Proof of Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations

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

The notation is that of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §background; in particular $x_{k}=\langle x,e_{k}\rangle$, $p_{n}(x)=(x_{1},\dots,x_{n})$ as in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, $\lVert\cdot\rVert$ is the Euclidean norm, and $c$ is a variance sequence, so $c_{k}>0$ for every $k$ by Variance Sequences and Their Truncations §variances. For a variance vector $w=(w_{1},\dots,w_{m})$ (The Diagonal Gaussian Density on Euclidean Space and Its Notation §variances), $\gamma_{w}$ is the diagonal Gaussian measure on $\mathbb{R}^{m}$ and $\rho_{w}$ its density; for $m=1$ and $w=(v)$ this is the measure written $\gamma_{v}$ in Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity. For $q\in\mathbb{N}$ and $k\in[q]$, $\pi_{k}:\mathbb{R}^{q}\to\mathbb{R}$ is the $k$-th coordinate function. Lebesgue measure on $\mathcal{B}(\mathbb{R}^{q})$ is $\lambda_{q}$ of Euclidean Space and Lebesgue Measure: Standing Notation §measure, that is of Lebesgue Measure on $\mathbb{R}^n$, and $\lambda=\lambda_{1}$.

Step 0 (Preliminaries on a length bound). Let $m$ be any length bound for $\alpha$ (Finitely Supported Multi-Indices: Order, Factorial, Powers and Weighted Order §multi-indices). For $k\in[m]$ put $f_{k}=H^{c_{k}}_{\alpha_{k}}$. By Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity §polynomial with $v=c_{k}$, $f_{k}$ is of class $C^{2}$ on $\mathbb{R}^{1}$, there are real numbers $b^{(k)}_{0},\dots,b^{(k)}_{\alpha_{k}}$ with $b^{(k)}_{\alpha_{k}}=1$ and $f_{k}(t)=\sum_{i=0}^{\alpha_{k}}b^{(k)}_{i}t^{i}$ for all $t$, and there is $M_{k}\in\mathbb{R}$ with $|f_{k}(t)|\le M_{k}(1+|t|^{\alpha_{k}})$ for all $t$; taking $t=0$ gives $0\le M_{k}$. Let $h^{m}_{\alpha}:\mathbb{R}^{m}\to\mathbb{R}$, $h^{m}_{\alpha}(y)=\prod_{k=1}^{m}f_{k}(y_{k})$; for $m=n$ this is the function $h_{\alpha}$ of the statement. By The Cylindrical Hermite Polynomials of a Diagonal Gaussian Measure on a Hilbert Space §hermite, computed with the length bound $m$, $H_{\alpha}(x)=\prod_{k=1}^{m}f_{k}(x_{k})=h^{m}_{\alpha}(p_{m}(x))$ for every $x\in X$. Steps 1 to 4 below are carried out for an arbitrary length bound $m$; for $m=n$ they give claim 1.

Step 1 ($h^{m}_{\alpha}$ is of class $C^{2}$, and its partial derivatives). The sets $\mathbb{R}^{m}$ and $\mathbb{R}^{1}$ are open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. Each $\pi_{k}$ is smooth on $\mathbb{R}^{m}$ by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, hence of class $C^{2}$ by Smooth Map on a Euclidean Open Set. By claim 2 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$ (with the order there taken to be $2$, $U=\mathbb{R}^{m}$, $V=\mathbb{R}^{1}$, $F=\pi_{k}$, $G=f_{k}$) the function $g_{k}=f_{k}\circ\pi_{k}$, $g_{k}(y)=f_{k}(y_{k})$, is of class $C^{2}$ on $\mathbb{R}^{m}$. Being of class $C^{2}$, $\pi_{k}$ and $f_{k}$ are of class $C^{1}$ on $\mathbb{R}^{m}$ and on $\mathbb{R}^{1}$ respectively, by claim 2 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. For $i\in[m]$ and $y\in\mathbb{R}^{m}$, every difference quotient of $\pi_{k}$ at $y$ in the $i$-th variable equals $1$ if $i=k$ and $0$ if $i\ne k$, so by Partial Derivative on a Euclidean Open Set (any $\delta$ serves) and Uniqueness of the Partial Derivative on a Euclidean Open Set, $\partial_{i}\pi_{k}(y)$ is $1$ if $i=k$ and $0$ otherwise. Claim 1 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$, whose sum has the single term $l=1$ (claim 1 of Properties of Finite Sums), gives $\partial_{i}g_{k}(y)=\partial_{1}f_{k}(y_{k})\,\partial_{i}\pi_{k}(y)$, that is $\partial_{i}g_{i}(y)=f_{i}'(y_{i})$ and $\partial_{i}g_{k}(y)=0$ for $k\ne i$, with $f'=\partial_{1}f$ as in Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity. For $i,k\in[m]$ let $f^{\langle i\rangle}_{k}=f_{k}'$ if $k=i$ and $f^{\langle i\rangle}_{k}=f_{k}$ if $k\ne i$; thus the family $(f^{\langle i\rangle}_{k})_{k\in[m]}$ is $(f_{k})_{k\in[m]}$ with its $i$-th member replaced by its derivative. For $l\in[m]$ let $G_{l}:\mathbb{R}^{m}\to\mathbb{R}$, $G_{l}(y)=\prod_{k=1}^{l}g_{k}(y)=\prod_{k=1}^{l}f_{k}(y_{k})$; then $G_{m}=h^{m}_{\alpha}$, and by claim 1 of Properties of Finite Products, $G_{1}=g_{1}$ and $G_{l+1}=G_{l}\,g_{l+1}$ whenever $l+1\in[m]$. Let $A$ be the set of $l\in\mathbb{N}$ such that, if $l\le m$, then $G_{l}$ is of class $C^{2}$ on $\mathbb{R}^{m}$ and, for all $i\in[m]$ and $y\in\mathbb{R}^{m}$, $\partial_{i}G_{l}(y)$ exists and equals $\prod_{k=1}^{l}f^{\langle i\rangle}_{k}(y_{k})$ if $i\le l$ and $0$ if $l<i$. We have $1\in A$: $G_{1}=g_{1}$ is of class $C^{2}$, $\partial_{1}g_{1}(y)=f_{1}'(y_{1})=\prod_{k=1}^{1}f^{\langle 1\rangle}_{k}(y_{k})$ and $\partial_{i}g_{1}(y)=0$ for $1<i$. Let $l\in A$; if $m<l+1$ then $l+1\in A$ trivially, so let $l+1\le m$, whence $l<m$. By claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, $G_{l+1}=G_{l}\,g_{l+1}$ is of class $C^{2}$, and by the product rule of claim 1 there, $\partial_{i}G_{l+1}(y)=\partial_{i}G_{l}(y)\,f_{l+1}(y_{l+1})+G_{l}(y)\,\partial_{i}g_{l+1}(y)$. If $i\le l$, the second term vanishes since $i\ne l+1$, and the first equals $\bigl(\prod_{k=1}^{l}f^{\langle i\rangle}_{k}(y_{k})\bigr)f^{\langle i\rangle}_{l+1}(y_{l+1})=\prod_{k=1}^{l+1}f^{\langle i\rangle}_{k}(y_{k})$ by claim 1 of Properties of Finite Products, as $f^{\langle i\rangle}_{l+1}=f_{l+1}$. If $i=l+1$, the first term vanishes since $l<i$, and, as $f^{\langle i\rangle}_{k}=f_{k}$ for $k\le l$, the second equals $\bigl(\prod_{k=1}^{l}f^{\langle i\rangle}_{k}(y_{k})\bigr)f_{i}'(y_{i})=\prod_{k=1}^{l+1}f^{\langle i\rangle}_{k}(y_{k})$, again by claim 1 there. If $l+1<i$, both terms vanish. Hence $l+1\in A$, and $A=\mathbb{N}$ by Principle of Induction for the Natural Numbers. Taking $l=m$: $h^{m}_{\alpha}$ is of class $C^{2}$ on $\mathbb{R}^{m}$, and

$$\partial_{i}h^{m}_{\alpha}(y)=\prod_{k=1}^{m}f^{\langle i\rangle}_{k}(y_{k})\qquad(y\in\mathbb{R}^{m},\ i\in[m]).\tag{D}$$

Step 2 (Growth). Let $y\in\mathbb{R}^{m}$, $u=\lVert y\rVert\ge0$, and let $w=\max\{1,u\}$ (Maximum of Two Elements of a Totally Ordered Set), so that $w=1$ if $u\le1$ and $w=u$ if $1<u$; in either case $1\le w$ and $u\le w$. Powers with exponent $0$ equal $1$ (Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §integers), and powers with exponents in $\mathbb{N}$ are handled by Properties of Natural Number Powers in a Field. Let $k\in[m]$. By Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §coordinate, $|y_{k}|\le u\le w$, hence $|y_{k}|^{\alpha_{k}}\le u^{\alpha_{k}}\le w^{\alpha_{k}}$ by claim 5 of Properties of Natural Number Powers in a Field (trivially if $\alpha_{k}=0$); and $1\le w^{\alpha_{k}}$, trivially if $\alpha_{k}=0$ and, if $\alpha_{k}\ge1$, because $w=w^{1}\le w^{\alpha_{k}}$ by claims 1 and 7 there. As $0\le M_{k}$,

$$|f_{k}(y_{k})|\le M_{k}(1+u^{\alpha_{k}})\le 2M_{k}w^{\alpha_{k}}\qquad(k\in[m]).$$

For $l\in[m]$ write $s_{l}=\sum_{k=1}^{l}\alpha_{k}$, and let $A$ be the set of $l\in\mathbb{N}$ such that, if $l\le m$, then $\bigl|\prod_{k=1}^{l}f_{k}(y_{k})\bigr|\le2^{l}\bigl(\prod_{k=1}^{l}M_{k}\bigr)w^{s_{l}}$. By the display and claim 1 of Properties of Finite Products, claim 1 of Properties of Finite Sums and claim 1 of Properties of Natural Number Powers in a Field, $1\in A$. Let $l\in A$ with $l+1\le m$. By claim 1 of Properties of Finite Products and claim 4 of Properties of the Absolute Value in an Ordered Field, $\bigl|\prod_{k=1}^{l+1}f_{k}(y_{k})\bigr|=\bigl|\prod_{k=1}^{l}f_{k}(y_{k})\bigr|\,|f_{l+1}(y_{l+1})|$; both factors are bounded by the nonnegative numbers $2^{l}\bigl(\prod_{k=1}^{l}M_{k}\bigr)w^{s_{l}}$ and $2M_{l+1}w^{\alpha_{l+1}}$ (nonnegative by claim 5 of Properties of Finite Products and claim 5 of Properties of Natural Number Powers in a Field), so the product is at most $2^{l+1}\bigl(\prod_{k=1}^{l+1}M_{k}\bigr)w^{s_{l+1}}$, by claim 1 of Properties of Finite Products, claim 1 of Properties of Finite Sums ($s_{l+1}=s_{l}+\alpha_{l+1}$), and claims 1 and 6 of Properties of Natural Number Powers in a Field ($2^{l+1}=2^{l}\cdot2$ and $w^{s_{l+1}}=w^{s_{l}}w^{\alpha_{l+1}}$, trivially when an exponent is $0$). So $l+1\in A$ (trivially so if $m<l+1$), and $A=\mathbb{N}$ by Principle of Induction for the Natural Numbers. Taking $l=m$, with $s_{m}=|\alpha|$ (Finitely Supported Multi-Indices: Order, Factorial, Powers and Weighted Order §order) and $M=2^{m}\prod_{k=1}^{m}M_{k}$, we get $|h^{m}_{\alpha}(y)|\le Mw^{|\alpha|}$. Finally $w^{|\alpha|}\le1+u^{|\alpha|}$: if $u\le1$ then $w^{|\alpha|}=1$ (claim 2 of Properties of Natural Number Powers in a Field, or $|\alpha|=0$), and if $1<u$ then $w^{|\alpha|}=u^{|\alpha|}$. Hence

$$|h^{m}_{\alpha}(y)|\le M\bigl(1+\lVert y\rVert^{|\alpha|}\bigr)\qquad(y\in\mathbb{R}^{m}).$$

For $x\in X$, the coordinates of $0_{X}$ vanish by Elementary Identities in a Real Inner Product Space §zero, so $p_{m}(0_{X})$ is the origin of $\mathbb{R}^{m}$, and the coordinate-map clause Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity with $x'=0_{X}$ gives $\lVert p_{m}(x)\rVert=\lVert p_{m}(x)-p_{m}(0_{X})\rVert\le|x-0_{X}|=|x|$; hence $\lVert p_{m}(x)\rVert^{|\alpha|}\le|x|^{|\alpha|}$ by claim 5 of Properties of Natural Number Powers in a Field (trivially if $|\alpha|=0$), and $|H_{\alpha}(x)|=|h^{m}_{\alpha}(p_{m}(x))|\le M(1+\lVert p_{m}(x)\rVert^{|\alpha|})\le M(1+|x|^{|\alpha|})$.

Step 3 (Polynomial form). We first record a padding fact, used again in Step 8. Let $q,r\in\mathbb{N}_{0}$ with $q\le r$, and let $t_{0},\dots,t_{r}$ be real numbers with $t_{i}=0$ whenever $q<i\le r$. Then

$$\sum_{i=0}^{q}t_{i}=\sum_{i=0}^{r}t_{i}=\sum_{j=1}^{r+1}t_{j-1},\tag{P}$$

the sums from $0$ being those of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §integers. Indeed, if $1\le q<r$, Splitting a Finite Sum at an Index (with $q$ and $r-q$ in place of $m$ and $n$ there) writes $\sum_{i=1}^{r}t_{i}$ as $\sum_{i=1}^{q}t_{i}$ plus a sum of zeros, which is $0$ by claim 7 of Properties of Finite Sums; if $q=0<r$, $\sum_{i=1}^{r}t_{i}=0$ by that claim; and if $q=r$ there is nothing to show. This gives the first equality. For the second, if $r\ge1$, Splitting a Finite Sum at an Index (with $1$ and $r$ in place of $m$ and $n$) and claim 1 of Properties of Finite Sums give $\sum_{j=1}^{r+1}t_{j-1}=t_{0}+\sum_{j=1}^{r}t_{j}$, and if $r=0$ both sides equal $t_{0}$ by that claim.

Now let $r=|\alpha|$; by Finitely Supported Multi-Indices: Order, Factorial, Powers and Weighted Order §order and claim 6 of Properties of Finite Sums, $\alpha_{k}\le r$ for every $k\in[m]$. For $k\in[m]$ and $i\in\{0,\dots,r\}$ let $\tilde b^{(k)}_{i}=b^{(k)}_{i}$ if $i\le\alpha_{k}$ and $\tilde b^{(k)}_{i}=0$ if $\alpha_{k}<i$. Fix $y\in\mathbb{R}^{m}$. By (P) with $q=\alpha_{k}$, $f_{k}(y_{k})=\sum_{j=1}^{r+1}\tilde b^{(k)}_{j-1}y_{k}^{j-1}$ for every $k\in[m]$, so all $m$ factors of $h^{m}_{\alpha}(y)$ are sums over the same range $[r+1]$. Hence Generalized Distributivity: Expanding a Product of Finite Sums, applied in the field of real numbers with $m$ factors, $r+1$ summands in each, and components $\tilde b^{(k)}_{j-1}y_{k}^{j-1}$ ($k\in[m]$, $j\in[r+1]$), followed by claim 2 of Properties of Finite Products in each term, gives

$$h^{m}_{\alpha}(y)=\sum_{\iota\in[r+1]^{m}}a_{\iota}\prod_{k=1}^{m}y_{k}^{\iota(k)-1},\qquad a_{\iota}=\prod_{k=1}^{m}\tilde b^{(k)}_{\iota(k)-1}.$$

The index set $[r+1]^{m}$ is nonempty and finite, the coefficients $a_{\iota}$ do not depend on $y$, and for each $\iota$ the tuple $(\iota(1)-1,\dots,\iota(m)-1)$ is a multi-index of length $m$. Writing the sum over $[r+1]^{m}$ along a bijection from an initial segment onto $[r+1]^{m}$, as in Sum over a Finite Index Set, exhibits $h^{m}_{\alpha}$ as a finite linear combination of monomials, that is, a polynomial function on $\mathbb{R}^{m}$ in the sense of Polynomial Functions Are Dense in the Square-Integrable Functions of a Diagonal Gaussian Measure on Euclidean Space.

Step 4 (Borel measurability and integrability). By claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, $h^{m}_{\alpha}$ is continuous from $(\mathbb{R}^{m},d_{E})$ to $(\mathbb{R},d_{\mathbb{R}})$, hence measurable with respect to $\mathcal{B}(\mathbb{R}^{m})$ and $\mathcal{B}(\mathbb{R})$ by claims 3 and 2 of Borel Measurability and Bounded Integration on a Metric Space, the Borel $\sigma$-algebra of the metric space $(\mathbb{R}^{m},d_{E})$, generated by its open subsets, being $\mathcal{B}(\mathbb{R}^{m})$ by Euclidean Space and Lebesgue Measure: Standing Notation §borel; $p_{m}$ is Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity; so $H_{\alpha}=h^{m}_{\alpha}\circ p_{m}$ is Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space. Let $p\ge1$ be real. The truncation $c^{(m)}$ is a variance vector and $(p_{m})_{\#}\gamma_{c}=\gamma_{c^{(m)}}$, by Variance Sequences and Their Truncations §truncations and Diagonal Gaussian Measures on a Hilbert Space §measure. The function $|h^{m}_{\alpha}|^{p}$ is nonnegative and measurable by Power-Integrable Functions and the p-Seminorm §measurable-power, and $|H_{\alpha}|^{p}=|h^{m}_{\alpha}|^{p}\circ p_{m}$. By claim 2 of Image Measures, Measures with Densities, and Change of Variables (the push-forward being the image measure, Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward), Step 3 and Polynomial Functions Are Dense in the Square-Integrable Functions of a Diagonal Gaussian Measure on Euclidean Space §integrable with $r=p$,

$$\int_{X}|H_{\alpha}|^{p}\,d\gamma_{c}=\int_{\mathbb{R}^{m}}|h^{m}_{\alpha}|^{p}\,d\gamma_{c^{(m)}}<\infty .$$

So $H_{\alpha}$ is $p$-integrable in the sense of Power-Integrable Functions and the p-Seminorm §space, and its class lies in $L^{p}(\gamma_{c})$ by The Lebesgue Space of Power-Integrable Functions §space and Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §lebesgue. With $m=n$, Steps 0 to 4 prove claim 1.

Step 5 (Claim 2). By Step 1 and claim 2 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, $h_{\alpha}=h^{n}_{\alpha}$ is of class $C^{1}$ on $\mathbb{R}^{n}$. Let $x\in X$, $a=p_{n}(x)$ and $\xi=\sum_{i=1}^{n}\partial_{i}h_{\alpha}(a)\,e_{i}\in X$. By A Real-Valued C^1 Function is Differentiable at Every Point, $h_{\alpha}$ is differentiable at $a$ with derivative matrix the row with entries $\partial_{i}h_{\alpha}(a)$. By claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, the Euclidean norm of a point $(s)$ of $\mathbb{R}^{1}$ is the unique $r\ge0$ with $r^{2}=s^{2}$, which is $|s|$ by claim 1 of Properties of the Absolute Value in an Ordered Field; so differentiability at $a$ says that for every $\varepsilon>0$ there is $\delta>0$ with $|h_{\alpha}(a+\eta)-h_{\alpha}(a)-\sum_{i=1}^{n}\partial_{i}h_{\alpha}(a)\eta_{i}|\le\varepsilon\lVert\eta\rVert$ whenever $\eta\in\mathbb{R}^{n}$ and $0<\lVert\eta\rVert<\delta$, and the inequality is trivial for $\eta=0$. Given $\varepsilon$, take such $\delta$ and let $z\in X$ with $|z|<\delta$; put $\eta=p_{n}(z)$. By the symmetry and additivity of the inner product (Real Inner Product Space §inner-product), $p_{n}(x+z)=a+\eta$; by Step 2, $\lVert\eta\rVert\le|z|<\delta$; and by Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space §combinations and symmetry, $\langle\xi,z\rangle=\sum_{i=1}^{n}\partial_{i}h_{\alpha}(a)z_{i}$. Hence

$$\bigl|H_{\alpha}(x+z)-H_{\alpha}(x)-\langle\xi,z\rangle\bigr|=\Bigl|h_{\alpha}(a+\eta)-h_{\alpha}(a)-\sum_{i=1}^{n}\partial_{i}h_{\alpha}(a)\eta_{i}\Bigr|\le\varepsilon\lVert\eta\rVert\le\varepsilon|z| .$$

Since $X$ is open in itself (Real Hilbert Spaces: Standing Notation and Background §topology), $H_{\alpha}$ is differentiable at $x$ with gradient $\xi$ in the sense of Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §differentiable, so $DH_{\alpha}(x)=\xi$ by Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §gradient, and as $x$ is arbitrary $H_{\alpha}$ is differentiable on $X$ (Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §on-set). Let $k\in\mathbb{N}$. By Partial Derivatives along an Orthonormal Basis of a Function Differentiable on a Hilbert Space §partial and Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space §combinations, $\partial_{k}H_{\alpha}(x)=\langle\xi,e_{k}\rangle=\sum_{i=1}^{n}\partial_{i}h_{\alpha}(a)\langle e_{i},e_{k}\rangle$. The basis is orthonormal (Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §space, Orthonormal Basis of a Real Hilbert Space §basis, Orthogonality, Orthogonal Complement and Orthonormal Families in a Real Inner Product Space §orthonormal), so $\partial_{k}H_{\alpha}(x)=\partial_{k}h_{\alpha}(a)$ if $k\le n$ and $\partial_{k}H_{\alpha}(x)=0$ if $k>n$. If $k>n$ then $\alpha_{k}=0$, as $n$ is a length bound, and the claimed value is $0$. Let $k\le n$. By (D) with $m=n$ and $i=k$, $\partial_{k}H_{\alpha}(x)=\prod_{j=1}^{n}f^{\langle k\rangle}_{j}(x_{j})$, whose $k$-th factor is $f_{k}'(x_{k})$. If $\alpha_{k}=0$, then $f_{k}'(x_{k})=0$ by Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity §derivative, so $\partial_{k}H_{\alpha}(x)=0$ by claim 4 of Properties of Finite Products. Let $\alpha_{k}\ge1$ and $\beta=\alpha-\varepsilon_{k}$. By Finitely Supported Multi-Indices: Order, Factorial, Powers and Weighted Order §unit, $n$ is a length bound for $\beta$, $\beta_{k}=\alpha_{k}-1$ and $\beta_{j}=\alpha_{j}$ for $j\ne k$. For $j\in[n]$ put $s_{j}=H^{c_{j}}_{\beta_{j}}(x_{j})$, and $r_{j}=\alpha_{k}$ if $j=k$, $r_{j}=1$ if $j\ne k$. Then $f^{\langle k\rangle}_{j}(x_{j})=r_{j}s_{j}$ for every $j\in[n]$: for $j=k$ because $f_{k}'(x_{k})=\alpha_{k}H^{c_{k}}_{\alpha_{k}-1}(x_{k})$ by Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity §derivative, and for $j\ne k$ because $f^{\langle k\rangle}_{j}=f_{j}=H^{c_{j}}_{\beta_{j}}$. By claims 2 and 3 of Properties of Finite Products and The Cylindrical Hermite Polynomials of a Diagonal Gaussian Measure on a Hilbert Space §hermite (computed with the length bound $n$),

$$\partial_{k}H_{\alpha}(x)=\Bigl(\prod_{j=1}^{n}r_{j}\Bigr)\Bigl(\prod_{j=1}^{n}s_{j}\Bigr)=\alpha_{k}H_{\beta}(x)=\alpha_{k}H_{\alpha-\varepsilon_{k}}(x).$$

This proves claim 2.

Step 6 (A product formula for diagonal Gaussian integrals). Let $m\in\mathbb{N}$, let $w=(w_{1},\dots,w_{m})$ be a variance vector, and let $\varphi_{1},\dots,\varphi_{m}:\mathbb{R}\to\mathbb{R}$ be Borel with $\varphi_{k}$ integrable with respect to $\gamma_{(w_{k})}$. We show that $\Phi(y)=\prod_{k=1}^{m}\varphi_{k}(y_{k})$ is Borel on $\mathbb{R}^{m}$ and integrable with respect to $\gamma_{w}$, with

$$\int_{\mathbb{R}^{m}}\Phi\,d\gamma_{w}=\prod_{k=1}^{m}\int_{\mathbb{R}}\varphi_{k}\,d\gamma_{(w_{k})}.$$

Each $\pi_{k}$ is measurable for $\mathcal{B}(\mathbb{R}^{m})$ by claims 1 and 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, so each $\varphi_{k}\circ\pi_{k}$ is Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space, and $\Phi$ is Borel by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. By The Diagonal Gaussian Density on Euclidean Space and Its Notation §scaling and The Diagonal Gaussian Density on Euclidean Space and Its Notation §density (in dimensions $m$ and $1$, with the positive constant $\kappa$ and the square root fixed there), $\rho_{w}(y)=\exp\bigl(\sum_{k=1}^{m}(-\tfrac{y_{k}^{2}}{2w_{k}}-\log(\kappa\sqrt{w_{k}}))\bigr)$ and $\rho_{(w_{k})}(s)=\exp(-\tfrac{s^{2}}{2w_{k}}-\log(\kappa\sqrt{w_{k}}))$, so $\rho_{w}(y)=\prod_{k=1}^{m}\rho_{(w_{k})}(y_{k})$. Indeed, writing $\tau_{k}=-\tfrac{y_{k}^{2}}{2w_{k}}-\log(\kappa\sqrt{w_{k}})$, let $A'$ be the set of $l\in\mathbb{N}$ such that, if $l\le m$, then $\exp\bigl(\sum_{k=1}^{l}\tau_{k}\bigr)=\prod_{k=1}^{l}\exp(\tau_{k})$; then $1\in A'$ by claim 1 of Properties of Finite Sums and claim 1 of Properties of Finite Products, and if $l\in A'$ and $l+1\le m$ then, by the recursions in those claims and claim 1 of Basic Properties of the Exponential Function, $\exp\bigl(\sum_{k=1}^{l+1}\tau_{k}\bigr)=\exp\bigl(\sum_{k=1}^{l}\tau_{k}\bigr)\exp(\tau_{l+1})=\prod_{k=1}^{l+1}\exp(\tau_{k})$; so $A'=\mathbb{N}$ by Principle of Induction for the Natural Numbers, and $l=m$ gives the formula. These densities are positive and Borel by The Diagonal Gaussian Density on Euclidean Space: Regularity, Gradient, Normalization, Second Moments and Exponential Moments of Diagonal Quadratic Forms §regularity, and by Diagonal Gaussian Measures on Euclidean Space §measure and claim 3 of Image Measures, Measures with Densities, and Change of Variables, a Borel $f$ on $\mathbb{R}^{q}$ is integrable with respect to $\gamma_{u}$ if and only if $f\rho_{u}$ is integrable with respect to $\lambda_{q}$, with equal integrals, and $\int|f|\,d\gamma_{u}=\int|f|\rho_{u}\,d\lambda_{q}$ in $[0,\infty]$. Put $\psi_{k}=\varphi_{k}\rho_{(w_{k})}$, which is Borel and $\lambda$-integrable with $a_{k}=\int\psi_{k}\,d\lambda=\int\varphi_{k}\,d\gamma_{(w_{k})}$ and $b_{k}=\int|\psi_{k}|\,d\lambda=\int|\varphi_{k}|\,d\gamma_{(w_{k})}<\infty$. For $l\in[m]$ let $\Psi_{l}(y)=\prod_{k=1}^{l}\psi_{k}(y_{k})$ on $\mathbb{R}^{l}$, a Borel function as above. By claim 2 of Properties of Finite Products, $\Phi(y)\rho_{w}(y)=\prod_{k=1}^{m}\varphi_{k}(y_{k})\rho_{(w_{k})}(y_{k})=\Psi_{m}(y)$, so $\Phi\rho_{w}=\Psi_{m}$. Let $A$ be the set of $l\in\mathbb{N}$ such that, if $l\le m$, then $\Psi_{l}$ is $\lambda_{l}$-integrable with $\int|\Psi_{l}|\,d\lambda_{l}=\prod_{k\le l}b_{k}$ and $\int\Psi_{l}\,d\lambda_{l}=\prod_{k\le l}a_{k}$. We have $1\in A$, since $\lambda_{1}=\lambda$ (Lebesgue Measure on $\mathbb{R}^n$) and $\Psi_{1}=\psi_{1}$ by claim 1 of Properties of Finite Products. Let $l'\in A$ and $l=l'+1$; if $m<l$ then $l\in A$ trivially, so let $l\le m$; thus $l\ge2$, $l-1=l'\le m$, and the assertions for $l-1$ (the induction hypothesis) are available. Identifying $\mathbb{R}^{l}$ with $\mathbb{R}^{l-1}\times\mathbb{R}$ as in Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$, $\lambda_{l}$ is the product measure $\lambda_{l-1}\otimes\lambda$ of Existence and Uniqueness of the Product Measure on $\mathcal{B}_{l-1}\otimes\mathcal{B}(\mathbb{R})=\mathcal{B}(\mathbb{R}^{l})$, by Lebesgue Measure on $\mathbb{R}^n$, claim 1 of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$ and claim 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, and both factors are $\sigma$-finite by claim 1 of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$. Writing $y=(y',s)$, $\Psi_{l}(y',s)=\Psi_{l-1}(y')\psi_{l}(s)$ by claim 1 of Properties of Finite Products. The Tonelli part of Tonelli and Fubini Theorems, applied to $|\Psi_{l}|$, and Linearity and Monotonicity of the Lebesgue Integral §nonnegative (constant factors) give $\int|\Psi_{l}|\,d\lambda_{l}=\int|\Psi_{l-1}(y')|\,b_{l}\,\lambda_{l-1}(dy')=\prod_{k\le l}b_{k}<\infty$, so $\Psi_{l}$ is integrable (Measure Spaces and the Lebesgue Integral: Standing Notation §integral). For every $y'$ the section $s\mapsto\Psi_{l-1}(y')\psi_{l}(s)$ is integrable with integral $\Psi_{l-1}(y')a_{l}$, by Linearity and Monotonicity of the Lebesgue Integral §integrable; so the Fubini part of Tonelli and Fubini Theorems provides a $\lambda_{l-1}$-null set $N$ such that the function equal to $a_{l}\Psi_{l-1}$ off $N$ and to $0$ on $N$ is integrable with integral $\int\Psi_{l}\,d\lambda_{l}$. This function agrees with $a_{l}\Psi_{l-1}$ off the null set $N$, so by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison, Linearity and Monotonicity of the Lebesgue Integral §integrable and the induction hypothesis, $\int\Psi_{l}\,d\lambda_{l}=a_{l}\prod_{k<l}a_{k}=\prod_{k\le l}a_{k}$. Hence $l\in A$, and $A=\mathbb{N}$ by Principle of Induction for the Natural Numbers. With $l=m$: $\Phi\rho_{w}=\Psi_{m}$ is $\lambda_{m}$-integrable, so $\Phi$ is $\gamma_{w}$-integrable with $\int\Phi\,d\gamma_{w}=\int\Psi_{m}\,d\lambda_{m}=\prod_{k=1}^{m}a_{k}$, as asserted.

Step 7 (Claim 3). Let $m$ be the larger of $n$ and a length bound for $\beta$; it is a length bound for both $\alpha$ and $\beta$. By Step 4 (with $p=2$) the classes of $H_{\alpha}$ and $H_{\beta}$ lie in $L^{2}(\gamma_{c})$, so by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product, $H_{\alpha}H_{\beta}$ is integrable and $\langle H_{\alpha},H_{\beta}\rangle_{L^{2}(\gamma_{c})}=\int_{X}H_{\alpha}H_{\beta}\,d\gamma_{c}$. By Step 0, $H_{\alpha}H_{\beta}=\Phi\circ p_{m}$ with $\Phi(y)=h^{m}_{\alpha}(y)h^{m}_{\beta}(y)=\prod_{k=1}^{m}\varphi_{k}(y_{k})$ and $\varphi_{k}=H^{c_{k}}_{\alpha_{k}}H^{c_{k}}_{\beta_{k}}$, the last equality by claim 2 of Properties of Finite Products. By Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity §orthogonality with $v=c_{k}>0$, each $\varphi_{k}$ is Borel and integrable with respect to $\gamma_{(c_{k})}$, with $\int\varphi_{k}\,d\gamma_{(c_{k})}=\alpha_{k}!\,c_{k}^{\alpha_{k}}$ if $\alpha_{k}=\beta_{k}$ and $=0$ otherwise. By Step 6 with $w=c^{(m)}$, $\Phi$ is Borel and $\gamma_{c^{(m)}}$-integrable, and claim 2 of Image Measures, Measures with Densities, and Change of Variables with $(p_{m})_{\#}\gamma_{c}=\gamma_{c^{(m)}}$ (Step 4) gives

$$\int_{X}H_{\alpha}H_{\beta}\,d\gamma_{c}=\int_{\mathbb{R}^{m}}\Phi\,d\gamma_{c^{(m)}}=\prod_{k=1}^{m}\int_{\mathbb{R}}\varphi_{k}\,d\gamma_{(c_{k})}.$$

If $\alpha=\beta$, the right side is $\prod_{k=1}^{m}\alpha_{k}!\,c_{k}^{\alpha_{k}}=\bigl(\prod_{k=1}^{m}\alpha_{k}!\bigr)\bigl(\prod_{k=1}^{m}c_{k}^{\alpha_{k}}\bigr)=\alpha!\,c^{\alpha}$ by claim 2 of Properties of Finite Products and Finitely Supported Multi-Indices: Order, Factorial, Powers and Weighted Order §order (length bound $m$). If $\alpha\ne\beta$, there is $k$ with $\alpha_{k}\ne\beta_{k}$; necessarily $k\le m$, since both vanish beyond $m$; the $k$-th factor is $0$, and so is the product, by claim 4 of Properties of Finite Products. This proves claim 3.

Step 8 (Claim 4). Fix $v\ge0$. For $q\in\mathbb{N}_{0}$ let $(\mathrm{M}_{q})$ be the assertion that there are real numbers $a_{q,0},\dots,a_{q,q}$ with $t^{q}=\sum_{j=0}^{q}a_{q,j}H^{v}_{j}(t)$ for every $t\in\mathbb{R}$. Let $A$ be the set of $q\in\mathbb{N}$ such that $(\mathrm{M}_{q'})$ holds for every $q'\in\mathbb{N}_{0}$ with $q'\le q-1$. Since $t^{0}=1=H^{v}_{0}(t)$ by Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity §low-orders, $(\mathrm{M}_{0})$ holds, so $1\in A$. Let $q\in A$; we prove $(\mathrm{M}_{q})$, so that $q+1\in A$. By Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity §polynomial there are real $b_{0},\dots,b_{q}$ with $b_{q}=1$ and $H^{v}_{q}(t)=\sum_{i=0}^{q}b_{i}t^{i}$ for all $t$. For $i\in\{0,\dots,q-1\}$, $(\mathrm{M}_{i})$ holds as $q\in A$; put $\tilde a_{i,j}=a_{i,j}$ for $j\le i$ and $\tilde a_{i,j}=0$ for $i<j\le q-1$. Fix $t\in\mathbb{R}$. By (P) of Step 3 (with $i$ and $q-1$ in place of $q$ and $r$ there), $t^{i}=\sum_{j=1}^{q}\tilde a_{i,j-1}H^{v}_{j-1}(t)$ for each such $i$; by (P) with $q$ in place of both $q$ and $r$, claim 1 of Properties of Finite Sums and $b_{q}=1$,

$$H^{v}_{q}(t)=\sum_{i=1}^{q+1}b_{i-1}t^{i-1}=\sum_{i=1}^{q}b_{i-1}t^{i-1}+t^{q}.$$

By claim 3 of Properties of Finite Sums and Interchange of a Finite Double Sum, and claim 3 once more,

$$\sum_{i=1}^{q}b_{i-1}t^{i-1}=\sum_{i=1}^{q}\sum_{j=1}^{q}b_{i-1}\tilde a_{i-1,j-1}H^{v}_{j-1}(t)=\sum_{j=1}^{q}e_{j}H^{v}_{j-1}(t),\qquad e_{j}=\sum_{i=1}^{q}b_{i-1}\tilde a_{i-1,j-1}.$$

Put $a_{q,j}=-e_{j+1}$ for $j\le q-1$ and $a_{q,q}=1$; these do not depend on $t$. By the two displays, claim 3 of Properties of Finite Sums (with the scalar $-1$), claim 1 there and (P),

$$t^{q}=H^{v}_{q}(t)-\sum_{j=1}^{q}e_{j}H^{v}_{j-1}(t)=\sum_{j=1}^{q+1}a_{q,j-1}H^{v}_{j-1}(t)=\sum_{j=0}^{q}a_{q,j}H^{v}_{j}(t).$$

This is $(\mathrm{M}_{q})$, so $q+1\in A$, and $A=\mathbb{N}$ by Principle of Induction for the Natural Numbers. As every $q\in\mathbb{N}_{0}$ satisfies $q\le(q+1)-1$ with $q+1\in A$, $(\mathrm{M}_{q})$ holds for every $q\in\mathbb{N}_{0}$. Now apply this with $v=c_{k}$ and $q=\alpha_{k}$ for each $k\in[n]$, writing $a^{(k)}_{j}$ ($0\le j\le\alpha_{k}$) for the coefficients. Let $r=|\alpha|$, so that $\alpha_{k}\le r$ for $k\in[n]$ (Step 3), and put $\tilde a^{(k)}_{j}=a^{(k)}_{j}$ for $j\le\alpha_{k}$ and $\tilde a^{(k)}_{j}=0$ for $\alpha_{k}<j\le r$. Let $x\in X$. By (P) with $q=\alpha_{k}$, $x_{k}^{\alpha_{k}}=\sum_{j=1}^{r+1}\tilde a^{(k)}_{j-1}H^{c_{k}}_{j-1}(x_{k})$ for every $k\in[n]$, all $n$ sums now running over the same range $[r+1]$; so Generalized Distributivity: Expanding a Product of Finite Sums (in the field of real numbers, with $n$ factors and $r+1$ summands in each) and claim 2 of Properties of Finite Products give

$$\prod_{k=1}^{n}x_{k}^{\alpha_{k}}=\sum_{\iota\in[r+1]^{n}}d_{\iota}\prod_{k=1}^{n}H^{c_{k}}_{\iota(k)-1}(x_{k}),\qquad d_{\iota}=\prod_{k=1}^{n}\tilde a^{(k)}_{\iota(k)-1}.$$

Let $T$ be the set of $\iota\in[r+1]^{n}$ with $\iota(k)-1\le\alpha_{k}$ for every $k\in[n]$; it contains the constant tuple with value $1$. For $\iota\notin T$ some factor of $d_{\iota}$ is $0$, so $d_{\iota}=0$ by claim 4 of Properties of Finite Products, and by claim 3 of Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus the sum may be taken over $T$ instead of $[r+1]^{n}$. For $\iota\in T$ let $\beta^{\iota}\in\mathcal{A}$ have terms $\iota(1)-1,\dots,\iota(n)-1$ followed by zeros; $n$ is a length bound for it, $\beta^{\iota}_{k}\le\alpha_{k}$ for every $k\in\mathbb{N}$ (for $k>n$ both are $0$, as $n$ is a length bound for $\alpha$), $|\beta^{\iota}|=\sum_{k=1}^{n}(\iota(k)-1)\le\sum_{k=1}^{n}\alpha_{k}=|\alpha|$ by Finitely Supported Multi-Indices: Order, Factorial, Powers and Weighted Order §order and claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers, and $\prod_{k=1}^{n}H^{c_{k}}_{\iota(k)-1}(x_{k})=H_{\beta^{\iota}}(x)$ by The Cylindrical Hermite Polynomials of a Diagonal Gaussian Measure on a Hilbert Space §hermite. The coefficients $d_{\iota}$ do not depend on $x$, so, writing the sum over the nonempty finite set $T$ along a bijection from an initial segment onto $T$ as in Sum over a Finite Index Set, the function $x\mapsto\prod_{k=1}^{n}x_{k}^{\alpha_{k}}$ is the finite linear combination $\sum_{\iota\in T}d_{\iota}H_{\beta^{\iota}}$, which proves claim 4.