Infinite product representation of Bessel's function of the 2nd kind

[Talon44]

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.

 

[pasmith]

Abramowitz & Stegun at (9.5.10) gives $$J_\alpha(z) = \frac{(z/2)^\alpha}{\Gamma(\alpha + 1)} \prod_{k=1}^\infty \left(1 - \frac{z^2}{j_{\alpha,n}^2}\right)$$ where $0 < j_{\alpha,1} < j_{\alpha, 2} < \dots$ are the non-negative zeroes of $J_\alpha$. You can then understand how this representation is obtained, since $J_\alpha$ is $(z/2)^\alpha/\Gamma(z+1)$ times a power series in $z^2$ which equals 1 when $z = 0$, and by definition vanishes at $z = j_{\alpha,n}$. Naturally there is more work to do to show that this result does in fact hold all other values of $z$.

The Bessel function of the second kind is defined for non-integer $\alpha$ as $$Y_\alpha(z) = \frac{J_\alpha(z)\cos \alpha \pi - J_{-\alpha}(z)}{\sin \alpha \pi}.$$ As in this case $Y_\alpha$ is a linear combination of the linearly independent solutions $J_{\pm \alpha}$ it does not receive separate analysis.

For integer $n$, $$Y_n(z) = \lim_{\alpha \to n} \frac{J_\alpha(z)\cos \alpha \pi - J_{-\alpha}(z)}{\sin \alpha \pi}$$ is the linearly independent solution which is singular at the origin, since $J_{\pm n}$ are not linearly independent but satisfy $J_{-n} = (-1)^nJ_n$. Wikipedia gives a representation of $Y_n$ which is essentially $$Y_n(z) = z^{-n}\sum_{k=0}^{n-1}b_nz^{2k} + \frac 2\pi J_n(z)\ln(z/2) + z^{n} \sum_{k=0}^\infty a_nz^{2k}$$ which could be similarly manipulated into an infinite product; however I don't think that the zeros of the last series are tabulated, making it less useful in practical terms. (The zeros of $Y_n$ itself are tabulated, as for example in Abramowitz & Stegun.)

 

[Talon44]

I will need to spend some time wrapping my head around that, but I wanted to thank you for taking the time to reply. Solving heat/diffusion problems in cylindrical geometry requires manipulating these Bessel functions and I just don't have a lot of formal experience with them. I was doing pretty well but then got stuck on the hollow cylinder.

Anyway, I will take some time with your response. Thanks again.