Editing Hermite polynomials (section)

==Generalizations==
The probabilist's Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution, whose density function is
<math display="block">\frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}},</math>
which has expected value 0 and variance 1.

Scaling, one may analogously speak of '''generalized Hermite polynomials'''<ref>{{Citation |last=Roman |first=Steven |date=1984 |title=The Umbral Calculus |series=Pure and Applied Mathematics |volume=111 |publisher=Academic Press |edition=1st |isbn=978-0-12-594380-2 |pages=87–93}}</ref>
<math display="block">\operatorname{He}_n^{[\alpha]}(x)</math>
of variance {{mvar|α}}, where {{mvar|α}} is any positive number. These are then orthogonal with respect to the normal probability distribution whose density function is
<math display="block">\frac{1}{\sqrt{2 \pi \alpha}} e^{-\frac{x^2}{2\alpha}}.</math>
They are given by
<math display="block">\operatorname{He}_n^{[\alpha]}(x) = \alpha^{\frac{n}{2}}\operatorname{He}_n\left(\frac{x}{\sqrt{\alpha}}\right) = \left(\frac{\alpha}{2}\right)^{\frac{n}{2}} H_n\left( \frac{x}{\sqrt{2 \alpha}}\right) = e^{-\frac{\alpha D^2}{2}} \left(x^n\right).</math>

Now, if 
<math display="block">\operatorname{He}_n^{[\alpha]}(x) = \sum_{k=0}^n h^{[\alpha]}_{n,k} x^k,</math>
then the polynomial sequence whose {{mvar|n}}th term is
<math display="block">\left(\operatorname{He}_n^{[\alpha]} \circ \operatorname{He}^{[\beta]}\right)(x) \equiv \sum_{k=0}^n h^{[\alpha]}_{n,k}\,\operatorname{He}_k^{[\beta]}(x)</math>
is called the [[Binomial type#Umbral composition of polynomial sequences|umbral composition]] of the two polynomial sequences. It can be shown to satisfy the identities
<math display="block">\left(\operatorname{He}_n^{[\alpha]} \circ \operatorname{He}^{[\beta]}\right)(x) = \operatorname{He}_n^{[\alpha+\beta]}(x)</math>
and
<math display="block">\operatorname{He}_n^{[\alpha+\beta]}(x + y) = \sum_{k=0}^n \binom{n}{k} \operatorname{He}_k^{[\alpha]}(x) \operatorname{He}_{n-k}^{[\beta]}(y).</math>
The last identity is expressed by saying that this [[parameterized family]] of polynomial sequences is known as a cross-sequence. (See the above section on Appell sequences and on the [[#Differential-operator representation|differential-operator representation]], which leads to a ready derivation of it. This [[binomial type]] identity, for {{math|1=''α'' = ''β'' = {{sfrac|1|2}}}}, has already been encountered in the above section on [[#Recursion relation]]s.)

==="Negative variance"===
Since polynomial sequences form a [[group (mathematics)|group]] under the operation of [[Binomial type#Umbral composition of polynomial sequences|umbral composition]], one may denote by
<math display="block">\operatorname{He}_n^{[-\alpha]}(x)</math>
the sequence that is inverse to the one similarly denoted, but without the minus sign, and thus speak of Hermite polynomials of negative variance. For {{math|α > 0}}, the coefficients of <math>\operatorname{He}_n^{[-\alpha]}(x)</math> are just the absolute values of the corresponding coefficients of <math>\operatorname{He}_n^{[\alpha]}(x)</math>.

These arise as moments of normal probability distributions: The {{mvar|n}}th moment of the normal distribution with expected value {{mvar|μ}} and variance {{math|''σ''<sup>2</sup>}} is
<math display="block">E[X^n] = \operatorname{He}_n^{[-\sigma^2]}(\mu),</math>
where {{mvar|X}} is a random variable with the specified normal distribution. A special case of the cross-sequence identity then says that
<math display="block">\sum_{k=0}^n \binom{n}{k} \operatorname{He}_k^{[\alpha]}(x) \operatorname{He}_{n-k}^{[-\alpha]}(y) = \operatorname{He}_n^{[0]}(x + y) = (x + y)^n.</math>