A base-
BBP-type formula is a convergent series formula of the type
(1)
where
and
are integer polynomials in (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.
(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
(33)
Borwein et al. (2004) have recently shown that 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
(34)
where
is the golden ratio, found by B. Cloitre (Cloitre;
Borwein and Chamberland 2005; Bailey et al. 2007, p. 277).
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852-855, 1997.Adamchik, V. and Wagon, S. "Pi: A 2000-Year Search
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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 Unpublished manuscript.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 in Golden Base." Unpublished manuscript. Undated.Finch,
S. R. "Archimedes' Constant." §1.4 in Mathematical
Constants. Cambridge, England: Cambridge University Press, pp. 17-28,
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S. "The Story Behind a Formula for Pi." sci.math and sci.math.symbolic
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