It is very difficult, sometimes impossible, to get a formula solution of recurrence relation, even for the case of homogeneous recurrence with one indice. In this paper, according to the principle of soluting algebraic equation, we present the formula of solution for a class of recurrnce relations with two indices by appling iteration and induction. It provides a concrete model to solve the concerning problems with modern computing tools.
Some atomic decomposition theorems for Banach-space-valued martingales are established. Using them, the embedding relationships between martingale spaces with small index are discussed. The results obtained here are connected closely with the p-uniform smoothness and q-uniform convexity of Banach space in which the martingales take values.
On the basis of author's former work, an explicit solution for a special class of homogeneous recurrence with two indices has been derived. It provides a concrete model to solve the concernced problems with computer. This consequence is of certain meaning both in theory and practice.
