Definition:Ramanujan Phi Function/One Argument
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Definition
When $n$ is omitted, the one argument Ramanujan phi function, $\map \phi a$, is defined as:
| \(\ds \map \phi a\) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \map \phi {a, n}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 + 2 \sum_{k \mathop = 1}^{\infty} \dfrac 1 {\paren {a k }^3 - a k}\) |
Also see
- Results about the Ramanujan phi function can be found here.
Source of Name
This entry was named for Srinivasa Aiyangar Ramanujan.
Sources
- 1985: Bruce C. Berndt: Ramanujan's Notebooks: Part I: Chapter $2$. Sums Related to the Harmonic Series or the Inverse Tangent Function
- Weisstein, Eric W. "RamanujanPhi-Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RamanujanPhi-Function.html