summation - Compute a sum involving binomial coefficients

Let $0 < a < b$ and $p_1 >0$ and $p_2>0$ be integers. The question is to prove the following identity:

\begin{equation}
\sum\limits_{j=a}^b
\left(\begin{array}{c} j \\ p_1 \end{array} \right)
\left(\begin{array}{c} j \\ p_2 \end{array} \right)
=

\sum\limits_{q=0}^{p_2} (-1)^{q+1}
\left[
\left(\begin{array}{c} a+q \\ p_1+q+1 \end{array} \right)
\left(\begin{array}{c} a \\ p_2-q \end{array} \right)
-
\left(\begin{array}{c} b+1+q \\ p_1+q+1 \end{array} \right)
\left(\begin{array}{c} b+1 \\ p_2-q \end{array} \right)
\right]
\end{equation}

Can this identity be generalised for $p_1,p_2$ being real?