#5688 simplify_sum gives wrong result for sums with finite-support binomial

[Dubey Vivek santoshkumar]

load(simplify_sum)$

Reported via SageMath issue #42894 and PR #42981.

Correct finite sum: 42

sum((3*k)*binomial(4, 3*k-6), k, 0, inf);
=> 3 * 'sum(binomial(4, 3*k-6)*k, k, 0, inf)  (unevaluated)

simplify_sum(sum((3*k)*binomial(4, 3*k-6), k, 0, inf));
=> 36   WRONG, should be 42

Larger case:

simplify_sum(sum((3*k)*binomial(11, 3*k-13), k, 0, inf));
=> 24   WRONG, should be 12630

The summand has finite support: binomial(n, 3*k-n-2) is nonzero only for 0 <= 3*k-n-2 <= n. simplify_sum converts the summand to a hypergeometric term and evaluates it without restricting to the finite support, producing a spurious contribution.

A similar issue was previously reported for sum(binomial(1,n), n, 0, oo) returning 3 instead of 2.

Closed form: S(n) = 2^n*(3*n+4)/6 - 2*(-1)^n/3

[Robert Dodier]

Thanks for the bug report. Can you clarify the final comment: "Closed form: S(n) = 2^n*(3*n+4)/6 - 2*(-1)^n/3"? I don't understand what that is referring to.