As you probably know, Plouffe invented a famous formula for pi which basically enables computing its Nth decimal place (but in radix 16 not 10) without computing previous decimals(!), in about O(N) steps and O(logN) memory. Other versions appeared collectively called BBP formulas. http://en.wikipedia.org/wiki/Bailey–Borwein–Plouffe_formula http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/bbp-formulas.pdf Constants now known to be in BBP class include pi, ln2, ln3, atan2, atan(4/5), pi*sqrt(2), pi*sqrt(3), sqrt(2)*ln(1+sqrt(2)), pi^2, (ln2)^2, pi*ln(2), CatalanConstant, zeta(3), zeta(5). Anyhow, this has probably been thought of before, but I did not see it in those sources... so I'd like to ask: WHICH FUNCTIONS F(x) have Maclaurin series which yield BBP formulae as their special cases when x=1/radix? In MAPLE notation, define (where A>=0 and B>0 and C>0 and k>=0 and D,E,F... are integers and x is complex or real) F0 := (A,B,C,x) -> x^A / (1-x^B)^C; F1 := (A,B,C,D,x) -> int( x^D * F0(A,B,C,x), x ); F2 := (A,B,C,D,E,x) -> int( x^E * F1(A,B,C,D,x), x ); F3 := (A,B,C,D,E,F,x) -> int( x^F * F2(A,B,C,D,E,x), x ); etc. [You must not take D,E,F... too negative or the integrals will not converge near 0.] THEOREM: All rational-function linear combinations of the Fk(A,B,C,..., x) are BBP class (where the rational functions of x have integer coefficients). For example, (x+3)/(3*x) * F2(2,5,1,0,0,x) + (11/7) * F1(3,4,2,0,x) and F2(3,4,1,-1,-1,x) are in BBP class, and if you ask MAPLE what these are, it will do the integrals and spit out some enormous closed form formula for you. I think this set includes all the formulas in the Bailey compendium above (though I'm not sure of that). But despite its enormity, the set I have described is not the full BBP class. For example Goofy(x) = sum( x^n / (n^2+1), n=0..infinity) = 3F2([1, i, -i], [1 + i, 1 - i], x) is in BBP class but not in my set. A larger set, which comes a lot closer to being the whole BBP class, is as follows. Redefine F0 to be the class of rational functions of x (with integer coefficients of course). Now for k>0 redefine Fk to be the class of rational-fn linear combinations of functions in F[k-1] and integrals(dx) of rational-fn linear combinations of functions from class F[k-1]. Disallow members including integrals which are nonanalytic at x=0. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)