BesselPolynomials

The Bessel polynomials are an orthogonal sequence of polynomials defined by

$$y_n(x) = \sum_{k=0}^n \frac{(n+k)!}{(n-k)!k!}\left(\frac{x}{2}\right)^k$$

The reverse Bessel polynomials[^2^][^8^][^3^][^15^] are similarly defined by:

$$\theta_n(x) = x^n y_n(1/x) = \sum_{k=0}^n \frac{(n+k)!}{(n-k)!k!}\frac{x^{n-k}}{2^k}$$

The coefficients of the second definition are the same as the first but in reverse order. For example, the third-degree Bessel polynomial is

$$y_3(x) = 15x^3 + 15x^2 + 6x + 1$$

while the third-degree reverse Bessel polynomial is

$$\theta_3(x) = x^3 + 6x^2 + 15x + 15$$

Properties

Definition in terms of Bessel functions

The Bessel polynomial may also be defined using Bessel functions from which the polynomial draws its name.

$$y_n(x) = x^n\theta_n(1/x)$$ $$y_n(x) = \sqrt{\frac{2}{\pi x}} e^{1/x} K_{n+\frac{1}{2}}(1/x)$$ $$\theta_n(x) = \sqrt{\frac{2}{\pi}} x^{n+1/2} e^x K_{n+ \frac{1}{2}}(x)$$

where $K_n(x)$ is a modified Bessel function of the second kind, $y_n(x)$ is the ordinary polynomial, and $\theta_n(x)$ is the reverse polynomial[^2^][^7^][^34^]. For example[^4^]:

$$y_3(x) = 15x^3 + 15x^2 + 6x + 1 = \sqrt{\frac{2}{\pi x}} e^{1/x} K_{3+\frac{1}{2}}(1/x)$$

Definition as a hypergeometric function

The Bessel polynomial may also be defined as a confluent hypergeometric function[^5^][^8^]:

$$y_n(x) = {}_2F_0(-n,n+1;;-x/2) = \left(\frac{2}{x}\right)^{-n} U\left(-n,-2n,\frac{2}{x}\right) = \left(\frac{2}{x}\right)^{n+1} U\left(n+1,2n+2,\frac{2}{x}\right)$$

A similar expression holds true for the generalized Bessel polynomials (see below)[^2^][^35^]:

$$y_n(x;a,b) = {}_2F_0(-n,n+a-1;;-x/b) = \left(\frac{b}{x}\right)^{n+a-1} U\left(n+a-1,2n+a,\frac{b}{x}\right)$$

The reverse Bessel polynomial may be defined as a generalized Laguerre polynomial:

$$\theta_n(x) = \frac{n!}{(-2)^n} L_n^{-2n-1}(2x)$$

from which it follows that it may also be defined as a hypergeometric function:

$$\theta_n(x) = \frac{(-2n)_n}{(-2)^n} {}_1F_1(-n;-2n;2x)$$

where $(-2n)_n$ is the Pochhammer symbol (rising factorial).

Generating function

The Bessel polynomials, with index shifted, have the generating function

$$\sum_{n=0}^\infty \sqrt{\frac{2}{\pi}} x^{n+\frac{1}{2}} e^x K_{n-\frac{1}{2}}(x) \frac{t^n}{n!} = 1 + x\sum_{n=1}^\infty \theta_{n-1}(x) \frac{t^n}{n!} = e^{x(1-\sqrt{1-2t})}$$

Differentiating with respect to $t$, cancelling $x$, yields the generating function for the polynomials ${\theta_n}_{n\ge0}$

$$\sum_{n=0}^\infty \theta_{n}(x) \frac{t^n}{n!} = \frac{1}{\sqrt{1-2t}} e^{x(1-\sqrt{1-2t})}$$

Similar generating function exists for the $y_n$ polynomials as well[^1^][^106^]:

$$\sum_{n=0}^\infty y_{n-1}(x) \frac{t^n}{n!} = \exp\left(\frac{1-\sqrt{1-2xt}}{x}\right)$$

Upon setting $t = z - xz^2/2$, one has the following representation for the exponential function[^1^][^107^]:

$$e^z = \sum_{n=0}^\infty y_{n-1}(x) \frac{(z-xz^2/2)^n}{n!}$$

Recursion

The Bessel polynomial may also be defined by a recursion formula:

$$y_0(x) = 1$$ $$y_1(x) = x + 1$$ $$y_n(x) = (2n-1)x y_{n-1}(x) + y_{n-2}(x)$$

and

$$\theta_0(x) = 1$$ $$\theta_1(x) = x + 1$$ $$\theta_n(x) = (2n-1) \theta_{n-1}(x) + x^2 \theta_{n-2}(x)$$

Differential equation

The Bessel polynomial obeys the following differential equation:

$$x^2 \frac{d^2 y_n(x)}{dx^2} + 2(x+1) \frac{dy_n(x)}{dx} - n(n+1) y_n(x) = 0$$

and

$$x \frac{d^2 \theta_n(x)}{dx^2} - 2(x+n) \frac{d \theta_n(x)}{dx} + 2n \theta_n(x) = 0$$

Orthogonality

The Bessel polynomials are orthogonal with respect to the weight $e^{-2/x}$ integrated over the unit circle of the complex plane[^1^][^104^]. In other words, if $n \neq m$,

$$\int_0^{2\pi} y_n\left(e^{i\theta}\right) y_m\left(e^{i\theta}\right) ie^{i\theta} \mathrm{d}\theta = 0$$