Editing Hermite polynomials (section)

=== Recursion relation ===
Following recursion relations of Hermite polynomials, the Hermite functions obey
<math display="block">\psi_n'(x) = \sqrt{\frac{n}{2}}\,\psi_{n-1}(x) - \sqrt{\frac{n+1}{2}}\psi_{n+1}(x)</math>
and
<math display="block">x\psi_n(x) = \sqrt{\frac{n}{2}}\,\psi_{n-1}(x) + \sqrt{\frac{n+1}{2}}\psi_{n+1}(x).</math>

Extending the first relation to the arbitrary {{mvar|m}}th derivatives for any positive integer {{mvar|m}} leads to
<math display="block">\psi_n^{(m)}(x) = \sum_{k=0}^m \binom{m}{k} (-1)^k 2^\frac{m-k}{2} \sqrt{\frac{n!}{(n-m+k)!}} \psi_{n-m+k}(x) \operatorname{He}_k(x).</math>

This formula can be used in connection with the recurrence relations for {{math|''He<sub>n</sub>''}} and {{math|''ψ''<sub>''n''</sub>}} to calculate any derivative of the Hermite functions efficiently.