I'm trying to find a closed form expression for the following product: $f(k) = \prod \limits_{m=1 \\m \neq k}^{N} \frac{1}{1-(\frac{k}{m})^{\alpha/2}}$.

This can be simplified or approximated into: $f(a) = \prod \limits_{m=1}^{N} \frac{1}{1-a \; m^{-\alpha/2}}$.

A hint or an expression for either would be great. A limit when N tends to infinity can also be useful. Also, it would be helpful if there is any book with tables of products of sequences.

  • $\begingroup$ Sorry for the typo, I corrected the question $\endgroup$
    – A. Shokair
    Oct 23 '18 at 14:54
  • $\begingroup$ See this page for your last question www-elsa.physik.uni-bonn.de/~dieckman/InfProd/InfProd.html $\endgroup$
    – Yuriy S
    Oct 23 '18 at 14:57
  • $\begingroup$ The condition $m \neq k$ should still be included in the "simplified" form $\endgroup$
    – Yuriy S
    Oct 23 '18 at 15:13
  • $\begingroup$ thank you Yuriy for the link, I'm looking at it. $\endgroup$
    – A. Shokair
    Oct 23 '18 at 15:19
  • $\begingroup$ It was very hard to solve the first one, and I found no known formula that deals with an exception in the product $\endgroup$
    – A. Shokair
    Oct 23 '18 at 15:20

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