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niemand_sequence [2016/07/14 01:26] nikolaj |
niemand_sequence [2016/07/22 15:13] nikolaj |
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| == Exponential function == | == Exponential function == | ||
| + | >todo: compare the following construction with the two q-exponentials | ||
| + | |||
| $a_{n,N} = \left(\prod_{k=1}^n\left(1-\dfrac{k-1}{N}\right)\right) \dfrac{1}{n!}\left(\dfrac{C(N)\,k\,z}{1-\frac{k\zeta}{N}}\right)^n $ | $a_{n,N} = \left(\prod_{k=1}^n\left(1-\dfrac{k-1}{N}\right)\right) \dfrac{1}{n!}\left(\dfrac{C(N)\,k\,z}{1-\frac{k\zeta}{N}}\right)^n $ | ||
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| ${\mathrm e}_N'(\zeta)=k\,{\mathrm e}_N(\zeta)$ | ${\mathrm e}_N'(\zeta)=k\,{\mathrm e}_N(\zeta)$ | ||
| + | |||
| + | (only for $z=\zeta$, not for all of $z$ like the exponential function) | ||
| and with $c(N)=1$, we have | and with $c(N)=1$, we have | ||
