First,we give the expansion of the AndrewsAskey integral with applications inbasic hypergeometric series.Using the generalized AndrewsAskey integral,we derivethe expansions of qPfaffSaalschǖtz formula.We also show the applications of thegeneralized AndrewsAskey integral in U(n+1) qseries and obtain the expansions of U(n+1)binomial theorems and some other new gidentities.Then,by means of the AndrewsAskey integral,a probabilistic distributionW(x;q)are been defined.We find the qintegral representation of the AlSalamCarlitzpolynomials.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 wellknown qbinomial theorem and the qGauss theorem.By theprobabilistic method and the AlSalamCarlitz polynomials,we obtain the expansion of the RogersRamanujan identity.Using W(x;q),Lebesgue’s dominated convergence theorem and analytic continuation theorem,we give the expansions of the followingidentities:qbinomial theorem,the qGauss theorem,3(?)2 transformation formula,Carlitz identity,Jackson transformation formula and qKarlssonMinton formula.Finally,the convergence of gintegral 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.
