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JFM 34.0498.02
Elliott, E. B.
A formula including {\it Legendre}'s $EK' + KE' - KK'=\frac 12 \pi$.
(English)
[J] Messenger (2) 33, 31-32 (1903).

Man setze \align L(a,b,c;k)& =\int_0^K \text{sn}^au \text{\,cn}^bu \text{\,dn}^cudu, \text{\,mod.\,}k, \\ & =\int_0^1 z^a(1-z^2)^{\frac 12(b-1)}(1-k^2z^2)^{\frac 12(c-1)} dz, \\ & =\tfrac 12 \int_0^1 x^{\frac 12(a-1)} (1-x)^{\frac 12(b-1)} (1-k^2x)^{\frac 12(c-1)} dx, \endalign so gilt die Formel: $$L(a,b,c+2;k)L(c,b,a;k') + L(a,b,c;k)L(c,b,a+2;k')$$ $$-L(a,b,c;k)L(c,b,a; k')$$ $$=\frac{\varGamma \left( \frac{a+1}{2} \right) \varGamma \left( \frac{b+1}{2} \right) \varGamma \left( \frac{c+1}{2} \right) }{ 4\varGamma \left( \frac{a+b+c+3}{2} \right)}\,.$$
(Data of JFM: JFM 34.0498.02; Copyright 2005 Jahrbuch Database used with permission)
[Lampe, Prof. (Berlin)]

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