I 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 <br /> 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 &lt; j_{\alpha,1} &lt; j_{\alpha, 2} &lt; \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 <br /> 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, <br /> 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 <br /> 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.