Pi Squared (original) (raw)

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Campbell (2022) used the WZ method to obtain the sum

(pi^2)/4=sum_(n=1)^infty(16^n(n+1)(3n+1))/(n(2n+1)^2(2n; n)^3), (1)

where (n; k) is a binomial coefficient.

There is a series of BBP-type formulas for pi^2 in powers of (-1)^k,

2^k,

some of which are noted by Bailey et al. (1997), and 3^k,

A dilogarithm identity is given by

pi^2=36Li_2(1/2)-36Li_2(1/4)-12Li_2(1/8)+6Li_2(1/(64)), (14)

where Li_n is the polylogarithm, which is equivalent to

pi^2=12Li_2(1/2)+6(ln2)^2 (15)

(Bailey et al. 1997).

See also

Pi, Pi Formulas

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References

Bailey, D. H.; Borwein, P.; and Plouffe, S. "On the Rapid Computation of Various Polylogarithmic Constants." Math. Comput. 66, 903-913, 1997.Campbell, J. M. "WZ Proofs of Identities From Chu and Kiliç, With Applications." Appl. Math. E-Notes, 22, 354-361, 2022.

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Pi Squared

Cite this as:

Weisstein, Eric W. "Pi Squared." FromMathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PiSquared.html

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