Difference between revisions of "Product representation of q-exponential E sub 1/q"
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The following formula holds for $0 \leq |q| \leq 1$: | The following formula holds for $0 \leq |q| \leq 1$: | ||
$$E_{\frac{1}{q}}(z) = \displaystyle\prod_{k=0}^{\infty} \left[ 1 + (1-q)zq^k \right],$$ | $$E_{\frac{1}{q}}(z) = \displaystyle\prod_{k=0}^{\infty} \left[ 1 + (1-q)zq^k \right],$$ | ||
| − | where $E_{\frac{1}{q}}$ denotes the [[Q-exponential E sub 1/q]]. | + | where $E_{\frac{1}{q}}$ denotes the [[Q-exponential E sub 1/q|$q$-exponential $E_{\frac{1}{q}}$]]. |
==Proof== | ==Proof== | ||
Revision as of 02:36, 20 June 2016
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 $E_{\frac{1}{q}}$.
Proof
References
- 2012: Thomas Ernst: A Comprehensive Treatment of q-Calculus ... (previous): (6.155)
