Editing Hermite polynomials (section)

===Orthogonality===
{{math|''H<sub>n</sub>''(''x'')}} and {{math|''He<sub>n</sub>''(''x'')}} are {{mvar|n}}th-degree polynomials for {{math|''n'' {{=}} 0, 1, 2, 3,...}}. These [[orthogonal polynomials|polynomials are orthogonal]] with respect to the ''weight function'' ([[measure (mathematics)|measure]])
<math display="block">w(x) = e^{-\frac{x^2}{2}} \quad (\text{for }\operatorname{He})</math> or <math display="block">w(x) = e^{-x^2} \quad (\text{for } H),</math>
i.e., we have
<math display="block">\int_{-\infty}^\infty H_m(x) H_n(x)\, w(x) \,dx = 0 \quad \text{for all }m \neq n.</math>

Furthermore,
<math display="block">\int_{-\infty}^\infty H_m(x) H_n(x)\, e^{-x^2} \,dx = \sqrt{\pi}\, 2^n n!\, \delta_{nm},</math>
and
<math display="block">\int_{-\infty}^\infty \operatorname{He}_m(x) \operatorname{He}_n(x)\, e^{-\frac{x^2}{2}} \,dx = \sqrt{2 \pi}\, n!\, \delta_{nm},</math>
where <math>\delta_{nm}</math> 
is the [[Kronecker delta]].

The probabilist polynomials are thus orthogonal with respect to the standard normal probability density function.