Q-exponential E sub 1/q

The $E_{\frac{1}{q}}$ function is defined by the formula $$E_{\frac{1}{q}}(z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{q^{ {k \choose 2} }}{[k]_q!} z^k.$$

Properties

Theorem: The following formula holds: $$D_q E_{\frac{1}{q}}(az)=aE_{\frac{1}{q}}(qaz),$$ where $D_q$ denotes the q-difference operator and $E_{\frac{1}{q}}$ denotes the $q$-exponential $E_{\frac{1}{q}}$.

Proof: █