Proof of Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity

Each result cited is universally quantified over the data in its own statement.

Conventions. Natural numbers, factorials and binomial coefficients are read in $\mathbb{R}$ through the canonical map, as in The Real Numbers: Standing Notation and Background §numbers. The arithmetic and order manipulations of real numbers, natural powers and finite sums made below without further comment (reindexing a finite sum along a bijection, splitting off or adding zero terms, regrouping, the triangle inequality for finite sums, and the power rules $(ab)^{k}=a^{k}b^{k}$, $a^{k}a^{l}=a^{k+l}$, $(a^{k})^{l}=a^{kl}$, $|a^{k}|=|a|^{k}$ with $a^{0}=1$) are those in force by The Real Numbers: Standing Notation and Background §background. The data $v,w\ge0$ and $n\in\mathbb{N}_{0}$ of the statement are arbitrary, so every claim, once proved, is available for every choice of nonnegative variances, every degree and every point, and it is used in that form below. For $m\in\mathbb{N}_{0}$ and $j\in J_{m}$ put

$$a_{m,j}=\frac{m!}{2^{j}\,j!\,(m-2j)!},$$

so that by Hermite Polynomials with a Given Variance §hermite, for every real $u\ge0$ and every $t\in\mathbb{R}$,

$$H^{u}_{m}(t)=\sum_{j\in J_{m}}a_{m,j}\,(-u)^{j}\,t^{m-2j}.$$

We call this the defining formula. Note $a_{m,0}=m!/m!=1$ for every $m$. The set $J_{m}$ is downward closed in $\mathbb{N}_{0}$ (if $2j\le m$ and $i\le j$ then $2i\le m$), finite and nonempty, so $J_{m}=\{0,1,\dots,\ell_{m}\}$ with $\ell_{m}$ its largest element. We also use the factorial rule (F): for every $k\in\mathbb{N}$, $k!=k\,(k-1)!$, where $(k-1)!=0!=1$ if $k=1$; for $k=1$ this reads $1!=1=1\cdot0!$ by Recursion for the Factorial of a Natural Number §recursion, and for $k\ge2$ it is the recursion of Recursion for the Factorial of a Natural Number §recursion applied to $k-1$.

Derivatives of powers. By claim 1 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line, $\mathbb{R}$ is an interval every point of which is an interior point. Let $e\in\mathbb{N}_{0}$ and $t\in\mathbb{R}$. If $e=0$ the map $x\mapsto x^{e}$ is the constant $1$, which is differentiable at $t$ with derivative $0$ by claim 1 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives; if $e=1$ its derivative at $t$ is $1=1\cdot t^{0}$, and if $e\ge2$, writing $e=S(e-1)$, its derivative at $t$ is $e\,t^{e-1}$, both by claim 1 of Derivative of a Polynomial Function on the Real Line. By claim 2 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line these derivatives are the partial derivatives $\partial_{1}$: so $\partial_{1}(x\mapsto x^{e})(t)$ is $0$ if $e=0$ and $e\,t^{e-1}$ if $e\ge1$. Consequently, for real numbers $\lambda_{j}$ ($j\in J_{m}$) and exponents $e_{j}\in\mathbb{N}_{0}$, the function $x\mapsto\sum_{j\in J_{m}}\lambda_{j}x^{e_{j}}$, which after the enumeration $k\mapsto k-1$ of $J_{m}$ by $[\ell_{m}+1]$ is a finite linear combination indexed by $[\ell_{m}+1]$, is differentiable at every $t$ with derivative $\sum_{j\in J_{m}}\lambda_{j}\,\partial_{1}(x\mapsto x^{e_{j}})(t)$, by Derivative of a Finite Linear Combination of Real Functions (with $I=\mathbb{R}$, and $t-1<t<t+1$), and this is again its partial derivative $\partial_{1}$ at $t$ by claim 2 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line. We refer to this as termwise differentiation.

Claim 1. Since $2j\le m$ is required, $J_{0}=J_{1}=\{0\}$, $J_{2}=J_{3}=\{0,1\}$ and $J_{4}=\{0,1,2\}$. Further $a_{2,1}=\frac{2!}{2\cdot1\cdot0!}=1$, $a_{3,1}=\frac{3!}{2\cdot1\cdot1!}=3$, $a_{4,1}=\frac{4!}{2\cdot1\cdot2!}=6$ and $a_{4,2}=\frac{4!}{4\cdot2!\cdot0!}=3$. The defining formula therefore gives, for every $t$, $H^{v}_{0}(t)=t^{0}=1$, $H^{v}_{1}(t)=t$, $H^{v}_{2}(t)=t^{2}+(-v)=t^{2}-v$, $H^{v}_{3}(t)=t^{3}+3(-v)t=t^{3}-3vt$ and $H^{v}_{4}(t)=t^{4}+6(-v)t^{2}+3(-v)^{2}=t^{4}-6vt^{2}+3v^{2}$. For the variance $0$ one has $(-0)^{0}=1$ and $(-0)^{j}=0$ for $j\ge1$, so only $j=0$ contributes and $H^{0}_{n}(t)=a_{n,0}t^{n}=t^{n}$.

Claim 2. Each summand $t\mapsto a_{n,j}(-v)^{j}t^{n-2j}$ of the defining formula is a scalar multiple of a power (of the constant $1$ when $n-2j=0$), hence a polynomial function on $\mathbb{R}$ by claims 1 and 2 of Constants, Powers, Sums, Scalar Multiples and Products of Polynomial Functions. Along the enumeration $J_{n}=\{0,\dots,\ell_{n}\}$ let $\mathcal{B}$ be the set of $k\in\mathbb{N}$ such that $k>\ell_{n}+1$ or the partial sum $t\mapsto\sum_{j=0}^{k-1}a_{n,j}(-v)^{j}t^{n-2j}$ is a polynomial function. The partial sum with $k=1$ is the summand $j=0$, so $1\in\mathcal{B}$. If $k\in\mathcal{B}$ and $k+1\le\ell_{n}+1$, then $k\le\ell_{n}$, so $k\in J_{n}$ and the partial sum up to $k$ is the partial sum up to $k-1$ plus the summand $j=k$, a polynomial function by claim 2 of that lemma; if $k+1>\ell_{n}+1$ there is nothing to show. So $\mathcal{B}$ is closed under successors, and $\mathcal{B}=\mathbb{N}$ by Principle of Induction for the Natural Numbers. Taking $k=\ell_{n}+1$ shows that $H^{v}_{n}$ is a polynomial function. By claim 2 of Polynomial Functions on the Real Line are Smooth, $H^{v}_{n}$ is of class $C^{k}$ on $\mathbb{R}^{1}$ for every natural number $k$, in particular of class $C^{2}$. For $i\in\{0,\dots,n\}$ put $b_{i}=a_{n,j}(-v)^{j}$ if $i=n-2j$ for some $j\in J_{n}$ (such a $j$ is unique, namely $j=(n-i)/2$), and $b_{i}=0$ otherwise. The map $j\mapsto n-2j$ is a bijection from $J_{n}$ onto the set of those $i\in\{0,\dots,n\}$ with $n-i$ even, so reindexing and adding the zero terms gives $H^{v}_{n}(t)=\sum_{i=0}^{n}b_{i}t^{i}$ for every $t$; and $b_{n}=a_{n,0}(-v)^{0}=1$. Finally let $t\in\mathbb{R}$ and $0\le i\le n$. If $|t|\le1$ then $|t|^{i}\le1$; if $1\le|t|$ then $|t|^{i}\le|t|^{n}$, by claim 1 of Growth Bound for a Polynomial Function on the Real Line when $i\ge1$ and because $|t|^{0}=1\le|t|^{n}$ when $i=0$. In either case $|t|^{i}\le1+|t|^{n}$, and with $M=\sum_{i=0}^{n}|b_{i}|$,

$$|H^{v}_{n}(t)|\le\sum_{i=0}^{n}|b_{i}|\,|t|^{i}\le M\,(1+|t|^{n}).$$

Claim 3. By claim 1, $H^{v}_{0}$ is the constant $1$, so $(H^{v}_{0})'(t)=0$ by the case $e=0$ above. Let $n\in\mathbb{N}$. Termwise differentiation of the defining formula gives, for every $t$,

$$(H^{v}_{n})'(t)=\sum_{j\in J_{n},\ 2j\le n-1}a_{n,j}(-v)^{j}(n-2j)\,t^{n-2j-1},$$

the indices $j\in J_{n}$ with $2j=n$ contributing $0$ (their power is the constant $t^{0}$). The index set on the right is exactly $J_{n-1}$. For $j\in J_{n-1}$ one has $n-2j\ge1$, so (F) gives $(n-2j)!=(n-2j)\,(n-1-2j)!$ and $n!=n\,(n-1)!$, whence

$$a_{n,j}\,(n-2j)=\frac{n!}{2^{j}\,j!\,(n-1-2j)!}=n\,\frac{(n-1)!}{2^{j}\,j!\,((n-1)-2j)!}=n\,a_{n-1,j}.$$

Since $t^{n-2j-1}=t^{(n-1)-2j}$, the defining formula for $H^{v}_{n-1}$ gives $(H^{v}_{n})'(t)=n\sum_{j\in J_{n-1}}a_{n-1,j}(-v)^{j}t^{(n-1)-2j}=n\,H^{v}_{n-1}(t)$.

Claim 4. If $n=0$, claims 1 and 3 give $tH^{v}_{0}(t)-v\,(H^{v}_{0})'(t)=t=H^{v}_{1}(t)$. Let $n\in\mathbb{N}$; by claim 3, $v\,(H^{v}_{n})'(t)=n\,v\,H^{v}_{n-1}(t)$, so both assertions of claim 4 reduce to

$$H^{v}_{n+1}(t)=t\,H^{v}_{n}(t)-n\,v\,H^{v}_{n-1}(t),$$

which we prove by comparing the three sums written over the common index set $J_{n+1}$. For $j\in J_{n+1}$ put $D_{j}=2^{j}\,j!\,(n+1-2j)!$, a positive real number, so that $a_{n+1,j}=(n+1)!/D_{j}$.

First, $tH^{v}_{n}(t)=\sum_{j\in J_{n}}a_{n,j}(-v)^{j}t^{n+1-2j}$. Here $J_{n}\subseteq J_{n+1}$, and every $j\in J_{n+1}\setminus J_{n}$ satisfies $n+1-2j=0$. For $j\in J_{n}$, (F) gives $(n+1-2j)!=(n+1-2j)(n-2j)!$, so $a_{n,j}=(n+1-2j)\,n!/D_{j}$; and for $j\in J_{n+1}\setminus J_{n}$ the number $(n+1-2j)\,n!/D_{j}$ is $0$. Hence, adding zero terms,

$$tH^{v}_{n}(t)=\sum_{j\in J_{n+1}}\frac{(n+1-2j)\,n!}{D_{j}}\,(-v)^{j}\,t^{n+1-2j}.$$

Second, $-n\,v\,H^{v}_{n-1}(t)=\sum_{i\in J_{n-1}}n\,a_{n-1,i}(-v)^{i+1}t^{n+1-2(i+1)}$. The map $i\mapsto i+1$ is a bijection from $J_{n-1}$ onto $J_{n+1}\setminus\{0\}$, since $2i\le n-1$ if and only if $2(i+1)\le n+1$. For $j=i+1$, (F) gives $j!=j\,(j-1)!$, $2^{j}=2\cdot2^{j-1}$ and $n!=n\,(n-1)!$, and $(n-1)-2(j-1)=n+1-2j$, so $n\,a_{n-1,j-1}=2j\,n!/D_{j}$; this expression is $0$ for $j=0$. Hence

$$-n\,v\,H^{v}_{n-1}(t)=\sum_{j\in J_{n+1}}\frac{2j\,n!}{D_{j}}\,(-v)^{j}\,t^{n+1-2j}.$$

Adding the two displays termwise and using $(n+1-2j)\,n!+2j\,n!=(n+1)\,n!=(n+1)!$, by Recursion for the Factorial of a Natural Number §recursion, gives $\sum_{j\in J_{n+1}}a_{n+1,j}(-v)^{j}t^{n+1-2j}=H^{v}_{n+1}(t)$.

Claim 5. The second derivative $(H^{v}_{n})''$ is the partial derivative of the function $(H^{v}_{n})'$. If $n=0$, $(H^{v}_{0})'$ is the zero function by claim 3, a constant function, which is differentiable at every point with derivative $0$ by claim 1 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives (with $I=\mathbb{R}$), this derivative being its partial derivative $\partial_{1}$ by claim 2 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line; so both sides of claim 5 vanish. If $n=1$, $(H^{v}_{1})'=1\cdot H^{v}_{0}$ is the constant $1$, so $(H^{v}_{1})''=0$ and the left side is $-t=-1\cdot H^{v}_{1}(t)$ by claim 1. Let $n\ge2$. By claim 3, $(H^{v}_{n})'$ is the function $nH^{v}_{n-1}$, and by claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set (scalar multiples, on $\mathbb{R}^{1}$, which is open by claim 1 of Polynomial Functions on the Real Line are Smooth) and claim 3 again, $(H^{v}_{n})''(t)=n\,(H^{v}_{n-1})'(t)=n(n-1)H^{v}_{n-2}(t)$. Therefore, using claim 4 in its second form with $n-1\in\mathbb{N}$ in place of $n$,

$$v\,(H^{v}_{n})''(t)-t\,(H^{v}_{n})'(t)=-n\bigl(t\,H^{v}_{n-1}(t)-(n-1)\,v\,H^{v}_{n-2}(t)\bigr)=-n\,H^{v}_{n}(t).$$

Claim 6. Let $r,t\in\mathbb{R}$. For $j\in J_{n}$, $(-r^{2}v)^{j}=(r^{2})^{j}(-v)^{j}=r^{2j}(-v)^{j}$ and $(rt)^{n-2j}=r^{n-2j}t^{n-2j}$, and $r^{2j}r^{n-2j}=r^{n}$. The defining formula with variance $r^{2}v\ge0$ therefore gives $H^{r^{2}v}_{n}(rt)=\sum_{j\in J_{n}}a_{n,j}r^{n}(-v)^{j}t^{n-2j}=r^{n}H^{v}_{n}(t)$.

Claim 7. For $m,j\in\mathbb{N}_{0}$ with $j\le m$ write $C(m,j)=\binom{m}{j}$. From the factorial formula and (F) one obtains: $C(m,0)=C(m,m)=1$; Pascal's rule $C(m+1,j)=C(m,j)+C(m,j-1)$ for $1\le j\le m$, because $\frac{m!}{j!(m-j)!}+\frac{m!}{(j-1)!(m+1-j)!}=\frac{m!\,((m+1-j)+j)}{j!\,(m+1-j)!}=\frac{(m+1)!}{j!\,(m+1-j)!}$; and, for $m\ge1$, $j\,C(m,j)=m\,C(m-1,j-1)$ for $1\le j\le m$ and $(m-j)\,C(m,j)=m\,C(m-1,j)$ for $0\le j\le m-1$, because both sides equal $\frac{m!}{(j-1)!(m-j)!}$, respectively $\frac{m!}{j!(m-j-1)!}$.

Fix $s,t\in\mathbb{R}$ and abbreviate $A_{j}=H^{v}_{j}(s)$, $A'_{j}=(H^{v}_{j})'(s)$, $B_{k}=H^{w}_{k}(t)$, $B'_{k}=(H^{w}_{k})'(t)$, and $G_{m}=\sum_{j=0}^{m}C(m,j)A_{j}B_{m-j}$ for $m\in\mathbb{N}_{0}$. By claim 3, $A'_{0}=B'_{0}=0$, $A'_{j}=jA_{j-1}$ and $B'_{k}=kB_{k-1}$ for $j,k\ge1$; by claim 4, $A_{j+1}=sA_{j}-vA'_{j}$ and $B_{k+1}=tB_{k}-wB'_{k}$ for all $j,k\in\mathbb{N}_{0}$.

Recursion for $G$. Let $m\in\mathbb{N}$. By Pascal's rule and $C(m+1,0)=C(m,0)$, $C(m+1,m+1)=C(m,m)$,

$$G_{m+1}=\sum_{j=0}^{m}C(m,j)A_{j}B_{m+1-j}+\sum_{i=0}^{m}C(m,i)A_{i+1}B_{m-i},$$

the second sum arising from the terms $C(m,j-1)A_{j}B_{m+1-j}$, $1\le j\le m+1$, by the shift $i=j-1$. Inserting $B_{m+1-j}=tB_{m-j}-wB'_{m-j}$ in the first sum and $A_{i+1}=sA_{i}-vA'_{i}$ in the second gives $G_{m+1}=(s+t)G_{m}-vX-wY$ with $X=\sum_{j=0}^{m}C(m,j)A'_{j}B_{m-j}$ and $Y=\sum_{j=0}^{m}C(m,j)A_{j}B'_{m-j}$. In $X$ the term $j=0$ vanishes, and for $1\le j\le m$, $C(m,j)A'_{j}=j\,C(m,j)A_{j-1}=m\,C(m-1,j-1)A_{j-1}$, so after the shift $i=j-1$, $X=m\sum_{i=0}^{m-1}C(m-1,i)A_{i}B_{m-1-i}=mG_{m-1}$. In $Y$ the term $j=m$ vanishes, and for $0\le j\le m-1$, $C(m,j)B'_{m-j}=(m-j)C(m,j)B_{m-1-j}=m\,C(m-1,j)B_{m-1-j}$, so $Y=mG_{m-1}$. Hence

$$G_{m+1}=(s+t)\,G_{m}-m\,(v+w)\,G_{m-1}\qquad(m\in\mathbb{N}).$$

Induction. The quantities $G_{m}$ depend on the fixed points $s,t$; write $G_{m}(s,t)=\sum_{j=0}^{m}C(m,j)H^{v}_{j}(s)H^{w}_{m-j}(t)$ for $m\in\mathbb{N}_{0}$ and $s,t\in\mathbb{R}$, so that the recursion for $G$ holds for every pair $s,t$. Let $\mathcal{A}$ be the set of $m\in\mathbb{N}$ such that, for all $s,t\in\mathbb{R}$, $G_{m-1}(s,t)=H^{v+w}_{m-1}(s+t)$ and $G_{m}(s,t)=H^{v+w}_{m}(s+t)$. By claim 1, for all $s,t\in\mathbb{R}$, $G_{0}(s,t)=C(0,0)H^{v}_{0}(s)H^{w}_{0}(t)=1=H^{v+w}_{0}(s+t)$ and $G_{1}(s,t)=H^{v}_{0}(s)H^{w}_{1}(t)+H^{v}_{1}(s)H^{w}_{0}(t)=t+s=H^{v+w}_{1}(s+t)$, so $1\in\mathcal{A}$. Let $m\in\mathcal{A}$ and $s,t\in\mathbb{R}$. The recursion for $G$ at the points $s,t$ and claim 4 in its second form, for the variance $v+w\ge0$, the degree $m\in\mathbb{N}$ and the point $s+t$, give

$$G_{m+1}(s,t)=(s+t)H^{v+w}_{m}(s+t)-m\,(v+w)\,H^{v+w}_{m-1}(s+t)=H^{v+w}_{m+1}(s+t);$$

as $s,t$ were arbitrary, $m+1\in\mathcal{A}$. By Principle of Induction for the Natural Numbers, $\mathcal{A}=\mathbb{N}$. Taking $m=1$ gives claim 7 for $n=0$, and taking $m=n$ gives it for $n\in\mathbb{N}$.

Claim 8. Assume $v>0$, so that $(v)$ is a variance vector in $\mathbb{R}^{1}$; let $\rho$ be the diagonal Gaussian density with variances $(v)$ and $\lambda_{1}$ Lebesgue measure on $\mathcal{B}(\mathbb{R}^{1})$. By the definition of $\gamma_{v}$ it is the measure with density $\rho$ with respect to $\lambda_{1}$, so by claim 3 of Image Measures, Measures with Densities, and Change of Variables a Borel $f:\mathbb{R}\to\mathbb{R}$ is integrable with respect to $\gamma_{v}$ if and only if $f\rho$ is integrable with respect to $\lambda_{1}$, and then $\int f\,d\gamma_{v}=\int f\rho\,d\lambda_{1}$. By The Diagonal Gaussian Density on Euclidean Space: Regularity, Gradient, Normalization, Second Moments and Exponential Moments of Diagonal Quadratic Forms §regularity, $\rho$ is smooth, positive and Borel, and by The Diagonal Gaussian Density on Euclidean Space: Regularity, Gradient, Normalization, Second Moments and Exponential Moments of Diagonal Quadratic Forms §gradient, whose scaling map is $y\mapsto y/v$ in dimension $1$, $\partial_{1}\rho(y)=-v^{-1}y\,\rho(y)$ for every $y$.

Step 8a (polynomials are integrable). Every polynomial function $p$ on $\mathbb{R}$, $p(x)=c_{0}+\sum_{k=1}^{N}c_{k}x^{k}$, is a finite linear combination of the monomials $y\mapsto y^{k}$ ($k\in\mathbb{N}_{0}$) on $\mathbb{R}^{1}$, so by Polynomial Functions Are Dense in the Square-Integrable Functions of a Diagonal Gaussian Measure on Euclidean Space §integrable, with dimension $1$, variance vector $(v)$ and exponent $1$, $p$ is continuous and Borel and $\int|p|\,d\gamma_{v}<\infty$ (since $|p|^{1}=|p|$ by Properties of Real Powers of Nonnegative Real Numbers §agreement); hence $p$ is integrable with respect to $\gamma_{v}$, by the criterion of Measure Spaces and the Lebesgue Integral: Standing Notation §integral. Products and linear combinations of polynomial functions are polynomial functions by Constants, Powers, Sums, Scalar Multiples and Products of Polynomial Functions; in particular every $H^{v}_{m}$, every product $H^{v}_{m}H^{v}_{n}$, and $y\mapsto y\,p(y)$ for a polynomial function $p$, are Borel and $\gamma_{v}$-integrable. Moreover, for a polynomial function $p$, its partial derivative $p'=\partial_{1}p$ exists everywhere and is a polynomial function, by claim 2 of Derivative of a Polynomial Function on the Real Line together with claims 1 and 2 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line.

Step 8b (Gaussian integration by parts). Let $p$ be a polynomial function on $\mathbb{R}$. We show

$$\int_{\mathbb{R}}p'\,d\gamma_{v}=v^{-1}\int_{\mathbb{R}}y\,p(y)\,\gamma_{v}(dy).$$

Let $\chi_{R}$ ($R>0$) be the cutoffs of Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §cutoff in dimension $q=1$, with the constant $M_{1}\ge0$ given there; on $\mathbb{R}^{1}$ the Euclidean norm is the absolute value by One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §scalars. The set $U=\mathbb{R}^{1}$ is open by claim 1 of Polynomial Functions on the Real Line are Smooth, so Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set applies on it. The function $f=p\rho$ is smooth on $\mathbb{R}^{1}$ by claim 2 of Polynomial Functions on the Real Line are Smooth, The Diagonal Gaussian Density on Euclidean Space: Regularity, Gradient, Normalization, Second Moments and Exponential Moments of Diagonal Quadratic Forms §regularity and claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, hence of class $C^{1}$ by Smooth Map on a Euclidean Open Set (with $k=1$), and by claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set and the formula for $\partial_{1}\rho$, $\partial_{1}f=(p'-v^{-1}y\,p)\rho$. For $m\in\mathbb{N}$ the cutoff $\chi_{m}$ is smooth and compactly supported, so Integration by Parts on Euclidean Space Against a Compactly Supported Function of Class $C^{1}$ §parts gives that $f\,\partial_{1}\chi_{m}$ and $(\partial_{1}f)\chi_{m}$ are integrable with respect to $\lambda_{1}$ and

$$\int_{\mathbb{R}}(p'-v^{-1}y\,p)\rho\,\chi_{m}\,d\lambda_{1}=-\int_{\mathbb{R}}p\rho\,\partial_{1}\chi_{m}\,d\lambda_{1}.$$

By Step 8a, $|p|\rho$ and $g=|p'-v^{-1}y\,p|\rho$ are integrable with respect to $\lambda_{1}$. Since $|\partial_{1}\chi_{m}|\le M_{1}m^{-1}$, Linearity and Monotonicity of the Lebesgue Integral §integrable bounds the absolute value of the right side by $M_{1}m^{-1}\int|p|\rho\,d\lambda_{1}$, which tends to $0$ as $m\to\infty$ by claim 3(a) of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities and Arithmetic of Limits of Real Sequences §scalar. On the left, $|(p'-v^{-1}yp)\rho\chi_{m}|\le g$ because $0\le\chi_{m}\le1$, and for each $y$ one has $\chi_{m}(y)=1$ for every $m\ge|y|$, such $m$ existing by the Archimedean property, so the integrands converge pointwise to $(p'-v^{-1}yp)\rho$; they are Borel, being integrable. By Dominated Convergence Theorem the left side converges to $\int(p'-v^{-1}yp)\rho\,d\lambda_{1}$, which is therefore $0$ by uniqueness of limits. By claim 3 of Image Measures, Measures with Densities, and Change of Variables this says $\int(p'-v^{-1}yp)\,d\gamma_{v}=0$, and linearity (Linearity and Monotonicity of the Lebesgue Integral §integrable), all integrands being $\gamma_{v}$-integrable by Step 8a, gives the display.

Step 8c (two identities). Write $I(m,k)=\int H^{v}_{m}H^{v}_{k}\,d\gamma_{v}$. First, $I(0,0)=\int1\,d\gamma_{v}=\gamma_{v}(\mathbb{R})=1$ by Diagonal Gaussian Measures on Euclidean Space §measure. Second, let $k\in\mathbb{N}$. By claim 4 (first form, degree $k-1$), $H^{v}_{k}(y)=y\,H^{v}_{k-1}(y)-v\,(H^{v}_{k-1})'(y)$, and Step 8b with $p=H^{v}_{k-1}$ gives $\int yH^{v}_{k-1}\,d\gamma_{v}=v\int(H^{v}_{k-1})'\,d\gamma_{v}$; by linearity $\int H^{v}_{k}\,d\gamma_{v}=0$. Since $H^{v}_{0}=1$, this is $I(0,k)=I(k,0)=0$. Third, let $m,k\in\mathbb{N}$ and $p=H^{v}_{m}H^{v}_{k-1}$. By claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set and claim 3, $p'=mH^{v}_{m-1}H^{v}_{k-1}+H^{v}_{m}(H^{v}_{k-1})'$. Using claim 4 for $H^{v}_{k}$, then Step 8b and linearity,

$$I(m,k)=\int y\,p\,d\gamma_{v}-v\int H^{v}_{m}(H^{v}_{k-1})'\,d\gamma_{v}=v\int p'\,d\gamma_{v}-v\int H^{v}_{m}(H^{v}_{k-1})'\,d\gamma_{v}=v\,m\,I(m-1,k-1).$$

Step 8d (induction). Let $\mathcal{A}$ be the set of $m\in\mathbb{N}$ such that for every $k\in\mathbb{N}_{0}$, $I(m-1,k)=(m-1)!\,v^{m-1}$ if $k=m-1$ and $I(m-1,k)=0$ if $k\ne m-1$. By Step 8c, $I(0,0)=1=0!\,v^{0}$ and $I(0,k)=0$ for $k\ge1$, so $1\in\mathcal{A}$. Let $m\in\mathcal{A}$ and $k\in\mathbb{N}_{0}$. If $k=0$, then $k\ne m$ and $I(m,0)=0$ by Step 8c. If $k\ge1$, Step 8c and $m\in\mathcal{A}$ give $I(m,k)=v\,m\,I(m-1,k-1)$, which is $v\,m\,(m-1)!\,v^{m-1}=m!\,v^{m}$ when $k-1=m-1$, by (F), and $0$ otherwise. So $m+1\in\mathcal{A}$, and $\mathcal{A}=\mathbb{N}$ by Principle of Induction for the Natural Numbers. Applying this to $m+1\in\mathcal{A}$ for the $m$ of the statement gives the displayed formula of claim 8, and the case $m=0$, $n\in\mathbb{N}$ is the final assertion. Borel measurability and integrability of $H^{v}_{m}$ and $H^{v}_{m}H^{v}_{n}$ were shown in Step 8a.

Claim 9. Assume $v>0$ and $r^{2}+s^{2}=1$, and let $t,y\in\mathbb{R}$. The numbers $r^{2}v$ and $s^{2}v$ are nonnegative and $r^{2}v+s^{2}v=v$. Claim 7 with the variances $r^{2}v$, $s^{2}v$ and the points $rt$, $sy$, followed by claim 6 (with $r$ and with $s$), gives

$$H^{v}_{n}(rt+sy)=\sum_{j=0}^{n}\binom{n}{j}H^{r^{2}v}_{j}(rt)\,H^{s^{2}v}_{n-j}(sy)=\sum_{j=0}^{n}\binom{n}{j}\,r^{j}s^{n-j}H^{v}_{j}(t)\,H^{v}_{n-j}(y).$$

For fixed $t$ the right side, as a function of $y$, is a linear combination of the polynomial functions $H^{v}_{n-j}$, hence a polynomial function by Constants, Powers, Sums, Scalar Multiples and Products of Polynomial Functions, and so it is Borel and $\gamma_{v}$-integrable by Step 8a. By linearity and claim 8 with $m=0$, $\int H^{v}_{n-j}\,d\gamma_{v}=0$ unless $n-j=0$, in which case it equals $\int H^{v}_{0}\,d\gamma_{v}=1$. Hence

$$\int_{\mathbb{R}}H^{v}_{n}(rt+sy)\,\gamma_{v}(dy)=\binom{n}{n}\,r^{n}s^{0}H^{v}_{n}(t)=r^{n}H^{v}_{n}(t).$$