\(q\)-extension of the Euler formula and trigonometric functions. (English) Zbl 1188.33001

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.


33B10 Exponential and trigonometric functions
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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