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BBP-Type Formula

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A base-b BBP-type formula is a convergent series formula of the type

alpha=sum_(k=0)^infty(p(k))/(b^kq(k))
(1)

where p(k) and q(k) are integer polynomials in k (Bailey 2000; Borwein and Bailey 2003, pp. 54 and 128-129).

Bailey (2000) and Borwein and Bailey (2003, pp. 128-129) give a collection of such formulas. The following extends those compilations to include several additional BBP-type formulas.

pi=sum_(k=0)^(infty)1/(16^k)(4/(8k+1)-2/(8k+4)-1/(8k+5)-1/(8k+6))
(2)
=1/2sum_(k=0)^(infty)1/(16^k)(8/(8k+2)+4/(8k+3)+4/(8k+4)-1/(8k+7))
(3)
pi^2=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)]
(4)
=2/(27)sum_(k=0)^(infty)1/(729^k)[(243)/((12k+1)^2)-(405)/((12k+2)^2)-(81)/((12k+4)^4)-(27)/((12k+5)^2)-(72)/((12k+6)^2)-9/((12k+7)^2)-9/((12k+8)^2)-5/((12k+10)^2)+1/((12k+11)^2)]
(5)
ln(9/(10))=-sum_(k=1)^(infty)1/(10^k·k)
(6)
ln2=2/3sum_(k=0)^(infty)1/(9^k(2k+1))
(7)
(ln2)^2=1/(32)sum_(k=0)^(infty)[(64)/((6k+1)^2)-(160)/((6k+2)^2)-(56)/((6k+3)^2)-(40)/((6k+4)^2)+4/((6k+5)^2)-1/((6k+6)^2)]
(8)
ln3=1/(729)sum_(k=0)^(infty)1/(729^k)((729)/(6k+1)+(81)/(6k+2)+(81)/(6k+3)+9/(6k+4)+9/(6k+5)+1/(6k+6))
(9)
=sum_(k=0)^(infty)1/(4^k(2k+1))
(10)
ln10=1/(32)sum_(k=0)^(infty)1/((-64)^k)((64)/(12k+1)+(16)/(12k+2)+8/(12k+4)-(16)/(12k+5)+8/(12k+6)+(12)/(12k+7)-2/(12k+8)+4/(12k+9)+1/(12k+10)-3/(12k+11)-1/(12k+12))
(11)
=1/2sum_(k=0)^(infty)1/((-4)^k)(6/(4k+1)-3/(4k+3)-1/(4k+4))
(12)
=1/(16)sum_(k=0)^(infty)1/(16^k)((24)/(4k+1)+(20)/(4k+2)+6/(4k+3)+1/(4k+4))
(13)
=1/8sum_(k=0)^(infty)1/(16^k)((16)/(8k+1)+8/(8k+2)-8/(8k+3)+4/(8k+4)-4/(8k+5)+2/(8k+6)+2/(8k+7)+1/(8k+8))
(14)
=2/9sum_(k=0)^(infty)1/(81^k)(9/(4k+1)+2/(4k+2)+1/(4k+3))
(15)
=2/(729)sum_(k=0)^(infty)1/(6561^k)((729)/(8k+1)+(162)/(8k+2)+(81)/(8k+3)+9/(8k+3)+2/(8k+3)+1/(8k+7))
(16)
pisqrt(2)=sum_(k=0)^(infty)1/((-8)^k)(4/(6k+1)+1/(6k+3)+1/(6k+5))
(17)
=1/(64)sum_(k=0)^(infty)1/((-512)^k)((256)/(18k+1)+(64)/(6k+3)+(64)/(18k+5)-(32)/(18k+7)-8/(18k+9)-8/(18k+11)+4/(18k+13)+1/(18k+15)+1/(18k+17))
(18)
=4sum_(k=0)^(infty)(-1)^k(1/(4k+1)+1/(4k+3))
(19)
=4sum_(k=0)^(infty)(-1)^k(1/(12k+1)+1/(12k+3)-1/(12k+5)-1/(12k+7)+1/(12k+9)+1/(12k+11))
(20)
=sum_(k=0)^(infty)(-1)^k(3/(20k+1)+3/(20k+3)+2/(20k+5)-3/(20k+7)+3/(20k+9)+3/(20k+11)-3/(20k+13)+2/(20k+17)+3/(20k+19))
(21)
=1/8sum_(k=0)^(infty)1/(64^k)((32)/(12k+1)+8/(2k+3)+8/(12k+5)-4/(12k+7)-1/(12k+9)-1/(12k+11))
(22)
pisqrt(3)=1/4sum_(k=0)^(infty)1/(64^k)((20)/(6k+1)+6/(6k+2)-1/(6k+3)-3/(6k+4)-1/(6k+5))
(23)
=1/9sum_(k=0)^(infty)1/(729^k)((81)/(12k+1)-(54)/(12k+2)-9/(12k+4)-(12)/(12k+6)-3/(12k+7)-2/(12k+8)-1/(12k+10))
(24)
=1/(36)sum_(k=0)^(infty)1/(729^k)((81)/(12k+1)+(27)/(12k+2)-(162)/(12k+3)-9/(12k+4)+(27)/(12k+5)+(24)/(12k+6)-3/(12k+7)+7/(12k+8)+6/(12k+9)+3/(12k+10)-1/(12k+11))
(25)
=1/9sum_(k=0)^(infty)1/(729^k)((81)/(12k+1)+(189)/(12k+2)+(45)/(12k+4)+(27)/(12k+5)+(24)/(12k+6)-3/(12k+7)+1/(12k+8)+1/(12k+10)-1/(12k+11))
(26)
piln2=1/(256)sum_(k=0)^(infty)1/(4096^k)[(4096)/((24k+1)^2)-(8192)/((24k+2)^2)-(26112)/((24k+3)^3)+(15360)/((24k+4)^2)-(1024)/((24k+5)^2)+(9984)/((24k+6)^2)+(11520)/((24k+8)^2)+(2368)/((24k+9)^2)-(512)/((24k+10)^2)+(768)/((24k+12)^2)-(64)/((24k+13)^2)+(408)/((24k+15)^2)+(720)/((24k+16)^2)+(16)/((24k+17)^2)+(196)/((24k+18)^2)+(60)/((24k+20)^2)-(37)/((24k+21)^2)]
(27)
K=sum_(k=0)^(infty)((-1)^k)/((2k+1)^2)
(28)
=1/(1024)sum_(k=0)^(infty)1/(4096^k)[(3072)/((24k+1)^2)-(3072)/((24k+2)^2)-(23040)/((24k+3)^2)+(12288)/((24k+4)^2)-(768)/((24k+5)^2)+(9216)/((24k+6)^2)+(10368)/((24k+8)^2)+(2496)/((24k+9)^2)-(192)/((24k+10)^2)+(768)/((24k+12)^2)-(48)/((24k+13)^2)+(360)/((24k+15)^2)+(648)/((24k+16)^2)+(12)/((24k+17)^2)+(168)/((24k+18)^2)+(48)/((24k+20)^2)-(39)/((24k+21)^2)]
(29)
Cl_2(1/3pi)=sqrt(3)sum_(k=0)^(infty)[1/((6k+1)^2)+1/((6k+2)^2)-1/((6k+4)^2)-1/((6k+5)^2)]
(30)
=(sqrt(3))/9sum_(k=0)^(infty)((-1)^k)/(27^k)[(18)/((6k+1)^2)-(18)/((6k+2)^2)-(24)/((6k+3)^2)-6/((6k+4)^2)+2/((6k+5)^2)].
(31)

where K is Catalan's constant, Cl_2(pi/3) is the hyperbolic volume of the figure eight knot complement, Cl_2(x) is Clausen's integral, and Cl_2(pi/3) is also the hyperbolic volume of the knot complement of the figure eight knot.

Another example is the Dirichlet L-series

L_(-7)(2)=sum_(n=0)^infty[1/((7n+1)^2)+1/((7n+2)^2)-1/((7n+3)^2)+1/((7n+4)^2)-1/((7n+5)^2)-1/((7n+6)^2)]
(32)

(Bailey and Borwein 2005; Bailey et al. 2007, pp. 5 and 62).

Note that this sort of sum is closely related to the polygamma function since, for example, the above sum can also be written

L_(-7)(2)=1/(49)[psi_1(1/7)+psi_1(2/7)-psi_1(3/7)+psi_1(4/7)-psi_1(5/7)-psi_1(6/7)].
(33)

Borwein et al. (2004) have recently shown that pi has no Machin-type BBP arctangent formula that is not binary, although this does not rule out a completely different scheme for digit-extraction algorithms in other bases.

A beautiful example of a BBP-type formula in a non-integer base is

pi^2=50sum_(k=0)^infty1/(phi^(5k))[(phi^(-2))/((5k+1)^2)-(phi^(-1))/((5k+2)^2)-(phi^(-2))/((5k+3)^2)+(phi^(-5))/((5k+4)^2)+(2phi^(-5))/((5k+5)^2)],
(34)

where phi is the golden ratio, found by B. Cloitre (Cloitre; Borwein and Chamberland 2005; Bailey et al. 2007, p. 277).

See also

Apéry's Constant, BBP Formula, Catalan's Constant, Digit-Extraction Algorithm, Dirichlet L-Series, Inverse Sine, Natural Logarithm of 2, Pi, Pi Formulas, Spigot Algorithm, Zero

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References

Adamchik, V. and Wagon, S. "A Simple Formula for pi." Amer. Math. Monthly 104, 852-855, 1997.Adamchik, V. and Wagon, S. "Pi: A 2000-Year Search Changes Direction." http://www-2.cs.cmu.edu/~adamchik/articles/pi.htm.Bailey, D. H. "A Compendium of BBP-Type Formulas for Mathematical Constants." 28 Nov 2000, updated 8 Apr 2023. https://www.davidhbailey.com/dhbpapers/bbp-formulas.pdf.Bailey, D. H. and Borwein, J. M. "Experimental Mathematics: Examples, Methods, and Implications." Not. Amer. Math. Soc. 52, 502-514, 2005.Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, pp. 31-33 and 222, 2007.Bailey, D. H.; Borwein, P. B.; and Plouffe, S. "On the Rapid Computation of Various Polylogarithmic Constants." Math. Comput. 66, 903-913, 1997.Borwein, J. and Bailey, D. "Other BBP-Type Formulas" and "Does Pi Have a Nonbinary BBP Formula?" §3.6 and 3.7 in Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 127-133, 2003.Borwein, J. M.; Borwein, D.; and Galway, W. F. "Finding and Excluding b-ary Machin-Type Individual Digit Formulae." Canad. J. Math. 56, 897-925, 2004.Borwein, J. M. and Chamberland, M. "A Golden Example." Unpublished manuscript. Feb. 7, 2005.Cloitre, B. "A BBP Formula for pi^2 in Golden Base." Unpublished manuscript. Undated.Finch, S. R. "Archimedes' Constant." §1.4 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 17-28, 2003.Gourévitch, B. "L'univers de pi. §6: Formules BBP en base 2: s in N, v=p/q, x=1/(2^n) dans Psi." http://www.pi314.net/hypergse6.php.Plouffe, S. "The Story Behind a Formula for Pi." sci.math and sci.math.symbolic newsgroup posting. 23 Jun 2003.

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BBP-Type Formula

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Weisstein, Eric W. "BBP-Type Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BBP-TypeFormula.html

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