FINITE CLASS OF CLASSICAL ORTHOGONAL POLYNOMIALS DEFINED ON THE POSITIVE REAL LINE

Abstract

The study of the class of orthogonal polynomials is one of the interesting research area for many decades, mainly due to its classical nature and its applications to various branches of mathematics itself. The present work deals with two of the less-known class of classical orthogonal polynomials that are of hypergeometric type. The relation between orthogonal polynomials and hypergeometric functions is evident in the literature and hence some information about hypergeometric functions is provided at the beginning. The introduction continued with the class of orthogonal polynomials. Definition of various required concepts and their related properties that are prerequisite for this work given. A brief account of the related work made by various authors in the direction of the present work are also given. Standard notations are used for all the definitions and classes that exist in the literature. Explanations for new notations are given wherever they appear for the first time. At first the derivative of the finite class of orthogonal polynomials /V4P'q)(x) that are orthogonal with respect to the weight function which is related to the probability den-sity function of the F distribution are considered. These polynomials are denoted by 4'1)(x). For this derivative class, besides orthogonality various other related proper-ties such as the normal form, the self adjoint form, recurrence relation, and Rodrigues' formula are found. These orthogonal polynomials system are mapped onto themselves by means of the Fourier transformation. a related complicated integral is also evaluated. The corresponding Gaussian quadrature formulae related to the class 471)(x) are also given. Examples are provided to support the advantages of considering this derivative class of the finite class of orthogonal polynomials and from the numerical observations, it is clear that the error will be less for the derivative class Zri(P'qi) (x) compare to the class of orthogonal polynomials M(P)(x). The derivatives of the class of orthogonal polynomials N(P)(x) with respect to the weigh. function which is related to the inverse gamma distribution over the infinite interval oo) are also considered. General properties for this derivative class, denoted by (x), similar to the study of Zri(731)(x) are carried out. The relation between these polynomials and Laguerre polynomials is also established. The corresponding Gaussian' quadrature formulae related to the class RTh(P)i(x) are also given. Orthogonality property related to the Fourier transform for this class of functions is also given. Examples are given to discuss the advantages and constraints of considering the derivative class R(P)i (x) of the finite class of orthogonal polynomials in comparison to the class 1\c(!) (x). In the sequel, the generalization of the the classical Riemann-Liouville fractional integral operator by taking two integral transforms involving Gaussian hypergeometric functions in the kernel are considered. The impact of fractional calculus on the class ./14')(x) are considered. This finite class of classical orthogonal polynomials defined on the positive real line for fractional order is generalized by considering the Riemann-Liouville type operator on these polynomials. Various properties like explicit representation in terms of hypergeometric functions, differential equations, recurrence relations are derived. To complete the study of fractional calculus on the other finite class of orthogonal poly-nomials on the positive real line, the properties of the MP) (x) based on a fractional operator are derived. Reasons for not obtaining all results to the class of polynomials AT,-,P)(x), similar to the class of polynomials M(P.)(x) are established. Various interesting properties for the two finite class of orthogonal polynomials M4') (x) and N,C,,P)(x) related to the results in the classical theory are verified. These results motivate further investigation. Further, some mixed recurrence relations for the two finite classes of classical orthogonal polynomials (COPS) MV)(x) and M,P)(x) using certain contiguous relations for Gaussian and confluent hypergeometric functions are derived. These mixed recurrence relations are useful in observing interlacing of zeros of these finite class of classical orthogonal polynomials of the same or adjacent degree as one or both of the parameters are shifted continuously within a certain range. Some numerical examples are also provided where interlacing property of zeros occurs. ii Proceeding in a slightly different direction, the following connection coefficient is con-sidered. These connection coefficients are useful in studying various properties of the corresponding class of orthogonal polynomials. The connection coefficients that takes the classes Jacobi, Laguerre, Hermite, /1/4"1)(x) and N,P)(x) to the class /11CP'q)(x) are obtained. Similarly the connection coefficients that takes the classes Jacobi, Laguerre, Hermite, /1/4")(x) and N,(f)(x) to the class N,;,'') (x) are obtained. Interesting obser-vations are deduced. Further using the recurrence relation for the monic orthogonal polynomials of the class M,P)(x), certain g-sequences or chain sequences appearing in continued fraction expansion of certain analytic self maps of the unit disc 1z1 < 1 are found. These continued fractions appear in the Nevanlinna theory of moment prob-lems of certain analytic functions. Certain bounds for the resulting chain sequences are obtained. The parameters obtained from the recurrence relation of the monic or-thogonal polynomials of N,Cf)(x) are useful in finding a spring-mass system studied for non-classical orthogonal polynomials as well. The results related to the g-sequence are extended to the class of Laguerre polynomials also. Finally results appeared in this work so far are revisited. Open problems for future research are underlined wherever possible. These include investigation of the range of parameters for which one has the orthogonality of linear combinations of these polynomi-als, the Fisher information related to the probability measure, Cramer-Rao information plane and the results for the chain sequences related to Jacobi polynomials and the class of polynomials NI,C,,")(x). Besides given in the end of each chapters, concluding remarks are also given in this chapter for completeness.