Re: Bug in Mathematica 7.0?

*To*: mathgroup at smc.vnet.net*Subject*: [mg99127] Re: [mg99116] Bug in Mathematica 7.0?*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Tue, 28 Apr 2009 04:43:11 -0400 (EDT)*References*: <200904270925.FAA24349@smc.vnet.net>

Not An Expert wrote: > Hello Mathematica-Experts, > > I'm using Mathematica 7.0 on a Windows XP machine and might have come across > a bug in Mathematica 7.0. > > I define the following function as a product of Gamma functions: > > A[M_, r_] := Product[Gamma[M + j]*Gamma[1/2 + j]*Gamma[r + 1/2 + j]/ > Gamma[r + M + j], {j, 0, M - 1}] > > To allow non-integer values M we re-write the above expression in terms > of the Barnes G-function and so we obtain (we've checked the algebra > many times!): This is the part that went wrong. > B[M_, r_] := BarnesG[2 + M]/BarnesG[2]*BarnesG[1/2 + M]/BarnesG[1/2] > *BarnesG[r + 1/2 + M]/ > BarnesG[r + 1/2]*BarnesG[r + M]/BarnesG[r + 2*M] > > As a test I've chosen M = 2 and r = 1 > and indeed I obtain > A[2, 1] = Pi^2/32 > and > B[2, 1] = Pi^2/32 > and so agreement in both functions. > > But choosing M = 3 and r = 1 gives > A[3, 1] = 9*Pi^3/2048 > B[3, 1] = 3^Pi^3/16384 > > So two different values for A[3, 1] and B[3, 1]. > Also A[3, 0] and B[3, 0] would give different answers - and there are > many other examples. > > Do you have any thoughts on what went wrong? > > Cheerio! Your translation from products of Gammas to BarnesG, specifically that first one, is not quite in agreement with what I find here. http://mathworld.wolfram.com/BarnesG-Function.html I obtain: aA[m_, r_] := Product[Gamma[m + j]*Gamma[1/2 + j]*Gamma[r + 1/2 + j]/ Gamma[r + m + j], {j, 0, m - 1}] bB[m_,r_] := (BarnesG[2*m]/BarnesG[m]) * (BarnesG[m+1/2]/BarnesG[1/2]) * (BarnesG[r+m+1/2]/BarnesG[r+1/2]) * (BarnesG[r+m]/BarnesG[r+2*m]) Test: In[44]:= Table[aA[m,r] - bB[m,r], {m,1,5}, {r,0,5}] Out[44]= {{0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}} Daniel Lichtblau Wolfram Research

**References**:**Bug in Mathematica 7.0?***From:*Not An Expert <not_an_expert@directbox.com>