Some Generalizations Of Q-integral And Fractional Q-integral Identities

q-Integral and fractional q-integral are hot topic in the field of mathematics.This thesis mainly studies the generalizations of several q-integral and fractional q-integral identities,and explores and generalizes the q-integral identities through the definition of fractional q-integral,q-difference equation and partial integral,which provides a foundation for further study of fractional q-integral theory and application.1.According to the definition of Riemann-Liouville fractional q-integral,we show the technique of iteration and generalize Riemann-Liouville fractional q-integrals involving Cauchy polynomials.In addition,we gain generalizations of Srivastava-Agarwal type generating functions by generalized fractional q-integrals involving Cauchy polynomials.Moreover,we derive generalizations for Rajkovi(?)-Marinkovi(?)-Stankovi(?) polynomials and generalizations of Jackson’s transformation formula involving Cauchy polynomial by fractional q-integrals.At last,We have also briefly considered relevant connection of the results,which we have established in this thesis with other known or new results.2.Based on the research status of q-difference equation of Al-Salam-Carlitz polynomial,motivated by methods of "Parameter Augmentation" and q-difference equation,the q-difference equation of q-exponential operator and the q-difference equation of q-Rogers-Szeg(?) polynomial are generalized with the motivation of parameter augitation method and q-difference equation.Using the method of q-difference equation,the partial generating function of Al-Salam-Carlitz polynomials and the proof of Rogers formula are generalized.We calculate some identities of qexponential operators involving the new generalized Al-Salam-Carlitz polynomials,and count generalizations for new generalized Al-Salam-Carlitz polynomials.We deduce two bilinear generalizations for generalized Al-Salam-Carlitz polynomials by q-difference equations,and obtain a transformational identity from q-difference equations.3.Through the research of double q-integral,the double and triple forms of Askey-Roy integral are deduced by analogy.Some basic equations of double Askey-Roy integral and triple Askey-Roy integral are obtained by using this formula,and the application of double Askey-Roy integral in q-Chu-Vandermonde equation and U(n+1)type formula is also obtained.