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I would like a closed form for the formula Sum[Binomial[k - b, n] Binomial[n + b, a], {n, 0, k}] where variables k, a, b are supposed to be integers. Simply evaluating this formula returns

Binomial[b, a] Hypergeometric2F1[1 + b, b - k, 1 - a + b, -1] -
    Binomial[-b + k, 1 + k] Binomial[1 + b + k, a] HypergeometricPFQ[{1, 1 + b, 2 + b + k}, {2 + k, 2 - a + b + k}, -1]

which works fine when a <= b but returns Indeterminate when b < a even though the original formula doesn't. For example:

Unevaluated@
  Sum[Binomial[k - b, n] Binomial[n + b, a], {n, 0, k}] /. {k -> 6, a -> 2, b -> 1}
  (* 160 *)

while

Sum[Binomial[k - b, n] Binomial[n + b, a], {n, 0, k}] /. {k -> 6, a -> 2, b -> 1}
   (* Indeterminate *)

However, the evaluated formula does give the right answer in the Limit.

Additionally, FullSimplify even returns 0 under the assumption that b < a:

FullSimplify[
 Unevaluated@Sum[Binomial[k - b, n] Binomial[n + b, a], {n, 0, k}], 
 Element[k | a | b, Integers] && k >= 0 && 0 <= a <= k && 0 <= b < a]
   (* 0 *)

So my question is: is this a bug? Is there a way to get Mathematica to try to simplify the sum over the whole domain?

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    – bbgodfrey
    Dec 15, 2021 at 21:44

1 Answer 1

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Indeed, the formula as returned by Mathematica does have the blemish you describe, due to the hypergeometric functions becoming infinite. This is actually a common problem encountered by people who use Sum[] to evaluate sums involving binomial coefficients. However, at least for this case, by using the Olver hypergeometric function instead (or what Mathematica calls Hypergeometric2F1Regularized[]), we can convert that result into something more useful. To wit,

(b! Hypergeometric2F1Regularized[1 + b, b - k, 1 - a + b, -1] +
 b Binomial[k - b, k] k! (b + k + 1)!
 HypergeometricPFQRegularized[{1, 1 + b, 2 + b + k}, {2 + k, 2 - a + b + k}, -1])/a!

is an expression equivalent to the one produced by Sum[] but (AFAICT) does not suffer indeterminacy problems.

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