On Tue, May 29, 2012 at 2:11 PM, Bill Gosper <billgosper@gmail.com> wrote:
Whoa, DLMF 15.7.5&6 imply (untested) c*Log[Hypergeometric2F1[a, b, c, z]] == Integrate[cfk[(a + k)*(b + k)*(1 - z)*z, c + k - (1 + a + b + 2*k)*z,{k, 0, ∞}]/((1 - z)*z), z] --rwg
Oops, no. The "F"s in 15.7.5 are not italic. That means they're (bagbiting) REGULARIZED (= divided by Γ(c))). The correct formula is Log[Hypergeometric2F1[a, b, c, z]] == Integrate[cfk[(a + k)*(b + k)*(1 - z)*z, c + k - (1 + a + b + 2*k)*z,{k, 0, ∞}]/((1 - z)*z), z] --rwg
On Tue, May 29, 2012 at 1:31 PM, Bill Gosper <billgosper@gmail.com> wrote:
Besides the traditional definite In[13]:= FunctionExpand[Integrate[t^(b - 1)*(1 - t)^(c - b - 1)/(1 - z*t)^a, {t, 0, 1}], 0 < z < 1]
Out[13]= ConditionalExpression[( Gamma[b] Gamma[-b + c] Hypergeometric2F1[a, b, c, z])/Gamma[c], Re[b] < Re[c] && (Re[z] < 1 || z \[NotElement] Reals) && Re[b] > 0]
(it ignored the 0 < z < 1), we have the indefinite Hypergeometric2F1[A, B, C, z] == E^Integrate[ContinuedFractionK[(A + n)*(B + n), (C + n - (1 + A + B + 2*n)*t)/Sqrt[(1 - t)*t], {n, 0,∞}]/Sqrt[(1 - t)*t], {t, 0, z}] --rwg Testing this is hard: As of 8.04, ContinuedFractionK is numerically and symbolically unusable.