Let G be the gamma function, and b be a constant in (-2,inf). Let

H(n, i) = G(i+1+b) * G(n-i+1+b) / [G(i+1) * G(n-i+1)]

for integers n > i > 0. Let

S(n) = \sum_{i=1}^{i=n-1} H(n, i).

Let x_ n = H(n,1) / S(n). Note x_ 2 = 1, x_ 3 = 1/2 for all b.

I am convinced that as n -> inf, x_ n -> 0 for b >= -1, and x_ n -> (-b-1)/2 for -2 < b < -1. I can prove the b >= -1 case, but not the other, except when b=-3/2. Can anyone help with a proof?

I have found recursive relationships between x_ n and S(n):

S(n+1) = (1/(n+1)) [n + 2 + 2b + 2(n+b)x_ n/n] S(n)

x_ {n+1} = (n+b)/n x_ n S(n) / S(n+1) = (n+b)(n+1)x_ n / [n(n+2+2b) + 2(n+b)x_ n]

which may be of use. One way to deal with the b >= -1 case is use the latter to relate 1/x_ {n+1} and 1/x_ n and show this tend to infinity.

For background, see section 4 of Probability Distributions on Cladograms (1996) by David Aldous In Random Discrete Structures (its available free online)

Graham