On some problems of polynomials and fractional calculus

Abstract

The present thesis embodies the researches carried out at the Aligarh Muslim University, Aligarh. The thesis has been divided into eight chapters. Besides the introductory first chapter, second chapter deals with the generalized heat polynomials of several variables, third and fourth chapters concerns with the study of heat type polynomials suggested by Jacobi polynomials and Gaussand#8242; hypergeometric polynomials.The next two chapter i.e. fifth and sixth deal with the generalization of Bessel-Maitland function and a new class of polynomials associated with differential operator and with a generalization of Bessel-Maitland function. The second last chapter gives an integral representations for the product of certain families of hypergeometric polynomials. The last chapter introduce fractional derivatives, integrals and generating functions for certain class of hypergeometric polynomials. Chapter-1, covers a comprehensive account of the historical origin of various polynomials and up to date development made in these areas. This chapter contains some preliminary concepts and important well known results needed in the subsequent text. The chapter also briefly introduces the classical orthogonal polynomials, generating function relations, fractional derivatives and integrals, integral transformations and its applications in different field of science and technology. Further, a brief chapter wise summary of chapter II through VIII have also be given in this introductory chapter. Chapter-II, deals with the study of generalized heat polynomials of several variables. Certain linear and bilinear generating functions, special properties, a finite sum property, expansion in terms of Hermite and Legendre polynomials, integral representations, fractional integrals and derivatives, Sch¨afli s contour integral and Laplace transform have been obtained in this chapter. Contents of this chapter are published/accepted in the form of two papers.