A TWO- PARAMETER GENERALISATION OF THE GAMMA FUNCTION AND ITS PROPERTIES

Abstract

This thesis presents a significant contribution to the field of special functions by introducing and thoroughly analyzing a novel two-parameter generalization of the Gamma function: the ( ,v)-generalizedGammafunction.Theresearchsuccess fully addressed a critical gap in the literature by unifying previously disparate generalizations, such as the-analogue and v-analogue Gamma functions. The thesis meticulously presented the ( , v)-generalizedGamma function through amodified integral representationandderivedseveral keyproperties, including recurrence relations and integral representations. A core strength of the work lies in its exploration of the properties and associated inequalities of the ( ,v) generalized Gamma function. Key analytical tools employed include H¨older’s inequality and Young’s inequality, alongwith techniques of integrationand di↵erentiation. The proofs provided are rigorous and clearly presented.