# Q-exponential e sub 1/q

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The $q$-exponential $e_{\frac{1}{q}}$ is an entire function and is defined for $0 < |q| < 1$ by $$e_{\frac{1}{q}}(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{q^{ {k \choose 2} }}{(1;q)_k} z^k,$$ where ${n \choose 2}$ denotes the binomial coefficient and $(1;q)_k$ is the q-shifted factorial.