerpt: The various types of special functions have become essential tools for scientists and engineers. One of the important classes of special functions is of the hypergeometric type. It includes all classical hypergeometric functions such as the well-known Gaussian hypergeometric functions, the Bessel, Macdonald, Legendre, Whittaker, Kummer, Tricomi and Wright functions, the generalized hypergeometric functions ? Fq, Meijer's G -function, Fox's H -function, etc. Application of the new special functions allows one to increase considerably the number of problems whose solutions are found in a closed form, to examine these solutions, and to investigate the relationships between different classes of the special functions. This book deals with the theory and applications of generalized associated Legendre functions of the first and the second kind, P m, n ? ( z ) and Q m, n ? ( z ), which are important representatives of the hypergeometric functions. They occur as generalizations of classical Legendre functions of the first and the second kind respectively. The authors use various methods of contour integration to obtain important properties of the generalized associated Legnedre functions as their series representations, asymptotic formulas in a neighborhood of singular points, zero properties, connection with Jacobi functions, Bessel functions, elliptic integrals and incomplete beta functions. The book also presents the theory of factorization and composition structure of integral operators associated with the generalized associated Legendre function, the fractional integro-differential properties of the functions P m, n ? ( z ) and Q m, n ? ( z ), the classes of dual and triple integral equations associated with the function P m, n -1/2+i? (cha) etc. Contents: A General Information on Legendre Functions; The Generalized Associated Legendre Functions; The Series Representations of the Generalized Associated Legendre Functions; Relations Between Different Solutions of the Generalized Legendre Equation. Wronskians of Linearly Independent Solutions; Relations Between Contiguous Generalized Associated Legendre Functions; Differential Operators Generated by the Generalized Associated Legendre Equation; Asymptotic Formulas for the Generalized Associated Legendre Functions in a Neighborhood of Singular Points; Asymptotic Representations of the Generalized Associated Legendre Functions as the Functions of Parameters; Integral Representations of the Generalized Associated Legendre Functions of the First Kind; Integral Representations of the Generalized Associated Legendre Functions of the Second Kind; Zeros of the Generalized Associated Legendre Functions; Connection of the Generalized Associated Legendre Functions with the Jacobi Functions; and other topics. Readership: Graduate students and researchers in mathematics, physics and engineer