Saturday, October 1, 2016
Entry 32
It is well-known that1π=2√2992∞∑k=0(4k)!k!429⋅70⋅13k+1103(3964)kBut it turns out we can also use the square root √3964=±3962 as a median point for two other pi formulas. First express the above as1π=192√2(3962)3/2∞∑k=0(4k)!k!42⋅58⋅15015k+72798(3964)kthen1π=192√2(3962−16)3/2∞∑k=0(2kk)k∑j=0(kj)(2k−2jk−j)(2jj)58⋅15015k+(72798+37)(−3962+16)k1π=192√2(3962+16)3/2∞∑k=0(2kk)k∑j=0(kj)(2k−2jk−j)(2jj)58⋅15015k+(72798−37)(3962+16)k where (nk) is the binomial coefficient. Note that they have a beautifully symmetric form and how the same integers (which figure in Pell equations as discussed in Entry 1) appear in all three formulas. These two are level-8 Ramanujan-Sato series.
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