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My attempts to search via Google seem to be failing, so I thought of asking here.

All the derivatives of the function

$r_n(z):=\frac{J_n(z)}{J_{n-1}(z)}$

where $J_n(z)$ is the Bessel function of the first kind are expressible in terms of $r_n(z)$, for instance $\frac{\mathrm{d}}{\mathrm{d}z}r_n(z)=r_n(z)^2-\frac{2n-1}{z}r_n(z)+1$ . I've been trying to derive a (linear?) differential equation (hopefully just second-order) that might be satisfied by $r_n(z)$, but my manipulative ability does not seem to be up to snuff.

Probably my problem can be resolved in two ways:

  1. Are there any papers where generating functions/differential equations of ratios of Bessel functions have been studied? ; or
  2. How can I derive a differential equation for $r_n(z)$ with the knowledge that all higher derivatives are expressible in terms of $r_n(z)$?

(On the other hand, the difference equation for $r_n(z)$ (and thus its continued fraction representation) is easily derived, so no problem for me there.)

I will be interested in any input. Thanks!

EDIT:

Per Pietro's request, I now tip my hand and reveal my reason for interest: I saw this paper many years ago on a neat method for computing the first few roots of the Bessel function of the first kind. Some time later, I came across J.F. Traub's "Iterative Methods for the Solution of Nonlinear Equations", where he shows the construction of iteration functions involving derivatives and can be constructed to have quadratic, cubic... convergence. (Newton's method is but the first member of this family). I also came across this short note by D.J. Hofsommer on how one might profitably exploit the methods derived by Traub if the function of interest satisfies a simple differential equation (Essentially, one just constructs the Newton correction $u=\frac{f(z)}{f^{\prime}(z)}$, and the high-order iteration functions are merely a series in powers of $u$). That got me wondering on how one might recursively generate iteration functions with increasing order of convergence for the case of finding the roots of the Bessel function. (On another note, I was able to successfully use the ideas of Traub and Hofsommer for the generation of Gaussian quadrature rules, e.g. Legendre, Lobatto, Radau, and was hoping things might be just as successful for Bessel function root-finding).

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A note: I am not of course counting "trivial" differential equations like the derivative formula I gave in the post. So probably something like a relation between $r_n(z)$ and its first few derivatives. –  J. M. Aug 22 '10 at 3:46
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Would you clarify why the Riccati equation you wrote is not satisfacory for you? –  Pietro Majer Aug 22 '10 at 8:05
    
Pietro: I suppose I should've inserted the adjective "linear" somewhere... :) I'll edit my post and detail my motivation further. –  J. M. Aug 22 '10 at 9:36

3 Answers 3

up vote 8 down vote accepted

No, you will not find a linear ordinary differential equation (with polynomial coefficients) for $r_n$. This is because $J_{n-1}$ has infinitely many zeroes which are not cancelled out by the zeroes of $J_n$, so that $r_n$ has infinitely many poles. Holonomic functions, aka solutions of LODEs with polynomial coefficients, can only have a finite number of singularities.

The transformation between LODEs and Riccati equations involves transformations between an equation for a quantity $y(z)$ [your $r_n$] and an LODE for $-u'/u$. So what you'll get is indeed an LODE for a single Bessel $J$ function!

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Thank you Jacques! I suppose I can't do better than ELF and GNOME then. :) –  J. M. Aug 22 '10 at 13:39

As noted by Pietro, the nonlinear differential equation you gave is a Riccati equation. Next, there is a standard method (due to Ince?) to convert a Riccati equation into a second-order linear equation.

http://en.wikipedia.org/wiki/Riccati_equation

Try it, and see if that is what you want.

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The equation I got after trying that transformation out (I came upon this earlier after Pietro pointed it out to me) has the regular solution $\frac{J_{n-1}(x)}{x^{n-1}}$ ... using that would require me to reckon the Bessel functions themselves from the ratio (resulting in more lines of code, sigh...). Am I to imply that I have to use the Bessel functions themselves, and not the ratios, to fully exploit Hofsommer's observation? –  J. M. Aug 22 '10 at 13:13
    
...and Jacques's answer makes the answer to my question in the previous comment "yes". –  J. M. Aug 22 '10 at 13:45

I do not know how helpful this would be to you however it was very helpful to understand the physics and numerics of Bessel. if you are studying elastic wave propagation. The solution of the differential equations of potential wave is the cylindrical Bessel:

$ <math>r^2 \frac{d^2 R}{dr^2} + r \frac{dR}{dr} + (r^2 - \alpha^2)R = f(r)</math>$

for an arbitrary real integer number α (the '''order''' of the Bessel function). In solving problems in cylindrical coordinate systems, Bessel functions are of integer order (α = ''n''). Since this is a second-order differential equation, there must be two [[linearly independent]] solutions. My solutions use Bessel J(n,.) and Hankel H(n,.) (as previously mentioned)

The potential is assumed for each media to be:

$<math>\phi=\left(a_{1}J_{n}(K*r)+a_{2}*H_{n}(K*r)\right)*e^{in\theta} ,</math>$

$<math>\psi(t) = \left(a_{3}J_{n}(k*r)+a_{4}*H_{n}(k*r)\right)*e^{in\theta} \,</math> $

However for numerical stability: They (ref. 1) normalize the potential for each layer and for each nth iteration The potential will have Hankel function equal to 1 at the inner radius, while Bessel J will be multiplied by Hankel at outer radius.

$<math>\phi=\left(a_{1}J_{n}(K*r)*H_{n}(K*r_{out})+a_{2}*\frac{H_{n}(K*r)}{H_{n}(K*r_{in})}\right)*e^{in\theta},</math>$

$<math>\psi(t)=\left(a_{3}J_{n}(k*r)*H_{n}(k*r_{out})+a_{4}*\frac{H_{n}(k*r)}{H_{n}(K*r_{in})}\right)*e^{in\theta},</math>$

For more details: Bessel functions in wave propagation and scattering Reference: David C. Ricks and Henrik Schmidt, "A numerically stable global matrix method for cylindrically layered shells excited by ring forces" 1994

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What you wrote does not appear to be related to answering the question. In particular, at no point do you do even mention the function $r_n$. The part of your first sentence where you said "it was very helpful" does not seem to make any sense in the context of this web page. –  S. Carnahan Dec 2 '11 at 9:53
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@ S. Carnahan, the use of Bessel ratios prove to be very useful. I was sharing my own experience that the ratio could be used to reduce the overflow. At higher order n and small arguments, very small real and large imaginary. This resulted in numerical instability. In other words the ratio can be used to normalize the potentials as: $<math>\phi=\left(a_{1}J_{n}(K*r)*H_{n}(K*r_{out})+a_{2}*\frac{H_{n}(K*r)}{H_{n}‌​(K*r_{in})}\right)*e^{in\theta},</math>$ $<math>\psi(t)=\left(a_{3}J_{n}(k*r)*H_{n}(k*r_{out})+a_{4}*\frac{H_{n}(k*r)}{H_‌​{n}(K*r_{in})}\right)*e^{in\theta},</math>$ –  Chad Dec 3 '11 at 3:49

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