Duplicate Form Of The Generalized Carlitz Inversions And Summation Formulae

The proof of q-hypergeometric identities have been paid a great attention,and people used different kinds of methods to study q-hypergeometric identities.Among them,the inversion method,namely using inversion relations to discover and prove identities,is one of the common methods.The key step of applying inverse relations to derive identities is how to get the inverse relations and whether we can find the identities that satisfy the inverse relations.In chapter 1,we look back on the history of hypergeometric series,q-hypergeometric series and inversion relations,and introduce some basic definitions and notations.In chapter 2,we present a simple proof of the duplicate form of the generalized Gould-Hsu inversion.With the same method,we construct and prove the duplicate form of the generalized Carlitz inversion.Applying the inversion to the suitable identities,we obtain several q-hypergeometric identities and limiting cases.