Research On Applications Of The Andrews-Askey Integral

First,we give the expansion of the Andrews-Askey integral with applications inbasic hypergeometric series.Using the generalized Andrews-Askey integral,we derivethe expansions of q-Pfaff-Saalschǖtz formula.We also show the applications of thegeneralized Andrews-Askey integral in U(n+1) q-series and obtain the expansions of U(n+1)binomial theorems and some other new g-identities.Then,by means of the Andrews-Askey integral,a probabilistic distributionW(x;q)are been defined.We find the q-integral representation of the Al-Salam-Carlitzpolynomials.Using this representation,some expectation formulas for W(x;q)are been derived.We construct the sequences of random variables to give the probabilisticderivations of the well-known q-binomial theorem and the q-Gauss theorem.By theprobabilistic method and the Al-Salam-Carlitz polynomials,we obtain the expansion of the Rogers-Ramanujan identity.Using W(x;q),Lebesgue's dominated convergence theorem and analytic continuation theorem,we give the expansions of the followingidentities:q-binomial theorem,the q-Gauss theorem,3(?)2 transformation formula,Carlitz identity,Jackson transformation formula and q-Karlsson-Minton formula.Finally,the convergence of g-integral are been discussed.We obtain some inequalities about r+1(?)r,r(?)r and r(?)r.Using the inequalities,we give some convergencetheorems for (?)z~α·r+1(?)r d_qz,(?)z~α·r(?)r d_qz and (?)z~α·r(?) d_qz.