# Closed-form expression for a hypergeometric series

What is the closed-form expression for

$${}_2 F_1 \left(1+2\lceil n/2\rceil,-n;1/2;-z/4\right)$$

According to the book Concrete Mathematics (R.Graham, D.Knuth, O.Patashnik 2nd), the authors say the general sum of $\sum_k {\ n-k \choose k}z^k$ leads to the closed form of the above series.

I understand $\sum_k {\ n-k \choose k}z^k= \frac{1}{\sqrt{1+4z}}((\frac{1+\sqrt{1+4z}}{2})^{n+1}-(\frac{1-\sqrt{1+4z}}{2})^{n+1})$, but I can not see how this sum leads to the closed-form of the the hypergeometric series.

• The program reduce thinks that -z/4 should be -4z. Leaving that aside reduce produces a less complicated answer. In fact once you convert the binomial to pochammer expressions you will have the result; after converting (n-k) summation from falling factorial to pochhammer rising. Now the conversion to the power expressions is interesting to me since I have been working with generating functions and combinatorial sums: see eq: 1.120-- dsi.unifi.it/~resp/GouldBK.pdf . A relation between ordinary second order recurrence and that sum? Maybe – rrogers Dec 23 '15 at 15:40