Second solutions

Posted on 27 August 2026 by John

This post provides a couple examples to go along with two earlier posts.

The pattern we’re illustrating is families of polynomials p_n(x) that each satisfy a differential equation and a three-term recurrence. The differential equations have a second solution q_n(x) that is the larger solution with respect to x but the smaller solution with respect to n.

In both the examples below p_n(x) is a polynomial, and so bounded on the interval [−1, 1], and q_n(x) is not a polynomial, with singularities at ±1. This is analogous to the previous examples with Bessel functions J_n(x) and Q_n(x) that satisfy the same differential equation but have contrasting behavior with respect to x versus n.

Legendre polynomials

The differential equation

$(1-x^2)\,y^{\prime\prime} - 2x\,y^\prime + n(n+1)\,y = 0$

has two solutions for each n, P_n(x) and Q_n(x).

The solutions P_n(x) are the Legendre polynomials. The solutions Q_n(x) are not polynomials but involve a term log((1 + x)/(1 − x)) that blows up at 1 and −1. But for fixed x and increasing n, P_n(x) grows exponentially and Q_n(x) decays exponentially, provided |x| > 1.

Chebyshev polynomials

The differential equation

$(1-x^2)\,y^{\prime\prime} - x\,y^\prime + n^2\,y = 0$

has two solutions for each n, T_n(x) and V_n(x).

The solutions T_n(x) are the Chebyshev polynomials. The solutions V_n(x) are not polynomials but involve a term √(x² — 1) that become vertical up at 1 and −1. But for fixed x with |x| > 1 and increasing n, T_n(x) grows exponentially and V_n(x) decays exponentially.

Junk solutions

Posted on 26 August 2026 by John

When you’re interested in studying a family of functions, it can be useful to look at a differential equation that the functions solve. This is a theme I’ve written about several times, most recently here and here, but also three years ago here.

Orthogonal polynomials are mathematically elegant as well as very useful in applications [1]. Various families of orthogonal polynomials satisfy various differential equations. These equations have a polynomial and non-polynomial solutions. What use are the latter?

If the differential equation modeled something physical, then the second solution would be necessary to have a complete basis of solutions. But if the differential equation is only instrumental in studying the orthogonal polynomials, what use is a non-polynomial solution?

These non-polynomial solutions turn out to be useful. Just as “junk” DNA turned out not to be junk, these “junk” solutions are important. Junk DNA doesn’t directly code for proteins, but it regulates DNA that does code for proteins and serves other purposes. Similarly, these non-polynomial solutions carry information related to the polynomial solutions.

For example, orthogonal polynomials are used to construct numerical integration methods, such as Gaussian quadrature, and the associated non-polynomial solutions describe the error in these integration methods. Incidentally, Gaussian quadrature is based on Legendre polynomials, mentioned in the previous post. For every family of orthogonal polynomials there is a corresponding integration method. See these notes.

Another tie-in to recent posts is that these non-polynomial solutions are the minimal solution to the polynomial family’s three-term recurrence, the solution that takes extra care to compute numerically.

This post has been very high-level, alluding to ideas without going into details. I’d like to write future posts that go into more depth regarding the ideas introduced here.

[1] “Real analysts cannot do without Fourier, complex analysts cannot do without Laurent, and numerical analysts cannot do without Chebyshev [polynomials].” — Lloyd N. Trefethen”

Ultraspherical

Posted on 26 August 2026 by John

When I hear the term ultraspherical I think of something extremely spherical. For example, a baseball is spherical, but a billiard ball is more spherical. Maybe a highly polished billiard ball is ultraspherical.

Using this line of thought, the term ultraspherical polynomial is inexplicable. This is an example of the arcane terminology I wrote about recently. In this post I’ll explain what it conveys.

A spherical polynomial is a polynomial that naturally falls out of solving Laplace’s equation in spherical coordinates, using separation of variables. Legendre polynomials are spherical polynomials.

Gegenbauer polynomials are so called because a man named Gegenbauer studied them, just as Legendre polynomials take their name from Legendre. Gegenbauer polynomials are also called ultraspherical polynomials. Why is that?

There are two possible reasons. I’m not sure which is the historical reason, but both are plausible and are useful mnemonics.

Ultraspherical polynomials are not extremely spherical, they’re beyond spherical in some sense. More modern terminology uses the hyper- prefix rather than ultra-, which helps a bit.

Ultraspherical polynomials are beyond spherical in two ways. Gegenbauer polynomials are a generalization of Legendre polynomials, so they’re beyond Legendre polynomials in this sense.

More importantly, Gegenbauer polynomials fall out of solving Laplace’s equation on a hypersphere, i.e. a sphere in ℝⁿ for n > 3, just as Legendre polynomials fall out of the case n = 3. It makes sense to call these polynomials hyperspherical because they fall out of solving an equation on a hypersphere. Unfortunately the classical term is ultraspherical rather than hyperspherical.

I think, but I’m not sure, that at one time higher dimensional spheres were called hyperspheres, but the the higher dimensional analog of spherical coordinates was called ultraspherical coordinates. If so, it would be understandable that the adjective modifying coordinates would be applied to the polynomials that result from solving equations in these coordinates.

Numerical (in)stability of recurrence relations

Posted on 24 August 2026 by John

The previous post gave several examples of three-term recurrence relations for special functions. These relations can be computationally useful, but they have to be applied carefully.

Several years ago I wrote a post on stable and unstable recurrences. In that post I show that the stability of the recurrence relation for Bessel functions produces depends on which kind of Bessel function and which direction the recurrence is applied.

In the forward direction, computing higher order values from lower order values, works well for Bessel functions of the second kind Y_n but not for Bessel functions of the first kind J_n. In the reverse direction, the recurrence is stable for J_n but not for Y_n.

I didn’t explain in that post why this is. In this post I will.

Second order linear difference equations have two independent solutions, just like second order linear differential equations. For both kinds of equations, all solutions are linear combinations of the two solutions. Suppose one solution grows with n and the other decays. You may want to compute the decaying solution, but in doing so you might pick up a small component of the growing solution due to rounding error. This post illustrates this phenomena for differential equations, and this post illustrates it for difference equations.

When you look at a plot of Bessel functions in a text book, you’ll probably see a few plots of J_n(x) andY_n(x) for a few small values of n. The functions seem to behave roughly the same way, like sine and cosine. And that’s true, as functions of x.

But it’s not true for J_n(x) andY_n(x) as functions of n for fixed x. As n increases, J_n(x) decays to zero and Y_n(x) goes off to −∞.

That’s the source of numerical instability. And there will be similar instability problems for other recurrences where the ratios of the two independent solutions goes to zero or infinity as a function of n.

There are techniques for computing the solution that does not diverge, the so-called minimal solution, such as Miller’s algorithm mentioned here.

Three-term recurrences

Posted on 24 August 2026 by John

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 and possibly 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.

NB: before using any of the recurrences below, see the next post for a numerical pitfall to avoid.

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.

The von Mises-Fisher distribution

Posted on 24 August 2026 by John

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.

What exactly is modified about a modified Bessel function?

Posted on 23 August 2026 by John

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.

Why special function terminology is arcane

Posted on 23 August 2026 by John

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 your instinct is that things with strange names are no longer important, your 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.

Inverse differential equations

Posted on 10 August 2026 by John

In science and engineering classes, you might describe a system using Newton’s laws and end up with a differential equation. You then solve the differential equation, analytically or numerically, to see how the solutions behave.

You might also do the opposite, especially in a mathematics class: look at what differential equation a set of functions satisfy in order to understand those functions.

Bessel functions came out of solving differential equations from astronomy. But then they turned out to be useful, not just in other areas of science, but in pure mathematics as well. So there are Bessel function users who came to the functions first and haven’t seen the differential equation they came from.

You can learn a lot about Bessel functions, and other special functions, by looking at their defining differential equation even if you’re not directly interested in the differential equation or the physical problem that motivated it.

Bessel functions satisfy

x² y″ + x y′ + (x² − n²) y = 0.

You can tell a lot about Bessel functions just by inspecting this equation without solving it. If we divide by x² and write the equation in the form

y″ + (p(x)/x) y′ + (q(x)/x²) y = 0

then p(x) = 1 and q(x) = x² − n². The indicial equation

r(r − 1) + p(0) r + q(0) = 0

reduces to

r² = n²

and so r = ±n. That alone tells us there are two solutions, one analytic at zero and one singular at zero. These are J_n and Y_n respectively. It also tells us the behavior of these functions as x goes to zero and as x goes to infinity. To find out more, look up “method of Frobenius.”

DNA and Bessel functions

Posted on 9 August 2026 by John

I was reading a book on the history of the discovery of the structure of DNA [1] and was surprised by a few passing references to Bessel functions.

According to Claude,

When X-rays are diffracted by a helical structure, the resulting diffraction pattern breaks into a series of horizontal “layer lines.” Cochran, Crick, and Vand showed mathematically that the diffracted amplitude on the n-th layer line is proportional to a Bessel function of the first kind, order n:

J_n(2π r R)

where r is the radius of the helix and R is the distance out from the center (the meridian) in the diffraction pattern.

The citation for this paragraph is a paper from 1952 [2] that amazingly is behind a paywall.

Update: Evan pointed out something fishy in the expression above: “Isn’t there a dimensional problem with J(2π r R)? Maybe 2πr/R instead?”

This was Claude’s response:

r is the radius of the helix, a real-space length (say, in Å).
R is the radial coordinate in reciprocal space — it has units of inverse length (Å⁻¹), because that’s how X-ray diffraction/Fourier space works (spatial frequency, not distance).

So Evan and Claude were both right. There is an implicit division happening, but crystallographers do write r R.

[1] Watson and Crick didn’t “discover DNA” as is commonly said. DNA was discovered in 1878. Watson and Crick discovered the structure of DNA in 1953.

[2] Cochran, W., Crick, F. H. C., & Vand, V. (1952). “The Structure of Synthetic Polypeptides. I. The Transform of Atoms on a Helix.” Acta Crystallographica, 5(5), 581–586.

Legendre polynomials

Chebyshev polynomials

Pure math perspective

Applied math perspective

Other modified functions

Related posts