#16652: expansion of psi(m/n)
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Reporter: rws | Owner:
Type: | Status: new
enhancement | Milestone: sage-6.3
Priority: major | Keywords: special, psi, digamma,
Component: symbolics | expansion
Merged in: | Authors:
Reviewers: | Report Upstream: N/A
Work issues: | Branch:
Commit: | Dependencies:
Stopgaps: |
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As example, expressions with several psi values of rational argument
result from infinite sums:
{{{
sage: sum((-1)^(k+1)/(4*k-3), k, 1, oo)
1/8*psi(5/8) - 1/8*psi(1/8)
}}}
`psi(m/n)`, m<k, has a closed form of finitely many terms of elementary
functions, so differences of psi values can yield nice expressions like
`1/8*psi(5/8) - 1/8*psi(1/8) = 1/(4*sqrt(2))*(pi+2*log(sqrt(2)+1))`
To arrive at such simplifications the expansion of `psi(m/n)` using Gauss'
Digamma Theorem should be implemented.
https://en.wikipedia.org/wiki/Digamma_function#Gauss.27s_digamma_theorem
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Ticket URL: <http://trac.sagemath.org/ticket/16652>
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