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!

  • $\begingroup$ You got that equation as it is stated or there is a background? $\endgroup$ – Phicar Jan 27 '17 at 17:57
  • $\begingroup$ 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. $\endgroup$ – Raizen Jan 29 '17 at 18:07

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