On Two Kinds Of Recurrence Relations And Their Applications

This paper is devoted to two kinds of recurrence relations and their applications. It is composed of the following two parts.The first part is mainly concerned with the so-called combinatorial sum where F(n, k) can be decomposed into the product of two sequences A(n, k) and B(n,k). Both two usually satisfy certain good recurrence relations. Arising form this problem is a recurrence relation which is helpful in finding a generalized Abel lemma. Some application to combinatorial identities are discussed.In the second part, we focus on another finite q-function equation built on the q-derivate operator, as follows: which contains f(a)= and tn(a)= (as;q)ntn as a special solution useful to basic hypergeometric series. As main results, we sketch its general solution and then show such solution is unique in context of hypergeometric series.