Difference between revisions of "Q-exponential E sub 1/q"
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| − | * {{BookReference|A Comprehensive Treatment of q-Calculus|2012|Thomas Ernst|prev=Q-difference equation for q-exponential E sub q|next= | + | * {{BookReference|A Comprehensive Treatment of q-Calculus|2012|Thomas Ernst|prev=Q-difference equation for q-exponential E sub q|next=Q-difference equation for q-exponential E sub 1/q}}: (6.153) |
[[Category:SpecialFunction]] | [[Category:SpecialFunction]] | ||
Revision as of 22:46, 16 June 2016
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: █
References
- 2012: Thomas Ernst: A Comprehensive Treatment of q-Calculus ... (previous) ... (next): (6.153)
