@dharr Thanks for your help. Could you explain why your code worked better than mine? I am not a frequent maple user, so I follow my intuition when I am writing the code, but it seems Maple has some tricks that we usually don't know. thanks.

@tomleslie thanks for your help. I am not a frequent user of maple. Your code is a great improvement, however, note that the final result was supposed to be a sum of 11 terms (v=0..10), but it does not seem to be doing the summation on 'v'. cheers

@Axel Vogt Thanks for your reply, I tried to implement your code but it is still very slow, please look at the code below.

Digits := 20; n1 := 225; x := 29; n2 := 111; y := 2; alpha1 := 12.1; beta1 := 5.1; alpha2 := 12.1; beta2 := 5.1; rho0 := .99;

const := 2.6342242569969157636*10^(-64);

evalf(beta1*beta2*GAMMA(beta2)*(Sum(binomial(n1-x, j)*rho0^(x+j+alpha1)*(Sum(binomial(n2-y, k)*(-1)^(j+k)*S(j, k), k = 0 .. n2-y)), j = 0 .. n1-x))/(const*GAMMA(1-beta1))):

eval(%, S = 'proc (j, k) options operator, arrow; evalf(Sum(rho0^(alpha1*h)*GAMMA(1-beta1+h)*GAMMA((y+k+x+j+alpha1+alpha2)/alpha2+alpha1*h/alpha2)/(factorial(h)*((x+j+alpha1)/alpha1+h)*GAMMA((y+k+x+j+alpha1+alpha2+alpha2*beta2)/alpha2+alpha1*h/alpha2)), h = 0 .. infinity)) end proc');

This is the same code you suggested, but still taking ages. Sometimes it`s very tricky how Maple sums up things.

Thanks in advance.

@tomleslie Fantastic. Many thanks.