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q-Exponential Function

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The exponential function has two different natural q-extensions, denoted e_q(z) and E_q(z). They are defined by

e_q(z)=sum_(n=0)^(infty)(z^n)/((q;q)_n)
(1)
=_1phi_0[0; -;q;z]
(2)
E_q(z)=sum_(n=0)^(infty)(q^(n(n-1)/2))/((q;q)_n)z^n
(3)
=_1phi_0[-; -;q;-z]
(4)
=(-z;q)_infty
(5)

(Koekoek and Swarttouw 1998, p. 18), where _rphi_s(a_1,...,a_r;b_1,...,b_s;q;z) is a q-hypergeometric function.

e_q(z) has the special form

e_q(z)=1/((z;q)_infty)
(6)

for |z|<1.

The q-exponential functions are related to the q-cosine and q-sine by

e_q(iz)=cos_q(z)+isin_q(z)
(7)
E_q(iz)=Cos_q(z)+iSin_q(z).
(8)

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