Editing Hermite polynomials (section)

==Differential-operator representation==
The probabilist's Hermite polynomials satisfy the identity<ref>{{cite book |last1=Rota |first1=Gian-Carlo |last2=Doubilet |first2=P. |title=Finite operator calculus |date=1975 |publisher=Academic Press |location=New York |isbn=9780125966504 |page=44}}</ref> <math display="block">\operatorname{He}_n(x) = e^{-\frac{D^2}{2}}x^n,</math> where {{mvar|D}} represents differentiation with respect to {{mvar|x}}, and the [[exponential function|exponential]] is interpreted by expanding it as a [[power series]]. There are no delicate questions of convergence of this series when it operates on polynomials, since all but finitely many terms vanish.

Since the power-series coefficients of the exponential are well known, and higher-order derivatives of the monomial {{math|''x''<sup>''n''</sup>}} can be written down explicitly, this differential-operator representation gives rise to a concrete formula for the coefficients of {{math|''H<sub>n</sub>''}} that can be used to quickly compute these polynomials.

Since the formal expression for the [[Weierstrass transform]] {{mvar|W}} is {{math|''e''<sup>''D''<sup>2</sup></sup>}}, we see that the Weierstrass transform of {{math|({{sqrt|2}})<sup>''n''</sup>''He<sub>n</sub>''({{sfrac|''x''|{{sqrt|2}}}})}} is {{math|''x<sup>n</sup>''}}. Essentially the Weierstrass transform thus turns a series of Hermite polynomials into a corresponding [[Maclaurin series]].

The existence of some formal power series {{math|''g''(''D'')}} with nonzero constant coefficient, such that {{math|1=''He<sub>n</sub>''(''x'') = ''g''(''D'')''x<sup>n</sup>''}}, is another equivalent to the statement that these polynomials form an [[Appell sequence]]. Since they are an Appell sequence, they are ''a fortiori'' a [[Sheffer sequence]].

{{Further|Weierstrass transform#The inverse transform}}