[seqfan] Re: Curious Sums

Joerg Arndt arndt at jjj.de
Sun Oct 8 18:34:28 CEST 2017


* Paul Hanna <pauldhanna.math at gmail.com> [Oct 06. 2017 18:10]:
> SeqFans,
>       Here are some curious sums I found that I'd like to share with you.
> 
> (1) Sum_{n=-oo..+oo}  x^n * (1 - x^n)^n  =  0.
> 
> (2) Sum_{n=-oo..+oo, n<>0}  x^n * (1 - x^(n-1))^n / n  =  -log(1-x).
> 
> These series are the motivation behind sequence https://oeis.org/A291937 ,
> in which I've recorded some related identities.

Impressive.

Both
  P(x) = x + x^2 + 3*x^3 + 5*x^5 - x^6 + 7*x^7 - 8*x^8 + ...
and
  Q(x) = 1 + x - x^2 + x^3 - 3*x^4 + x^5 - 2*x^6 + x^7 - ...
should also be submitted!
Also M(x) and N(x).

Best regards,   jj


> [...]


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