Expanding a general term in the infinite product?

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January 5th 2010, 05:36 AM
tsunamijon

Expanding a general term in the infinite product?

I have the product

sin(x)/x
= (1 - x^2/pi^2)(1 - x^2/4pi^2)(1 - x^2/9pi^2)(1 - x^2/16pi^2)......

But i'm looking to expand the whole product and find only the term in x^k, for some general k. and find an expression for this general coefficient.

I know I can equate it to the Taylor series, but I want to do it this way to get a different expression. Please help, thanks!
January 9th 2010, 11:16 PM
chisigma

The function sinc(*) is defined as...

$sinc (x) = \left\{\begin{array}{cc}\frac{\sin \pi x}{\pi x}, &\mbox{if } x\ne 0\\1, & \mbox{if } x=0\end{array}\right.$ (1)

... and, in my opinion, it is a powerful kay to open many 'doors' in the mathematical world. It admits the 'simplest existing' infinite product expansion...

$sinc (x) = \prod_{n=1}^{\infty} (1-\frac{x^{2}}{n^{2}})$ (2)

... and from (2) we can derive many interesting results. As example let suppose to compute the log of the sinc(*). With easy steps we obtain...

$\ln sinc (x) = \sum_{n=1}^{\infty} \ln (1-\frac{x^{2}}{n^{2}}) = - \sum_{k=1}^{\infty} \frac{x^{2k}}{k}\cdot \sum_{n=1}^{\infty} \frac{1}{n^{2k}}= - \sum_{k=1}^{\infty} \frac{x^{2k}\cdot \zeta(2k)}{k}$ (3)

... where $\zeta(*)$ is the 'Riemann Zeta Function'. The (3) suggests as alternative way for the computation of $\zeta(2k)$ the following...

$\zeta(2k)= - \frac{k}{(2k)!} \cdot \frac{d^{2k}}{dx^{2k}} \ln sinc (x)_{x=0}$ (4)

This is only one of the possibilities that the sinc (*) opens to You (Itwasntme) ...

Kind regards

$\chi$ $\sigma$
January 10th 2010, 04:35 PM
tsunamijon

Hi, thanks for the answer. It's pretty much what I was looking for. Do u have any research related to this information? Or promising lines of attack to the Riemann hypothesis?