BBP-Type Formula
A base-b BBP-type formula is a convergent series formula of the type
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.
![画像: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)]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/BBP-TypeFormula/Inline13.svg){kind=link}
![画像: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)]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/BBP-TypeFormula/Inline16.svg){kind=link}
![画像: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)]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/BBP-TypeFormula/Inline25.svg){kind=link}
![画像: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)]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/BBP-TypeFormula/Inline82.svg){kind=link}
![画像: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)]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/BBP-TypeFormula/Inline88.svg){kind=link}
![画像:sqrt(3)sum_(k=0)^(infty)[1/((6k+1)^2)+1/((6k+2)^2)-1/((6k+4)^2)-1/((6k+5)^2)]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/BBP-TypeFormula/Inline91.svg){kind=link}
![画像:(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)].](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/BBP-TypeFormula/Inline94.svg){kind=link}
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, ZeroExplore with Wolfram|Alpha
More things to try:
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. http://crd.lbl.gov/~dhbailey/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.Referenced on Wolfram|Alpha
BBP-Type FormulaCite this as:
Weisstein, Eric W. "BBP-Type Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BBP-TypeFormula.html