Some Applications Of The (f,g)-inversion

The key of applying inverse pairs to derive identities is to get the inverse pairs and find the identities that satisfy the inverse relations.In this paper,we mainly study(f,g)-inversion.By taking some special value to derive(f,g)inverse pairs and applying them to hypergeometric series,harmonic number and basic hypergeometric series,we obtain a series of identities.In chapter 1,we look back on the history of hypergeometric series,basic hypergeometric series and inversion relations.In chapter 2,by taking explicit functions and sequences in the(f,g)-inversion,we derive some inverse pairs,and then get some new inverse pairs by adding factors on the original inverse pairs.At last,we compute a lot of q-inverse pairs and get some combinatorial identities.In chapter 3,we combine the(f,g)inversion and the q-differential operators to derive some basic hypergeometric series identities.