---
tag: M5NXATQ48S
kind: concept
name: Bessel polynomials
msc: [33C47, 33C10]
sources:
  - url: https://en.wikipedia.org/wiki/Bessel_polynomials
same_as:
  wikidata: Q4896402
  wikipedia: Bessel polynomials
---

## Summary

The Bessel polynomials $y_n$ are polynomials with positive integer coefficients that arise from the [[modified Bessel functions|2ET068BZZH]] of half-integer order.
They solve a second-order differential equation with polynomial coefficients.
They are orthogonal for a weight on a circle in the complex plane, not on an interval of the real line.
Their reversed forms are the denominators of Bessel filters in electrical engineering.

## Explanations

### L1 human

Bessel functions of half-integer order can be written with exponentials and finitely many powers of $1/x$.
The polynomials that appear in this way are the Bessel polynomials.
The first ones are $y_0 = 1$, $y_1 = 1 + x$ and $y_2 = 1 + 3x + 3x^2$.
Engineers use them to design filters that delay all frequencies of a signal by almost the same time.