A q-analogue for partial-fraction decomposition of a rational function and its application

In this paper, by using the residue method of complex analysis, we obtain a q-analogue for partial-fraction decomposition of the rational function $\frac{x^M}{{(x+1)}_n^{\lambda }}$ $\frac{x^{M}}{(x+1)^{\lambda}_{n}}$ . As applications, we deduce the corresponding q-algebraic and q-combinatorial identities which are the q-extensions of Chu’ results.