# Double sum combinatorial identity

I would like to prove the following identity:

$$\sum_{p=0}^{k}\sum_{q=0}^{l}(-1)^{p+q}\big(\prod_{i=1}^{2k-l-1}(b-2p+q+i)\big)\big(\prod_{i=1}^{2l-k-1}(a-2q+p+i)\big)\binom{k}{p}\binom{l}{q}=0$$

Here $a,b$ are complex numbers while $k$ and $l$ are positive integers with the following restrictions: $l<2k$ and $k<2l$.

I tried proving it by writing the $LHS$ using the gamma function and then trying to do some residue calculus, but couldn't manage to solve it. Several trials in Mathematica confirm the identity while several other trials with $k$ and $l$ positive integers not satisfying the above restrictions give non-zero values which could possibly mean that the above identity may hold iff $l<2k$ and $k<2l$, but I am just interested in proving the identity as first stated, the other direction being just extra if indeed it would hold.

I would appreciate any help. Thank you for your cooperation!

• You got that equation as it is stated or there is a background? – Phicar Jan 27 '17 at 17:57
• I am trying to prove a duality relation for Horn's G3 series and at some point after lots of computation, I arrive at the LHS of the stated equation as being the coefficient of the monomial ((-x)^k)*((-y)^l) which I would like it to be 0 as computational trials suggest it should be. – Raizen Jan 29 '17 at 18:07