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Three-term recurrences

2026-08-24 22:14:55

There many examples of families of functions where each function can be computed as a linear combination of the two previous terms

$f_{n+1}(x) = a(x) f_n(x) + b(x) f_{n-1}(x)$

where a and b are functions of x but not on n. This is called a three-term recurrence formula.

It’s amazing how often you can run into three-term recurrence formulas. There are theorems that give conditions for such recurrences to hold, but I haven’t reached the bottom of that rabbit hole [1].

For this post I just want to give examples.

Bessel functions of the first and second kind:

$\begin{align*} J_{\nu+1}(x) &= \frac{2\nu}{x}\,J_\nu(x) - J_{\nu-1}(x) \\ Y_{\nu+1}(x) &= \frac{2\nu}{x}\,Y_\nu(x) - Y_{\nu-1}(x) \end{align*}$

Modified Bessel functions of the first and second kind:

$\begin{align*} I_{\nu+1}(x) &= I_{\nu-1}(x) - \frac{2\nu}{x}\,I_\nu(x) \\ K_{\nu+1}(x) &= K_{\nu-1}(x) + \frac{2\nu}{x}\,K_\nu(x) \end{align*}$

Chebyshev polynomials of the first and second kind:

$\begin{align*} T_{n+1}(x) &= 2x\,T_n(x) - T_{n-1}(x) \\ U_{n+1}(x) &= 2x\,U_n(x) - U_{n-1}(x) \end{align*}$

Hermite polynomials (physicists’ convention):

$H_{n+1}(x) = 2x\,H_n(x) - 2n\,H_{n-1}(x)$

Legendre polynomials:

$P_{n+1}(x) = \frac{2n+1}{n+1}\,x\,P_n(x) - \frac{n}{n+1}\,P_{n-1}(x)$

[1] See Bochner’s theorem for orthogonal polynomials, the Nikiforov–Uvarov method, and Infeld-Hull factorization.

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The von Mises-Fisher distribution

2026-08-24 22:14:25

Probability density function must integrate to 1, and so if you know a density function up to a constant, the constant is determined.

When you’re looking at a probability density f(x) for the first time, it helps to ignore the normalizing constant. Concentrate on the part of the function involving x and know that the normalizing constant is whatever it has to be. For example, about half of the ink that it takes to write down a beta or chi-squared density is devoted to the normalization constant; the rest of the expression is easier to understand.

This post will do the opposite of the advice above and focus on normalization constants because this ties into the previous post on modified Bessel functions.

The von Mises probability distribution on a circle has two parameters, μ and κ, and its density function is

$f(x \mid \mu, \kappa) = \frac{\exp(\kappa \cos(x - \mu))}{2\pi I_0(\kappa)}$

The normalizing constant is 2π I₀(κ). The factor of 2π is unsurprising for anything defined on a circle. The more interesting part is I₀, the modified Bessel function of order 0.

The von Mises-Fisher distribution is the generalization of the von Mises distribution to a sphere in p dimensions. The density function is

$f(\mathbf{x} \mid \boldsymbol{\mu}, \kappa) = C_{p}(\kappa) \exp \left( {\kappa \boldsymbol{\mu}^\mathsf{T} \mathbf{x} } \right)$

where the normalization constant C_p(κ) is

$C_{p}(\kappa)=\frac {\kappa^{p/2-1}} {(2\pi)^{p/2}I_{p/2-1}(\kappa)}$

where I_{p/2 − 1} is the modified Bessel function of order p/2 − 1. The values of x and μ are in bold face because they are now vectors, points on the unit sphere.

When p = 2, we have the “sphere” in two dimensions, i.e. the circle, and the von Mises-Fisher distribution reduces to the von Mises distribution. But where did the cosine go? The inner product of x and μ is the cosine of the angle between the two vectors.

When p = 3, obviously an important special case, the von Mises-Fisher distribution is known as the Fisher distribution. In that case the normalizing constant C₃(κ) can be written without using modified Bessel functions because when ν = ½ + n for an integer n, I_ν(x) is an elementary function.

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What exactly is modified about a modified Bessel function?

2026-08-24 02:40:57

Special functions often have arcane names that not very helpful without some context. The previous post goes into some reasons for this. This post will expand on a point at the end of the post about “modified” functions.

Things are given their names for reasons. Discovering those reasons may help you understand their motivation and use.

Pure math perspective

For each integer n, the modified Bessel function I_n is essentially the Bessel function J_n evaluated along the imaginary axis. Specifically,

$I_n(x) = i^{-n} J_n(ix)$

From a certain shallow perspective, that’s the end of the story: modified Bessel functions are modified in the sense that the argument is multiplied by i. And there’s a fiddly constant term up front for no apparent reason.

But of course that’s not the end of the story or else this wouldn’t be worth an entire post.

The equation above is analogous to the relationships between circular and hyperbolic functions

$\begin{align*} \sin(ix) &= i \sinh(x) \\ \cos(ix) &= \phantom{i} \cosh(x) \\ \tan(ix) &= \phantom{i} \tanh(x) \end{align*}$

These relationships are interesting because the circular and hyperbolic functions are independently meaningful. If you view these equations merely as definitions you lose their significance. Circular and hyperbolic functions were widely used before Euler discovered the connection between them.

Similarly, there’s a reason the modified Bessel functions were given a name their own. If you were led to Bessel functions and modified Bessel functions separately by different applications, you would regard the equation

$I_n(x) = i^{-n} J_n(ix)$

as a discovery rather than just a definition. The following section explains why someone would be interested in modified Bessel functions.

Before we move on, I’d like to explain the reason for the term i⁻ⁿ term. In general

$I_\nu(x) = \exp(\nu\pi i/2) J_n(ix)$

for all real ν. The reason for the exp(νπi/2) term is that it makes I_ν(x) real for all real x.

Applied math perspective

Bessel functions often arise from solving problems with radial symmetry. Solving the wave equation in cylindrical coordinates using separation of variables leads to Bessel’s differential equation

$x^2 y'' + x y' + (x^2 - \nu^2) y = 0$

and its solutions J_n and Y_n, Bessel functions of the first and second kind.

Solving the heat equation in cylindrical coordinates with separation of variables leads to the modified Bessel equation

$x^2 y^{\prime \prime} + x y^{\prime} - (x^2 + \nu^2) y = 0$

and its solutions I_n and K_n, the modified Bessel functions of the first and second kind.

This is the reason behind the complex analysis perspective above: the change of variables sending x to ix changes the sign of the x² term in Bessel’s equation.

Bessel functions describe radially symmetric oscillations, such as the vibrations of a drum head. Modified Bessel functions describe radially symmetric exponential growth or decay [1], such as in the heat in a cylinder.

Other modified functions

Struve functions are closely related to Bessel functions. The (modified) Struve functions also satisfy Bessel’s (modified) differential equation, but with a non-zero right hand side. The modified Struve functions are proportional to the unmodified Struve functions evaluated along the imaginary axis, with a proportionality constant that makes the modified Struve functions real for real arguments.

There’s a similar relationship between the Mathieu functions and modified Mathieu functions. The general pattern is that “modified” in the context of special functions means “evaluated at ix and multiplied by a constant to make the function real for real arguments.”

[1] The functions I_n grow exponentially and the functions K_n decay exponentially. For this reason, A&S didn’t tabulate I_n and K_n per se. Instead it tabulated e^−xI_n and e^xK_n because these functions varied less over their range.

The post What exactly is modified about a modified Bessel function? first appeared on John D. Cook.

Why special function terminology is arcane

2026-08-24 01:56:03

Special functions are special because they’re useful. They can also be shrouded in arcane terminology. These two facts are related.

The more widely useful a function is, the more likely it is that the function will be discovered independently multiple times. Independent discoveries lead to varying definitions and notations. For example, there are two widely used definitions of Hermite polynomials, one used in probability and another used in physics, that only differ by a scaling factor. This also explains why there are so many variations on the definitions of the Fourier transform and spherical coordinates.

Special functions were discovered and applied before they were studied systematically. As with most mathematics, practice preceded theory. In hindsight, some names and conventions were less than ideal, at least from the perspective of someone seeking to organize a theory.

Functions can have arcane names for several reasons, one being that their usefulness became apparent long ago. If you’re instinct is that things with strange names are no longer important, you’re instinct might be backward. The strange name may be an indication that something is so important that its usefulness became apparent long ago.

Sometimes special functions have bland, uninformative names because the names stuck before anybody could think of something better. Bob looks into an interesting family of functions [1], then later he finds another interesting family of functions. These become known as “Bob’s functions of the first kind” and “Bob’s functions of the second kind.” These names are quite understandable at the time, though in the future people will want to know what distinguishes the functions, other than the fact that Bob discovered them, and what the groupings have in common other than the order in which Bob found them.

I started this post intending to discuss modified Bessel functions and explain what exactly is modified about them, but my preface became its own post. “Modified” is an example of the bland terminology mentioned above. There are Bessel functions and modified Bessel functions. Without more context, the “modified” term isn’t very informative. But it does provide a clue that there’s some kind of close relationship between the modified and unmodified functions. That’ll be the topic of my next post.

[1] Math education doesn’t place much emphasis on history and motivation. You may have to do some digging to find out why Bob was interested in his functions. What else was Bob known for? Maybe they’re related.

The post Why special function terminology is arcane first appeared on John D. Cook.

The difference orbit inclination makes

2026-08-23 08:01:10

Suppose you wanted to find the distance between Earth and Mars over time. To first approximation, both planets orbit the sun in elliptic orbits in the same plane.

If you wanted to be more accurate, you’d need to take into account the fact that the orbit of Mars is tilted about 1.85° relative to the Earth’s orbit. How much difference does that make?

To simplify things, let’s assume the Earth orbits the sun in a circle of radius 1 and Mars orbits the sun in a circle of radius 1.5. The distance between Earth and Mars over time would be basically sinusoidal.

How much does inclination contribute to this distance? In other words, what is the difference between the distance accounting for the inclination of Mars’ orbit and the distance if we assume the two orbits are in the same plane?

This plot gives the answer.

The effect is not large, about three orders of magnitude smaller than the main effect, but it’s interesting how erratic it is.

The plots were made with the following code.

from numpy import *

R = 1.5
T = R**1.5 # Kepler's third law

def f(t, theta):
    return sqrt(
        (cos(t) - R*cos(t/T)*cos(theta))**2 +
        (sin(t) - R*sin(t/T))**2 +
        (R*sin(theta)*cos(t/T))**2
    )

The first plot graphs f(t, θ) and the second graphs f(t, θ) − f(t, 0).

The post The difference orbit inclination makes first appeared on John D. Cook.

Coming soon

2026-08-22 23:34:11

There’s a pizza shop near my home with a sign out front that says “Coming Soon.” When I drove by it this morning I thought about how you would model the time until an event happens that is “coming soon.”

Suppose I look at the sign one day and guess how many days until the pizza shop will open. When I drive by a week later and guess again, should my guess be smaller? You might argue that the shop will open some day, fixed in time but unknown to me, and so every day I’m one day closer to the eventual opening.

You might model the pizza shop opening like radioactive decay and say that the estimated number of days until it opens is always the same until the day it actually opens.

Now I think this shop has been “coming soon” for over a year. So instead of decreasing, every day I increase my estimate of the time until the shop opens. Something has gone wrong that the owners didn’t expect when they put up the sign.

Maybe the reasonable thing would be for estimated days until opening to decrease over time, but only up to a point. After some point, the longer a business has been “coming soon” the less like that it is coming soon, or coming at all.

This brings up an interesting point about modeling. There are two probability distributions at work: the probability that the shop will eventually open, and the time until opening assuming it eventually opens.

When the sign first goes up saying the business is coming soon, there’s some change that it is in fact not coming. Maybe you’re optimistic and think this probability is small, but it would seem unreasonable to think the probability is zero. That means the expected number of days until opening is always infinite. If there’s a probability ε that the shop never opens, the expected time to opening is

ε × ∞ + (1 − ε) × something = ∞.

The post Coming soon first appeared on John D. Cook.

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