The linear differential equation
in which
is a real constant, is called the Hermite equation .\, Its general solution is\,
\, with
and
arbitrary constants and the functions
and
presented as\\
\quad
\\
\quad
\\
It's easy to check that these power series satisfy the differential equation.\, The coefficients
in both series obey the recurrence formula
Thus we have the radii of convergence
Therefore the series converge in the whole complex plane and define entire functions.
If the constant
is a non-negative integer, then one of
and
is simply a polynomial function.\, The polynomial solutions of the Hermite equation are usually normed so that the highest degree term is
and called the Hermite polynomials.