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Thursday, September 29, 2016

Entry 29

The first Watson triple integral was discussed in Enry 26. It turns out all three integrals can be expressed simply by the Dedekind eta function η(τ)I1=1π3π0π0π0dxdydz1cosxcosycosz=4η4(i)I2=1π3π0π0π0dxdydz3cosxcosycosxcoszcosycosz=41/33η4(3)I3=1π3π0π0π0dxdydz1cosxcosycosz=26(1+2)1/3η4(6)
In terms of the gamma functionI1=Γ4(14)4π3I2=3Γ6(13)214/3π4I3=Γ(124)Γ(524)Γ(724)Γ(1124)166π3
. There are infinitely many Ramanujan-type formulas for the Ii a few of which are (1+2)Γ(38)4(2π)5/2I1=n=0(2n)!3n!61(26)n12I1=n=0(2n)!3n!61(29)n43I2=n=0(2n)!3n!61(28)n1+2+66I3=n=0(2n)!3n!61(23(1+2+6)3(1+2))n
There is one other formula that belongs to the family with denominators as powers of 216η(7)4=n=0(2n)!3n!61(212)n
but I don't know if this has an equivalent and analogous integral.

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