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.

--"confluent" means what? --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. 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 ?