The two Barnes constants can be replaced: Product[(3^(-(1/2) + 3*n)*Gamma[a + n]* Gamma[(1 - a)/2 - (1/6)*Sqrt[1 + 18*a - 27*a^2] + n]* Gamma[(1 - a)/2 + (1/6)*Sqrt[1 + 18*a - 27*a^2] + n])/(2*Pi* Gamma[3*n]), {n, Infinity}] == (E^((8/3)*Derivative[1]["Zeta"][-1])*(2* Pi*(-(1/3))!)^(1/3))/ (3^(5/36)*BarnesG[1 + a]* BarnesG[3/2 - a/2 - (1/6)*Sqrt[1 + 18*a - 27*a^2]]* BarnesG[3/2 - a/2 + (1/6)*Sqrt[1 + 18*a - 27*a^2]]) Zeta is in quotes to forestall conversion to a small mess featuring that idiot Glaisher symbol. Remove::rmptc: Symbol Glaisher is Protected and cannot be removed. >> I assume a SetDelayed back to Zeta'[-1] will cause death and destruction. --rwg 3^(5/36)? On Sat, Aug 1, 2015 at 1:50 PM, Bill Gosper <billgosper@gmail.com> wrote:
On Sat, Aug 1, 2015 at 2:15 AM, Bill Gosper <billgosper@gmail.com> wrote:
Mathematica should be able to get Product[(3^(-(1/2) + 3*n)*Gamma[2/21 + n]*Gamma[4/21 + n]*Gamma[5/7 + n])/ (2*Pi*Gamma[3*n]), {n, Infinity}] == (BarnesG[4/3]*BarnesG[5/3])/ (BarnesG[23/21]*BarnesG[25/21]*BarnesG[12/7])
In[885]:= N[Rule @@ %]
During evaluation of In[885]:= NIntegrate::slwcon:Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small. >>
During evaluation of In[885]:= NIntegrate::ncvb:NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in n near {n} = {7273.96592}. NIntegrate obtained -0.000408638628 and 8.024253968625528`*^-8 for the integral and error estimates. >>
Out[885]= 0.9878943769692339 -> 0.9878939404389084
This seems to generalize to Out[990]= Product[(3^(-(1/2) + 3*n)*Gamma[a + n]* Gamma[(1 - a)/2 - (1/6)*Sqrt[1 + 18*a - 27*a^2] + n]* Gamma[(1 - a)/2 + (1/6)*Sqrt[1 + 18*a - 27*a^2] + n])/(2*Pi* Gamma[3*n]), {n, Infinity}] == (BarnesG[4/3]*BarnesG[5/3])/ (BarnesG[1 + a]* BarnesG[3/2 - a/2 - (1/6)*Sqrt[1 + 18*a - 27*a^2]]* BarnesG[3/2 - a/2 + (1/6)*Sqrt[1 + 18*a - 27*a^2]])
Remarkably, Mathematica seems to be able to do the integer cases, even though all but one are complex: In[991]:= %990[[1]] /. a -> 1
Out[991]= 1
In[993]:= %990[[1]] /. a -> 3
Out[993]= ( BarnesG[4/3] BarnesG[5/3])/(2 BarnesG[-((I Sqrt[47])/3)] BarnesG[(I Sqrt[47])/3])
In[1008]:= %990[[1]] /. a -> 2
Out[1008]= (BarnesG[4/3] BarnesG[5/3])/( BarnesG[1/2 - (I Sqrt[71])/6] BarnesG[1/6 (3 + I Sqrt[71])])
In[1037]:= %990[[1]] /. a -> 4
Out[1037]= (BarnesG[4/3] BarnesG[5/ 3])/(12 BarnesG[-(1/2) - (I Sqrt[359])/6] BarnesG[ 1/6 I (3 I + Sqrt[359])]) --rwg
simply by expressing the finite product using BarnesG, then taking the limit using "Barnes' Stirling's formula". --rwg