<s>[INST] Estimation of Hypergeometric function ${_3F_2}$ Is there any way to estimate the following function, which is a result of sum of ratios of Gamma functions?
$$
{_3F_2}\begingroup
\renewcommand*{\arraystretch}
% your pmatrix expression
\left[
\begin{array}{c@{}c}
 \begin{array}{c}
         -q, \frac{M}{2}, \frac{1}{2}+\frac{M}{2}\\ 
                       \frac{1}{2}, -q-\frac{n-M}{2}+1
                      \end{array} ;& 1
 \end{array}\right]\endgroup,
$$
where $q, M, n \in N, n\geq M$
 [/INST] Here are explicit forms of
$$f(q,n,m)={_3F_2}\begingroup
\renewcommand*{\arraystretch}
% your pmatrix expression
\left[
\begin{array}{c@{}c}
 \begin{array}{c}
         -q, \frac{m}{2}, \frac{1}{2}+\frac{m}{2}\\ 
                       \frac{1}{2}, -q-\frac{n-m}{2}+1
                      \end{array} ;& 1
 \end{array}\right]\endgroup,$$
for small values of $q$:
$$f(1,n,m)=-\frac{m^2+n}{m-n},$$
$$f(2,n,m)=\frac{m^4+m^2 (6 n+8)+3 n (n+2)}{3 (m-n-2) (m-n)},$$
$$f(3,n,m)=-\frac{m^6+5 m^4 (3 n+8)+m^2 (45 n^2 +210n+184)+15 n (n+2) (n+4)}{15 (m-n-4) (m-n-2) (m-n)}.$$
For large $m$ or $n$ we have the limits
$$\lim_{m\rightarrow\infty}m^q f(q,n,m)=(-1)^q\frac{1}{(2q-1)!!},
$$
$$\lim_{n\rightarrow\infty}n\bigl(f(q,n,m)-1\bigr)=q(m+m^2).$$
</s>