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Hi everyone,

In the process of studying a problem in the Johnson association scheme I came across the following sum: $$\sum_{k\geq 0}(-1)^k\binom{n}{k}\binom{a-k}{a-b}\binom{c+k}{b}.$$ All the variables are non-negative integers. I've tried to no avail to simplify this expression using Gosper's algorithm, as well Wilf-Gosper (but it becomes unwieldy).

Is there perhaps a simpler form for this sum? Is there any connection with Eberlein polynomials?

Thanks in advance


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This is a hypergeometric function times a binomial coefficient, isn't it? To see exactly which one, one can do the usual procedure described in books, e.g. in A=B.

Then, perhaps, hypergeometric identities can be applied, but at least having your sum encoded like this might help.

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The sum, in fact, equals $\binom{a}{b}\binom{c}{b} _{3}F_{2}(c+1,−b,−n;c−b+1,−a;1)$. I've already tried looking up hypergeometric identities but couldn't find one that does the job and reduces it to something nicer. – Moshe Schwartz Jan 10 '13 at 17:26
Well, it might be it - no simpler formula. Often this is the case, e.g. check out recent work of A.Schrijver and his collaborators on improved bounds for codes, such as, However, often sums like yours come up as entries of a certain matrix, and this matrix might have a reasonably nice $LU$ decomposition. E.g. in your case $L$ might be having entries ${n \choose k}$. – Dima Pasechnik Jan 11 '13 at 1:48

Where did it come from? What kind of answer are you looking for and why do you expect that there would be one? In the case $n,a,b,c=9,4,3,6$ your sum has four non-zero terms and comes out to the negative of a prime number: $-3889.$ So there is likely not an expression as a simple product.

MUCH LATER On the other hand, a small search (most cases with $n \le 30,a \le b \le 50, c \le 50$) turned up a few hundred cases where the sum is prime, but all of them had either $b \in \{{2,3,4\}}$ or $a=b$ or $a=b+1$ so perhaps examination of the factorizations would be productive. Two more prime cases are $19,14,4,26$ and $19,14,4,23.$

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