1

Can someone explain the details to the equality: $$ x^n \sum_{k=0}^{\lfloor \frac{n}2\rfloor} \binom{n+1}{2k+1}(1-x^{-2})^k = \sum_{k=0}^{\lfloor \frac{n}2 \rfloor} \binom{2k-(n+1)}{k}(2x)^{n-2k}? $$

How is the LHS equal to the RHS?

Thanks!

EDIT: any hint or pointing me in the right direction would be greatly appreciated!

Mojtaba
  • 690
user09837836
  • 88
  • It's easier for people to provide useful answers if you can explain what you already know and what you've tried or what similar problems you can solve. – MJD Jun 10 '15 at 15:28
  • @MJD: Thanks. This is not a homework question but I'm just trying to understand what properties were used in order to derive from LHS to the RHS. I'm guessing some sort of combinatorial identities (or properties of binomials) were used, but I have never studied combinatorics. Either way, I'll do some literature search to find the right tools. – user09837836 Jun 10 '15 at 15:32
  • Where did you find this equation? – Robert Israel Jun 10 '15 at 15:40
  • @RobertIsrael I found it on the wiki: I've been trying to connect Chebyshev polynomials and hypergeometric functions, and in the midst of their derivation, the above equality appears without any explanations. – user09837836 Jun 10 '15 at 15:56

0 Answers0