Bessel polynomials

Definition

Let $n \ge 0$ be an integer. The Bessel polynomial of degree $n$ is the polynomial $y_n(x) = \sum_{k=0}^{n} \frac{(n + k)!}{(n - k)! \, k!} \left( \frac{x}{2} \right)^k.$ The reverse Bessel polynomial is $\theta_n(x) = x^n y_n(1/x)$ .

Remarks

Examples: $y_0 = 1$ , $y_1 = 1 + x$ and $y_2 = 1 + 3x + 3x^2$ . Check for the top coefficient of $y_2$ : $\frac{4!}{0! \, 2!} \cdot \frac14 = 3$ . Each $y_n$ solves $x^2 y'' + (2x + 2) y' - n(n + 1) y = 0$ ; for $y_1$ this reads $0 + (2x + 2) - 2(1 + x) = 0$ . Non-example: $1 + 2x$ has degree $1$ but is not a solution for $n = 1$ , since $(2x + 2) \cdot 2 - 2(1 + 2x) = 2 \ne 0$ . For $x > 0$ they are linked to the modified Bessel function of the second kind: $y_n(x) = \sqrt{\frac{2}{\pi x}} \, e^{1/x} K_{n + 1/2}(1/x)$ .

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Polynomial function concept
Bessel polynomials definition

Tag	`YQQW8X2ZZJ`
URI	`m2://YQQW8X2ZZJ`
Kind	definition
Writing	W0 stub
MSC	33C47, 33C10
Defines	Bessel polynomials
Uses	Polynomial function

Bessel polynomials

Definition

Remarks

Reading path

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