Study On Q-operator,q-fractional Integral,q-identities And Q-polynomials

In this dissertation,we study on q-differential operator,q-fractional calulus(q-fractional Libniz formula,q-fractional integral,q-fractional derivative)and some q-polynomials.Also we point out that q-operator plays an important role in deriving applications for basic hyperge-ometric series.In accordance with chapter order,the main results are as follows:We start by introducing some of the definitions and the basic identities on basic hyperge-ometric series.Specifically,q-derivative,q-integral,q-Libniz theorem,q-summation formulas and the most important of all is q-binomial theorem which plays a very vital part in deriving many new identities.Followed by introduction,we discuss few applications related to q-operator,researched by Chen and Liu,Zhang,V Y.B.Chen and Gu,Fang and they obtained many well known transformation,summation formulas and much more known and new identities for example extensions of the Askey-Wilson integral,the Askey-Roy integral,Sears' two-term summation formula,as well as the q-analogs of Barnes' lemmas,bivariate Rogers-Szego polynomials.In Chapter ?,we have derived two new extensions of Mehler formula by the method of q-exponential operator T(a,b,D_q).It proceeds by stating how Liu in 2015 by using method of q-partial differential equations,reproved Mehler formula for Hahn polynomials proved by Al-Salam and Carlitz(1965).The study about q-fractional calculus and some of its applications is discussed under Chapter IV.We discuss some properties for q-fractional integral,using which the q-fractional Libniz theorem is defined and hence we get some basic ideas on q-fractional derivative.In the end we study some applications related to q-Libniz theorem.Finally,we define a homogeneous q-differential operator E(bD_xy)on functions in two variable which turns out to be suitable for dealing with homogeneous form of q-polynomials.The operator defined here is mainly concerned in proving identities that involve trivariate Rogers-Szego polynomials rn(x,y,b).Also we derive some identities related to operator E(bD_xy)which is used in verifying well known identities like generating function,Rogers' formula,Mehler's formula of rn(x,y,b).The identities of trivariate Rogers-Szego polynomials can be considered as a general form of generalized Rogers-Szego polynomials rn(x,b),and classical Rogers-Szego polynomials hn(x|q),when y=0,and y = 0,b= 1 respectively.Further,we prove several extensions of trivariate Rogers-Szego polynomials rn(x,y,b)such as extended generating function,extended Rogers' formula and extended Mehler's formula.