Remarks on a sum involving binomial coefficients (Q2356304)

The sum \(S_p(a,b,t)=\frac 1b \sum_{k=0}^p \frac{\binom {p}{k}}{\binom{ak+b}{b}}t^k\) is studied, where \(p\) is a natural number, \(t\) is real, \(a\), \(b\) positive. It is shown that as a function of \(a\) it is completely monotonic for \(t>0\) and also completely monotonic as a function of \(b\) for \(f \geq -1\).