Editing Hermite polynomials (section)

===Hermite's differential equation===
The probabilist's Hermite polynomials are solutions of the [[differential equation]]
<math display="block">\left(e^{-\frac12 x^2}u'\right)' + \lambda e^{-\frac 1 2 x^2}u = 0,</math>
where {{mvar|λ}} is a constant. Imposing the boundary condition that {{mvar|u}} should be polynomially bounded at infinity, the equation has solutions only if {{mvar|λ}} is a non-negative integer, and the solution is uniquely given by <math>u(x) = C_1 \operatorname{He}_\lambda(x) </math>, where <math>C_{1}</math> denotes a constant.

Rewriting the differential equation as an [[Eigenvalue|eigenvalue problem]]
<math display="block">L[u] = u'' - x u' = -\lambda u,</math>
the Hermite polynomials <math>\operatorname{He}_\lambda(x) </math> may be understood as [[eigenfunction]]s of the differential operator <math>L[u]</math> . This eigenvalue problem is called the '''Hermite equation''', although the term is also used for the closely related equation
<math display="block">u'' - 2xu' = -2\lambda u.</math> 
whose solution is uniquely given in terms of physicist's Hermite polynomials in the form <math>u(x) = C_1 H_\lambda(x) </math>, where <math>C_{1}</math> denotes a constant, after imposing the boundary condition that {{mvar|u}} should be polynomially bounded at infinity.

The general solutions to the above second-order differential equations are in fact linear combinations of both Hermite polynomials and confluent hypergeometric functions of the first kind. For example, for the physicist's Hermite equation
<math display="block">u'' - 2xu' + 2\lambda u = 0,</math>
the general solution takes the form
<math display="block">u(x) = C_1   H_\lambda(x) + C_2  h_\lambda(x),</math>
where <math>C_{1}</math> and <math>C_{2}</math> are constants, <math>H_\lambda(x)</math> are physicist's Hermite polynomials (of the first kind), and  <math>h_\lambda(x)</math> are physicist's Hermite functions (of the second kind). The latter functions are compactly represented as <math> h_\lambda(x) = {}_1F_1(-\tfrac{\lambda}{2};\tfrac{1}{2};x^2)</math> where  <math>{}_1F_1(a;b;z)</math> are [[Confluent hypergeometric function|Confluent hypergeometric functions of the first kind]]. The conventional Hermite polynomials may also be expressed in terms of confluent hypergeometric functions, see below.

With more general [[boundary conditions]], the Hermite polynomials can be generalized to obtain more general [[analytic function]]s for complex-valued {{mvar|λ}}. An explicit formula of Hermite polynomials in terms of [[contour integral]]s {{harv|Courant|Hilbert|1989}} is also possible.