Product representation of q-exponential E sub 1/q

Theorem

The following formula holds for $0 \leq |q| \leq 1$: $$E_{\frac{1}{q}}(z) = \displaystyle\prod_{k=0}^{\infty} \left[ 1 + (1-q)zq^k \right],$$ where $E_{\frac{1}{q}}$ denotes the $q$-exponential $E_{\frac{1}{q}}$.

References

• 2012: Thomas Ernst: A Comprehensive Treatment of q-Calculus ... (previous) ... (next): ($6.155$)