Finite form of the quintuple product identity

where the q-shifted factorial is defined by (x; q)0 = 1 and (x; q)n = (1− x)(1 − qx) · · · (1− q x) for n = 1, 2, · · · with the following abbreviated multiple parameter notation [α, β, · · · , γ; q]∞ = (α; q)∞(β; q)∞ · · · (γ; q)∞. This identity has several important applications in combinatorial analysis, number theory and special functions. For the historical note, we refer the reader to the paper [1]. In this short note, we shall show that identity (1) follows surprisingly from the following algebraic identity. Theorem (Finite form of the quintuple product identity). For a natural number m and a variable x, there holds an algebraic identity: