Lusztig gives that when integer N＞M ≥ 0,i=j+1,there is a formula E(N)i,i+1E(M)j,j+1=Σ0≤r≤M(-1)r[r+N-M-1N-M-1]E(M-r)i,i+1E(M)j,j+1 E(r+N-M)i,i+1 called the higher order quantum Serre relations.This article uses the multiplication formula of quantum groups and mathematical induction to generalize this conclusion,and proves that this formula also holds when N＞M ≥ 0,|i-j|=1,it is helpful for studying the generators and relationships of quantum groups of integers,as well as multiplication formulas of other types of algebras.
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