From: Bill Gosper <billgosper@gmail.com> Bisecting the sum, plus minor fooling around in Mma gave Out[446]= LogIntegral[x] == CoshIntegral[Log[x]] + SinhIntegral[Log[x]] In[457]:= FullSimplify[%446, x > 1] Out[457]= True WDS> Ramanujan [notebook]: Li(x) = EulerMascheroniGamma + lnln(x) - sqrt(x) * SUM(n>0) a[n] * ln(x)^n where a[n] = (-1/2)^n / (n!) * SUM(0<=m<=floor((n-1)/2)) 2/(2*m+1) which I suppose has a nicer matrix product reformulation. <WDS Triangular 3x3, confluent. I seem to recall the nonexistence of confluent 2x2 path-invariant systems (except CF), but maybe luck is better with 3x3. WDS>--"confluent" means what?< Nothing like it sounds. It just means pFq when p≠q. They are limiting cases of p+1Fp: add √t to two numerator parameters, and t/z to a denominator parameter, and let t blow up. wds> --The matrix product reformulation has the advantage that it runs faster than the naive double sum. Of course, I admit, we can just be a teeny bit smarter about the double sum to evaluate N terms of the outer sum in O(N) steps. Also, I'd like to understand where this identity came from and if there are more.<wds You should at least get something nice from CoshIntegral[Log[x]] + SinhIntegral[Log[x]] It looks tedious to work the Mma derivation backwards due to intermediate swell. wds> The naive series for Li(x) is Li(expX) = Ei(X) = EulerMascheroniGamma + ln(X) + SUM(k>0) X^k / (k*k!) oho I see http://en.wikipedia.org/wiki/Exponential_integral mentions both this and Ramanujan's faster series, except they seem to have a sign error in the latter. Example questions: for which rationals R do we get nice series of the form Li(x) = EulerMascheroniGamma + lnln(x) + x^R * SUM(n>0) c[n] * ln(x)^n ? <wds I probably won't bite on this, except to say your coauthor Conway claims Mascheroni lacks the claim to fame.