Complete monotonicity of a function related to the binomial probability

Let k and n be integers with 0≤k≤n and p∈(0,1). We prove that the functionG(a)=Gk,n,p(a)=Γ(an+1)Γ(ak+1)Γ(a(n−k)+1)pak(1−p)a(n−k) is completely monotonic on (0,∞). This extends a result of Leblanc and Johnson, who showed in 2007 that the sequence {G(j)}j=1∞ is decreasing.