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

Published on August 23, 2024

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

Answer

Proving Bessel Functions and Have No Common Zeros (Except x=0)

This proof relies on the following properties of Bessel functions:
  1. Bessel’s differential equation:

    where is a solution.
  2. Recurrence relation:
Proof by contradiction:
  1. Assumption: Let’s assume that and have a common zero at where . This means:
    and .
  2. Using the recurrence relation: We can write the recurrence relation for and :
  3. Applying the assumption: Since and , the recurrence relations simplify to:
  4. Dividing the equations: Dividing the first simplified recurrence relation by the second, we get:
  5. Repeating the process: We can repeat steps 2-4 for and , then for and , and so on. This will lead to a chain of equalities:
  6. Final step: Notice that the last term in the chain involves and . Since we assumed , and Bessel functions have no repeated zeros (except at x=0), we know that and .
    This contradicts the chain of equalities we derived, meaning our initial assumption that and share a common zero at (where ) must be false.
Therefore, Bessel functions and have no common zeros except at .
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