wds>many formulas about gamma function are in my paper:http://rangevoting.org/WarrenSmithPages/homepage/gammprox.pdf _______________________________________________ Interesting stuff. Random remarks: P4 middle is garbled, at least in my Preview. Did you mean asymptosy vs asymptopia? P6, In[16]:= F[0, c] == Assuming[c > 0,Integrate[ x^(2*n + 1)/(1 + x^2)^(n + 1)*E^(-c*x) /. n -> 0, {x, 0, \[Infinity]}]] Out[16]= F[0,c] == (-Cos[c])*CosIntegral[c] + (1/2)*Sin[c]*(Pi - 2*SinIntegral[c]) In[12]:= G[0, c] == Assuming[c > 0, Integrate[ x^(2*n)/(1 + x^2)^(n + 1)*E^(-c*x) /. n -> 0, {x, 0, \[Infinity]}]] Out[12]= G[0, c] == CosIntegral[c]*Sin[c] + (1/2)*Cos[c]*(Pi - 2*SinIntegral[c]) In[14]:= H[0, c] == Assuming[c > 0, Integrate[ x^(2*n)/(1 + x^2)^(n)*E^(-c*x) /. n -> 0, {x, 0, \[Infinity]}]] Out[14]= H[0, c] == 1/c Eliminating G and H from the F recurrence, 4*(n-1)*n*F(n)=2*(n-1)*(6*n-5)*F(n-1)-(12*n^2-32*n+c^2+22)*F(n-2)+2*(n-2)*(2*n-3)*F(n-3) Mma gets only MeijerG for n>0 in all of these. P7 (40), see http://www.tweedledum.com/rwg/idents.htm near bottom of 1st screen. (42) and (43) raise the interesting question of evaluating Product[((1/(12*n)) + 1)*((n - 1/2)*(E^(1/n) - 1))^n,{n, 1, Infinity}] (i.e., prod(((n-1/2)*(%e^(1/n)-1))^n*(1+1/12/n),n,1,inf) --"cwg"