## $$q$$-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.

### MSC:

 33B10 Exponential and trigonometric functions 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$
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### References:

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