An s-fold integral of three polynomials against 1/(1 - x1 x2 ... xs) over the unit cube can be written as an explicit linear combination of the zeta-constants zeta(2), ..., zeta(s), with coefficients built from the polynomial coefficients and from generalized harmonic numbers. Two combinatorial lemmas produce those coefficients in closed form: a geometric-series expansion of the kernel, and a partial-fraction identity for 1/((r+k+1)^p (k+1)^s) proved by induction. Choosing the shifted Legendre polynomial and (1-x)^n as two of the three factors forces the integral below c* 2^(-2n), so the relations become a linear system whose determinant solution approximates zeta(s) by rational fractions; the freedom left in the third polynomial can be spent either on convergence rate or on cutting how many fractions must be computed.
Derives explicit formulas expressing the s-fold integral I_s (the integral over the unit cube of P_n(x1)Q_n(x2)T_n(x3)/(1 - x1 x2 ... xs)) as a linear combination of the zeta-constants zeta(2), zeta(…