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infinite_geometric_series [2016/07/22 15:19] nikolaj |
infinite_geometric_series [2019/09/23 21:19] nikolaj |
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== q-Integral == | == q-Integral == | ||
- | For a function $f$, the q-integral from $0$ to $1$ is defined as | + | For a function $f$, the q-integral from $0$ to $1$ ("$z$-integral" if we stick to our notation above) is defined as |
- | $\sum_{k=0}^\infty f(z^k)\,z^k=\dfrac{1}{1-z}\int_0^1 f(s){\mathrm d}_zs$ | + | $\sum_{k=0}^\infty f(z^k)\,z^k=\dfrac{1}{1-z}\cdot\int_0^1 f(s)\,{\mathrm d}_zs$ |
== Related notes == | == Related notes == | ||
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See also [[Niemand sequences]]. | See also [[Niemand sequences]]. | ||
+ | |||
+ | == Powers == | ||
+ | <code> | ||
+ | Sum[Binomial[-s, k] x^k, {k, 0, \[Infinity]}] | ||
+ | Series[(1 - x)^-s, {x, 0, 4}] | ||
+ | Binomial[-s, 3]; | ||
+ | % - (-s)!/(3! ((-s) - 3)!) // FullSimplify | ||
+ | %% + 1/6 (2 s + 3 s^2 + s^3) // FullSimplify | ||
+ | </code> | ||
=== References === | === References === |