Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $k,p$ be positive integers. Is there a closed form for the sums

$$\sum_{i=0}^{p} \binom{k}{i} \binom{k+p-i}{p-i}\text{, or}$$

$$\sum_{i=0}^{p} \binom{k-1}{i} \binom{k+p-i}{p-i}\text{?}$$

(where 'closed form' should be interpreted as usual, i.e. meaning free of sums and hypergeometric functions).

We know that the first sum has generating function $(1+z)^k/(1-z)^{k+1}$, and the second sum has generating function $(1+z)^{k-1}/(1-z)^{k+1}$, but that doesn't help me find a closed form so far.

share|improve this question
6  
I'm sure you already checked, but Mathematica gives $$\sum_{i=0}^p\binom{k}{i}\binom{k+p-i}{p-i} = \binom{k+p}{p}\ _2 F_1(-k, -p;-k-p;-1)$$ So having a closed form for the left side would imply a closed form for the hypergeometric function on the right. –  Aeryk Jul 15 '11 at 13:47
4  
Maple agrees, and also gives for the second sum $$\sum_{i=0}^p {k-1 \choose i} {k + p - i \choose p - i } = {k+p \choose p} {}_2F_1(-k+1,-p; -k-p; -1)$$ –  Robert Israel Jul 16 '11 at 0:17

1 Answer 1

up vote 8 down vote accepted

Your first sum is the Delannoy number $D(k,p)$. See OEIS sequence A008288

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.