I am trying to prove the following inequality: $$ \sum\limits_{k=0}^{m}\frac{(a)_k(a+\mu)_{m-k}}{(c)_k(c+\mu)_{m-k}}\binom{m}{k}(m-2k+\mu)\geq{0}, $$ where $(a)_0=1$, $(a)_k=a(a+1)\cdots(a+k-1)$ is rising factorial and $a\geq{c}\geq{1/2}$, $\mu\geq{0}$. It is easy to prove it for $c\geq{a}>0$ by Gauss pairing but this method fails for $c<a$. I also managed to prove it for integer $\mu$. I believe the general case can be handled once $0<\mu<1$ is proved.
Any help welcome.
$(a)_k(a+\mu)_{m-k}/[(c)_k(c+\mu)_{m-k}](m-2k+\mu)$and summation by parts. The resulting sum can be handled by Gauss pairing. Dmitry $\endgroup$