Geodesic
An Apéry-like difference equation for Catalan's constant
The electronic journal of combinatorics, Tome 10 (2003)

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Applying Zeilberger's algorithm of creative telescoping to a family of certain very-well-poised hypergeometric series involving linear forms in Catalan's constant with rational coefficients, we obtain a second-order difference equation for these forms and their coefficients. As a consequence we derive a new way of fast calculation of Catalan's constant as well as a new continued-fraction expansion for it. Similar arguments are put forward to deduce a second-order difference equation and a new continued fraction for $\zeta(4)=\pi^4/90$.
DOI : 10.37236/1707
Classification : 11Y60, 33C20, 33F10, 39A05 
W. Zudilin. An Apéry-like difference equation for Catalan's constant. The electronic journal of combinatorics, Tome 10 (2003). doi: 10.37236/1707
@article{10_37236_1707,
     author = {W. Zudilin},
     title = {An {Ap\'ery-like} difference equation for {Catalan's} constant},
     journal = {The electronic journal of combinatorics},
     year = {2003},
     volume = {10},
     doi = {10.37236/1707},
     zbl = {1093.11075},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1707/}
}
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