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Pi Squared

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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,

pi^2=12sum_(k=0)^(infty)((-1)^k)/((k+1)^2)
(2)
=sum_(k=0)^(infty)(-1)^k[(13)/((3k+1)^2)-(13)/((3k+2)^2)+4/((3k+3)^2)]
(3)
=12sum_(k=0)^(infty)(-1)^k[1/((5k+1)^2)-1/((5k+2)^2)+1/((5k+3)^2)-1/((5k+4)^2)+1/((5k+5)^2)]
(4)
=12sum_(k=0)^(infty)(-1)^k[1/((7k+1)^2)-1/((7k+2)^2)+1/((7k+3)^2)-1/((7k+4)^2)+1/((7k+5)^2)-1/((7k+6)^2)+1/((7k+7)^2)]
(5)
=sum_(k=0)^(infty)(-1)^k[(13)/((7k+1)^2)-(13)/((7k+2)^2)+(13)/((7k+3)^2)-(13)/((7k+4)^2)+(13)/((7k+5)^2)-(13)/((7k+6)^2)-(36)/((7k+7)^2)]
(6)
=sum_(k=0)^(infty)(-1)^k[(13)/((9k+1)^2)-(13)/((9k+2)^2)+4/((9k+3)^2)-(13)/((9k+4)^2)+(13)/((9k+5)^2)-4/((9k+6)^2)+(13)/((9k+7)^2)-(13)/((9k+8)^2)+4/((9k+9)^2)].
(7)

2^k,

pi^2=sum_(k=0)^(infty)1/(16^k)[8/((8k+1)^2)-8/((8k+2)^2)-4/((8k+3)^2)-8/((8k+4)^2)-2/((8k+5)^2)-2/((8k+6)^2)+1/((8k+7)^2)]
(8)
=9/8sum_(k=0)^(infty)1/(64^k)[(16)/((6k+1)^2)-(24)/((6k+2)^2)-8/((6k+3)^2)-6/((6k+4)^2)+1/((6k+5)^2)]
(9)
=1/8sum_(k=0)^(infty)1/(256^k)[(128)/((16k+1)^2)-(128)/((16k+2)^2)-(128)/((16k+3)^2)-(128)/((16k+4)^2)-(32)/((16k+5)^2)-(32)/((16k+6)^2)+(16)/((16k+7)^2)+8/((16k+9)^2)-8/((16k+10)^2)-4/((16k+11)^2)-8/((16k+12)^2)-2/((16k+13)^2)-2/((16k+14)^2)+1/((16k+15)^2)]
(10)
=9/(512)sum_(k=0)^(infty)1/(4096^k)[(1024)/((12k+1)^2)-(1536)/((12k+2)^2)-(512)/((12k+3)^2)-(384)/((12k+4)^2)+(64)/((12k+5)^2)+(16)/((12k+7)^2)-(24)/((12k+8)^2)-8/((12k+9)^2)-6/((12k+10)^2)+1/((12k+11)^2)]
(11)

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

pi^2=2/(27)sum_(k=0)^(infty)1/(729^k)[(243)/((12k+1)^2)-(405)/((12k+2)^2)-(81)/((12k+4)^2)-(27)/((12k+5)^2)-(72)/((12k+6)^2)-9/((12k+7)^2)-9/((12k+8)^2)-5/((12k+10)^2)+1/((12k+11)^2)]
(12)
=2sum_(k=0)^(infty)1/(531441^k)[(177147)/((24k+1)^2)-(295245)/((24k+2)^2)-(59049)/((24k+4)^2)-(19683)/((24k+5)^2)-(52488)/((24k+6)^2)-(6561)/((24k+7)^2)-(6561)/((24k+8)^2)-(3645)/((24k+10)^2)+(729)/((24k+11)^2)+(243)/((24k+13)^2)-(405)/((24k+14)^2)-(81)/((24k+16)^2)-(27)/((24k+17)^2)-(72)/((24k+18)^2)-9/((24k+19)^2)-9/((24k+20)^2)-5/((24k+22)^2)+1/((24k+23)^2)].
(13)

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." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PiSquared.html

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