# Bessel Functions: No Common Zeros for J_n(x) and J_{n+m}(x)

Published on August 23, 2024

This proof demonstrates that Bessel functions of different orders, $J_n(x)$ and $J_{n+m}(x)$, have no common zeros except at x=0. The proof uses the Bessel differential equation, recurrence relations, and a proof by contradiction to show that assuming a common zero leads to a contradiction.

## Question

For integers n >= 0 and m >= 1, the Bessel functions J_n(x) and J_{n+m}(x) have no common zeros except at x = 0.

How to proof it