Yet another sum involving binomial coefficients Let $k,p$ be positive integers. Is there a closed form for the sums 
$$\sum_{i=0}^{p} \binom{k}{i} \binom{k+p-i}{p-i}\text{, or}$$
$$\sum_{i=0}^{p} \binom{k-1}{i} \binom{k+p-i}{p-i}\text{?}$$
(where 'closed form' should be interpreted as a representation which is free of sums, binomial coefficients, or any other hypergeometric functions).

Using a computer algebra system I get
In[114]:= Sum[ Binomial[k, i] Binomial[k+p-i,p-i],{i,0,p}]
Out[114]= Binomial[k + p, p] Hypergeometric2F1[-k, -p, -k - p, -1]
and
In[115]:= Sum[Binomial[k - 1, i] Binomial[k + p - i, p - i], {i, 0, p}]
Out[115]= Binomial[k + p, p] Hypergeometric2F1[1 - k, -p, -k - p, -1]
