WDS> H(n) = SUM(1/j, j=1..n) is the nth "harmonic number." QUESTION: what if we want to generalize this to real or complex n, not merely integer n, similarly to how Euler converted factorial to gamma function? ANSWER: Let psi(x) = d/dx ( ln Gamma(x) ) = Gamma'x) / Gamma(x) be the "psi function"; then H(x) = psi(x) + EulerMascheroniConstant = 0.5772156649 is the sought-for generalization. Euler also had found the integral representation H(x) = INTEGRAL( (1 - t^x) / (1-t), t=0..1 ) which is equivalent to this. http://en.wikipedia.org/wiki/Harmonic_number#Special_values_for_fractional_a... discusses further, and many other things are "continuizable" in similar way (also discussed in the wikipedia article). <WDS------- Conway prefers "Euler's constant". This wiki seems to lack the general formula for H(p/q), the last entry in the numeric tables ending each tAoCP. Also, that first H_x formula could be enlivened: H_x = lim y->∞ H_{x,y}, where H{x,y} = H{y,x} := Sum(y/k/(k+y))_{k=1}^x (not Mma's braindead Sum semantics). Also, by http://dspace.mit.edu/bitstream/handle/1721.1/6088/AIM-304.pdf p25: HarmonicNumber[p]==-Sum[(3*k + p)* Binomial[k - p - 1, k]/k/(k + p)/Binomial[2*k, k], {k, ∞}] (Terminates after (integer) p), which Mma converts to two useless 3F2s unless integer 2p. --rwg