In the Sturm-Liouville Boundary Value Problem, there is a special case called Hermite's Differential Equation which arises in the treatment of the harmonic oscillator in quantum mechanics. Hermite's Differential Equation is defined as:
where is a real number. For is a non-negative integer, i.e., , the solutions of Hermite's Differential Equation are often referred to as Hermite Polynomials .
Rodrigues' Formula: The Hermite Polynomials can be expressed by Rodrigues' formula
Generating Function: The generating function of Hermite Polynomial is
Orthogonality: Hermite Polynomials , , form a complete orthogonal set on the interval with respect to the weighting function . It can be shown that:
By using this orthogonality, a piecewise continuous function can be expressed in terms of Hermite Polynomials:
This orthogonal series expansion is also known as Fourier-Hermite Series expansion or Generalized Fourier Series expansion.
Even/Odd Functions: Whether a Hermite Polynomial is an even or odd function depends on its degree .
Based on ,
is an even function, when is even.
is an odd function, when is odd.
Recurrence Relation: A Hermite Polynomial at one point can be expressed by neighboring Hermite Polynomials at the same point.