Restricted sum of product of binomial coefficients

Question

I am interested in understanding the sum $$\sum_{s=A}^B\binom{k}{s}\binom{n-k}{s}$$ where $0<k\le\frac{n}{2}$. There are some further dependencies ($\frac{A}{k}\approx p_n-\varepsilon$ and $\frac{B}{k}\approx p_n+\varepsilon$, for $p_n\approx\frac{1}{2}$ (limiting in $n$) and $\varepsilon$ constant but tiny), but for now I would be interested to understand this so long as $n$ and $k$ are allowed to grow. Is there a nice way to view this sum? some combinatorial interpretation? I have checked all eight Gould tables and not found anything that seems to handle this (or really any restricted sums where the dummy variable does not appear in the top of both binomial coefficients).

Answer 1

$$S=\sum_{s=A}^B\binom{k}{s}\binom{n-k}{s}$$ is given ,in terms of the generalized hypergeometric function, as $$S=\binom{k}{A} \binom{n-k}{A} \, _3F_2(1,A-k,A+k-n;A+1,A+1;1)-$$ $$\binom{k}{B+1} \binom{n-k}{B+1} \, _3F_2(1,B-k+1,B+k-n+1;B+2,B+2;1)$$ Let $$k=\alpha n \qquad\qquad A=\alpha n\left(\frac{1}{2}-\epsilon \right)\qquad\qquad B=\alpha n\left(\frac{1}{2}+\epsilon \right)$$ So $$S=f(\alpha,\epsilon,n)$$ which can be visualized using a three-dimensional contour plot (better to plot $\log(S)$) or cross-sections of it.

For $\epsilon=\frac 1{10}$, you will find below a table of values

$$\left( \begin{array}{cccc} n & \log\left(S_{\frac 18}\right) &\log\left(S_{\frac 14}\right)& \log\left(S_{\frac 12}\right)\\ 5 & 0.82016 & 1.49755 & 2.18211 \\ 10 & 2.08389 & 3.72836 & 5.29832 \\ 15 & 3.48649 & 6.13878 & 8.58230 \\ 20 & 4.96045 & 8.63052 & 11.9297 \\ 30 & 8.02461 & 13.7363 & 18.7052 \\ 40 & 11.1761 & 18.9297 & 25.5297 \\ 50 & 14.3783 & 24.1728 & 32.3776 \\ 60 & 17.6138 & 29.4483 & 39.2393 \\ 70 & 20.8731 & 34.7468 & 46.1100 \\ 80 & 24.1500 & 40.0625 & 52.9874 \\ 90 & 27.4407 & 45.3915 & 59.8697 \\ 100 & 30.7424 & 50.7311 & 66.7562 \\ \end{array} \right)$$

As one could expect, when $n$ becomes large, appears the linear trend in a logarithmic scale.