Prove the identity $ \sum\limits_{s=0}^{\infty}{p+s \choose s}{2p+m \choose 2p+2s} = 2^{m-1} \frac{2p+m}{m}{m+p-1 \choose p}$

Let $d_s = \binom{p+s}{s} \binom{2p+m}{2p+2s}$. Using the recurrence relations for binomial, the ratio of successive terms is: $$ \frac{d_{s+1}}{d_s} = \frac{\left(s - m/2\right)\left(s -(m-1)/2\right)}{ (s+1)(s+p+1/2) } = \frac{(s+a)(s+b)}{(s+1)(s+c)} $$ The hypergeometric certificate above means that $$ \sum_{s=0}^\infty d_s = d_0 \sum_{s=0}^\infty \frac{(a)_s (b)_s}{s! (c)_s} = \binom{2p+m}{2p} {}_2 F_1\left( -\frac{m}{2}, -\frac{m-1}{2} ; p+\frac{1}{2} ; 1\right) $$ where $a = -\frac{m}{2}$, $b=-\frac{m-1}{2}$ and $c=p+\frac{1}{2}$.

Using Gauss's theorem, valid for $c>a+b$: $$ {}_2 F_1\left( a, b; c; 1\right) = \frac{\Gamma(c) \Gamma(c-a-b)}{\Gamma(c-a) \Gamma(c-b)} $$ we obtain the required identity: $$ \sum_{s=0}^\infty \binom{p+s}{s} \binom{2p+m}{2p+2s} = \binom{2p+m}{2p} \frac{\Gamma\left(p+\frac{1}{2}\right) \Gamma\left( p+m \right)}{ \Gamma\left( p+\frac{m+1}{2} \right) \Gamma\left( p+\frac{m}{2} \right) } $$ Applying the duplication formula for $\Gamma(2p+m+1)$ and $\Gamma(2p+1)$ arising from $\binom{2p+m}{2p}$ we arrive at the result: $$ \sum_{s=0}^\infty \binom{p+s}{s} \binom{2p+m}{2p+2s} = 2^{m-1} (m+2p) \frac{\Gamma(m+p)}{\Gamma(m+1) \Gamma(p+1)} = 2^{m-1} \frac{m+2p}{m+p} \binom{m+p}{p} $$

Ok, here is an approach with generating functions. Let $$ g_1(z) = \sum_{s=0}^\infty \binom{p+s}{s} z^s = \frac{1}{\left(1-z\right)^{p+1}} $$

$$ g_2(z) = \sum_{s=0}^\infty \binom{2p+m}{s} z^s = \left(1+z\right)^{m+2p} $$ Now $$ \begin{eqnarray} \sum_{s=0}^\infty \binom{p+s}{s} \binom{2p+m}{2p+2s} &=& \sum_{s=0}^\infty \binom{p+s}{s} \binom{2p+m}{m-2s} = [z]^m g_1(z^2) g_2(z) = [z]^m \frac{\left(1+z\right)^{m+2p}}{(1-z^2)^{p+1}} \\ &=& [z]^m \frac{\left(1+z\right)^{m+p-1}}{\left(1-z\right)^{p+1}} \end{eqnarray} $$

Here is a verification:

In[27]:= With[{p = 5, 
  m = 7}, {SeriesCoefficient[(1 + z)^(m + 2 p)/(1 - z^2)^(
   p + 1), {z, 0, m}], 
  Sum[Binomial[p + s, s] Binomial[2 p + m, 2 p + 2 s], {s, 
    0, \[Infinity]}]}]

Out[27]= {71808, 71808}

Let's continue: $$ \begin{eqnarray} [z]^m \frac{\left(1+z\right)^{m+p-1}}{\left(1-z\right)^{p+1}} &=& \sum_{s=0}^\infty \binom{p+m-1}{m-s} \binom{p+s}{s} = \sum_{s=0}^\infty \binom{p+m-1}{p+s-1} \binom{p+s}{s}\\ &=& \sum_{s=0}^\infty \frac{(p+s) (m+p-1)!}{p! s! (m-s)!} = \sum_{s=0}^\infty \frac{p (m+p-1)!}{p! s! (m-s)!} + \sum_{s=0}^\infty \frac{s (m+p-1)!}{p! s! (m-s)!} \\ &=& \binom{m+p-1}{m} \left( \sum_{s=0}^\infty \binom{m}{s} + \sum_{s=0}^\infty \frac{s}{p} \binom{m}{s} \right) \\ &=& \binom{m+p-1}{m} \left( 2^m + 2^{m-1} \frac{m}{p} \right) \end{eqnarray} $$

I will try to give an answer using basic complex variables here. This calculation is very simple in spite of some more complicated intermediate expressions that appear.

Suppose we are trying to show that $$\sum_{q=0}^\infty {p+q\choose q} {2p+m\choose m-2q} = 2^{m-1} \frac{2p+m}{m} {m+p-1\choose p}.$$

Introduce the integral representation $${2p+m\choose m-2q} = \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{(1+z)^{2p+m}}{z^{m-2q+1}} \; dz.$$

This gives for the sum the integral (the second binomial coefficient enforces the range) $$\frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{(1+z)^{2p+m}}{z^{m+1}} \sum_{q=0}^\infty {p+q\choose q} z^{2q} \; dz \\ = \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{(1+z)^{2p+m}}{z^{m+1}} \frac{1}{(1-z^2)^{p+1}} \; dz \\ = \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{(1+z)^{p+m-1}}{z^{m+1}} \frac{1}{(1-z)^{p+1}} \; dz.$$

This is $$\frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{(2+z-1)^{p+m-1}}{z^{m+1}} \frac{1}{(1-z)^{p+1}} \; dz \\ = 2^{p+m-1} \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{(1+(z-1)/2)^{p+m-1}}{z^{m+1}} \frac{1}{(1-z)^{p+1}} \; dz \\ = 2^{p+m-1} \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{1}{z^{m+1}} \frac{1}{(1-z)^{p+1}} \sum_{q=0}^{p+m-1} {p+m-1\choose q} \frac{(z-1)^q}{2^q} \; dz \\ = 2^{p+m-1} \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{1}{z^{m+1}} \frac{1}{(1-z)^{p+1}} \sum_{q=0}^{p+m-1} {p+m-1\choose q} (-1)^q \frac{(1-z)^q}{2^q} \; dz \\ = 2^{p+m-1} \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{1}{z^{m+1}} \sum_{q=0}^{p+m-1} {p+m-1\choose q} (-1)^q \frac{(1-z)^{q-p-1}}{2^q} \; dz.$$

The only non-zero contribution is with $q$ ranging from $0$ to $p.$ This gives $$ 2^{p+m-1} \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{1}{z^{m+1}} \sum_{q=0}^p {p+m-1\choose q} (-1)^q \frac{1}{2^q} \frac{1}{(1-z)^{p+1-q}} \; dz$$ which on extracting coefficients yields $$2^{p+m-1} \sum_{q=0}^p {p+m-1\choose q} (-1)^q \frac{1}{2^q} {m+p-q\choose p-q}.$$

Introduce the integral representation $${m+p-q\choose p-q} = \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{(1+z)^{m+p-q}}{z^{p-q+1}} \; dz.$$

This gives for the sum the integral (the second binomial coefficient enforces the range) $$2^{p+m-1} \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{(1+z)^{m+p}}{z^{p+1}} \sum_{q=0}^\infty {p+m-1\choose q}\frac{(-1)^q}{2^q} \left(\frac{z}{1+z}\right)^q \; dz \\ = 2^{p+m-1} \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{(1+z)^{m+p}}{z^{p+1}} \left(1-\frac{1}{2}\frac{z}{1+z}\right)^{p+m-1} \; dz \\ = 2^{p+m-1} \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{1+z}{z^{p+1}} \left(1+z-1/2\times z\right)^{p+m-1} \; dz \\ = \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{1+z}{z^{p+1}} \left(2+z\right)^{p+m-1} \; dz.$$

Extracting coefficients now yields $${p+m-1\choose p} \times 2^{m-1} + {p+m-1\choose p-1} \times 2^m.$$

This symmetric form may be re-written in an asymmetric form as follows, $${p+m-1\choose p} \times 2^{m-1} + \frac{p}{m} {p+m-1\choose p} \times 2^m \\ = 2^{m-1} \times \left(1 + \frac{2p}{m}\right) {p+m-1\choose p}$$ as claimed.

The bonus feature of this calculation is that we evaluated two binomial sums instead of one.

We have not made use of the properties of complex integrals here so this computation can also be presented using just algebra of generating functions.

Apparently this method is due to Egorychev although some of it is probably folklore.