Closed form for the following sum of Binomial coefficients

Asked Dec 22 '23 at 19:28

Active Dec 22 '23 at 19:28

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I am trying to find a closed form for the following for any $p$ $$ \sum_{n = 0}^{m - 1} p^{n} \binom{2m}{n} $$

I typed this into Wolfram Alpha and got an answer in terms of the hypergeometric function, which I confirmed using this page and the first formula under the section named "Generating Functions" in this Wikipedia article; however, the hyperbolic geometric function diverges when $|p| > 1$ based on the the words under equation (9) in this link.

What would be a closed form expression for this finite summation?

asked Dec 22 '23 at 19:28

User

1

For $|p|>1$ rewrite the summation as $$S(p) = \sum_{n=0}^{2m} p^n{2m\choose n} - \sum_{n=m}^{2m}p^n{2m\choose n}$$ $$ = (1+p)^{2m} - p^{2m}\sum_{n=0}^m p^{-n}{2m\choose n}$$ Then the same formula applies for $|p^{-1}| < 1$ that you used for the original summation. – Ninad Munshi Dec 22 '23 at 19:38
@NinadMunshi Thank you very much! – User Dec 22 '23 at 19:43
See as well here. – Jean Marie Dec 22 '23 at 21:08

Closed form for the following sum of Binomial coefficients

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