\(q\)-extension of the Euler formula and trigonometric functions (Q1033712)

Replacing the factorial \(n!\) by the \(q\)-factorial \[ [n]_q!=\prod_{k=1}^n\frac{1-q^k}{1-q} \] the author defines the \(q\)-exponential function \[ e_q(z)=\sum_{n=0}^\infty\frac{z^n}{[n]_q!} \] and the related \(q\)-trigonometric functions \(\cos_q(z)\) and \(\sin_q(z)\) as the real and imaginary parts of \(e_q(iz)\). Then he derives \(q\)-analogues of classical formulae satisfied by these functions.