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

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

For integer [itex]n[/itex], [tex]
Y_n(z) = \lim_{\alpha \to n} \frac{J_\alpha(z)\cos \alpha \pi - J_{-\alpha}(z)}{\sin \alpha \pi}[/tex] is the linearly independent solution which is singular at the origin, since [itex]J_{\pm n}[/itex] are not linearly independent but satisfy [itex]J_{-n} = (-1)^nJ_n[/itex]. Wikipedia gives a representation of [itex]Y_n[/itex] which is essentially [tex]
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}[/tex] 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 [itex]Y_n[/itex] 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.