Opened 5 years ago

Last modified 5 years ago

#16652 new enhancement

expansion of psi(m/n)

Reported by: rws Owned by:
Priority: minor Milestone: sage-wishlist
Component: symbolics Keywords: special, psi, polygamma, expansion
Cc: Merged in:
Authors: Reviewers:
Report Upstream: N/A Work issues:
Branch: Commit:
Dependencies: Stopgaps:

Description

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

Change History (3)

comment:1 Changed 5 years ago by rws

  • Keywords polygamma added; digamma removed
  • Priority changed from major to minor

comment:2 Changed 5 years ago by vbraun_spam

  • Milestone changed from sage-6.3 to sage-6.4

comment:3 Changed 5 years ago by rws

  • Milestone changed from sage-6.4 to sage-wishlist
Note: See TracTickets for help on using tickets.