(from both my Macsyma and Mma 6.0): 3F2[a,b,c;d,e] = R(e) + r(e) Psi[1](e), for positive integer a,b,c, and d, where R and r are rational functions, and r(e) = 0 when max(a,b,c)>=d. E.g.,
hyper_f ([1, 1, 1], [3, e + 1], 1) = 2 e (e Psi (e) - 1). 3, 2 1 --rwg
Well, this is odd. As I went to implement this case it spontaneously started working: (c228) (hyper_f[1,1]([2,2,2],[3,x]),%% = hypersimp(%%)) (d228) hyper_f ([2, 2, 2], [3, x]) 3, 2 2 (x - 2) (x - 1) ((x - 3) Psi (x - 2) - 1) 1 = - ------------------------------------------- . x - 3 Note that the 3F2 series definition wants Re(x)>3 for convergence. (It's not obvious why it diverges for x=3+%i: The terms decrease like 1/n and the signs of the real and imaginary parts fluctuate. But as the term ratio approaches 1, its imaginary part fades away and with it the fluctuation frequency. The ever longer bursts of "parallel" terms slew the partial sums at decreasing frequency and nonvanishing amplitude.) But convergence is a red herring, since the numerical method (3 by 3 matrix product) gives consistent values for all x: (c232) expand(subst(1/2,x,d228)) 2 1 3 %pi 109 (d232) hyper_f ([2, 2, 2], [3, -]) = - ------ - --- 3, 2 2 4 15 (c233) dfloat(%) (d233) - 14.6688699674837d0 = - 14.6688699674837d0 (Note that the series terms are all positive.) Another EulerGamma series with degrees of freedom: For nonnegative integers p and m, (d197) Psi (n) = 0 k ==== \ j > (- 1) log(n - j - 1) binomial(p + k, p + j) m - 1 / ==== ==== \ j = 0 > -------------------------------------------------- / binomial(p + k + 1, p + 1) ==== k = 0 -------------------------------------------------------- p + 1 m + k ==== \ j > (- 1) binomial(m + k, j) log(n - m + j) inf / ==== ==== m \ j = 0 (- 1) > ---------------------------------------------- / binomial(m + k + 1, m + 1) ==== k = 0 + ----------------------------------------------------------- m + 1 m ==== \ j > (- 1) binomial(m + k, k + j) log(n - j) p - 1 / ==== ==== \ j = 1 - > ----------------------------------------------. / (k + 1) binomial(m + k + 1, k + 1) ==== k = 0 Some combinations of p and m converge faster than others. Note that replacing log by some other slowly growing (e.g., log^2) or decreasing (e.g., 1/x) function will compute some other f(n), independent of p and m. Random afterblurt: Does the sequence A081464 = 1,2,4,29,95,153,532,613,840,2033,..., = the n for which fractionpart((3/2)^n) form a decreasing sequence, characterize 3/2? If not, what set? 1,2,3,4,..., characterizes an infinite set of algebraics, (and 0<x<1, but I'd think most sequences would characterize the empty set. --rwg