- #1

Talon44

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- TL;DR Summary
- Looking for infinite product representation of Bessel's function of the 2nd kind

An infinite product representation of Bessel's function of the first kind is:

$$J_\alpha(z) =\frac{(z/2)^\alpha}{\Gamma(\alpha+1)}\prod_{n=1}^\infty(1-\frac{z^2}{j_{n,\alpha}^2})$$

Here, the ##j_{n,\alpha}## are the various roots of the Bessel functions of the first kind. I found this expression at a number of sources (including at Wikipedia). I am looking for an analogous expression for Bessel's function of the second kind but cannot find one. Is it more or less the same (just with different roots, obviously)? I am not sure how to derive such representations.

$$J_\alpha(z) =\frac{(z/2)^\alpha}{\Gamma(\alpha+1)}\prod_{n=1}^\infty(1-\frac{z^2}{j_{n,\alpha}^2})$$

Here, the ##j_{n,\alpha}## are the various roots of the Bessel functions of the first kind. I found this expression at a number of sources (including at Wikipedia). I am looking for an analogous expression for Bessel's function of the second kind but cannot find one. Is it more or less the same (just with different roots, obviously)? I am not sure how to derive such representations.