LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Gaussian orthogonality and the monomial expansion of the Hermite polynomials

Statement

Assume AC, and let $Z \sim N (0, 1)$ (Standard normal and normal laws). For the monic Hermite polynomials $H_{m}$ of The monic probabilists' Hermite polynomials:

(i) $E [H_{m} (Z)] = 0$ for every $m \geq 1$ , and $E [H_{m} (Z) H_{n} (Z)] = m! δ_{m n}$ for all $m, n \geq 0$ ;

(ii) for every $m \geq 0$ , $x^{m} = \sum_{j = 0}^{⌊ m / 2 ⌋} c_{m, j} H_{m - 2 j} (x)$ with $c_{m, 0} = 1$ and $c_{m, j} = \frac{m!}{2^{j} j! (m - 2 j)!} = (\binom{m}{2 j}) (2 j - 1)!! \in Z (0 \leq j \leq ⌊ m / 2 ⌋);$ equivalently, the monomials and the Hermite polynomials are related by a unitriangular change of basis in each finite degree;

(iii) consequently, for any $N \geq 1$ and any $m_{1}, \dots, m_{N} \in N$ , the mixed monomial $\prod_{k = 1}^{N} x_{k}^{m_{k}}$ is a $Z$ -linear combination of products $\prod_{k} H_{m_{k}^{'}} (x_{k})$ with $m_{k}^{'} \leq m_{k}$ and $m_{k}^{'} \equiv m_{k} (m o d 2)$ , the coefficient of $\prod_{k} H_{m_{k}}$ being $1$ .

Facts & Assumptions

Given: AC; a random variable $Z \sim N (0, 1)$ ; the polynomials $H_{m}$ defined by $H_{0} = 1$ , $H_{1} (x) = x$ and $x H_{m} = H_{m + 1} + m H_{m - 1}$ (The monic probabilists' Hermite polynomials); $φ_{Z} (t) = E [e^{i t Z}]$ is the characteristic function of $Z$ .

[F1]

For $Z \sim N (0, 1)$ and every real $t$ , $φ_{Z} (t) = e^{- t^{2} / 2}$ (Characteristic function of a normal law).

[F2]

If $E ∣ Y ∣^{k} < \infty$ then $φ_{Y} \in C^{k} (R)$ with $φ_{Y}^{(j)} (t) = E [(i Y)^{j} e^{i t Y}]$ for $0 \leq j \leq k$ ; in particular $φ_{Y}^{(j)} (0) = i^{j} E [Y^{j}]$ (Moments give derivatives of the characteristic function).

[F3]

For $Z \sim N (0, 1)$ , $E ∣ Z ∣^{2 m} = (2 m - 1)!!$ for every $m \geq 1$ (Gaussian even moments for Brownian increments); and $E ∣ X Y ∣ \leq (E X^{2})^{1 / 2} (E Y^{2})^{1 / 2}$ for square-integrable $X, Y$ (Cauchy-Schwarz for random variables).

[F4]

Derivative rules: if $g$ is differentiable then $t \mapsto g (- t)$ has derivative $- g^{'} (- t)$ , and $(f g)^{'} = f^{'} g + f g^{'}$ , $(α f)^{'} = α f^{'}$ (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[F5]

Factorials: $m!$ is the product of $1, \dots, m$ with $0! = 1$ , and $(\binom{m}{2 j}) = \frac{m!}{(2 j)! (m - 2 j)!}$ (The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ , $(\binom{n}{k}) k! (n - k)! = n!$ for $k \leq n$ ; hence $(\binom{n}{k}) k! = n^{\underline{k}}$ , the quotient $n! / (k! (n - k)!)$ is a natural number, and $(\binom{n}{k}) = (\binom{n}{n - k})$ ).

[F6]

Expectations of integrable variables are linear, monotone for real variables, and satisfy $∣ E U ∣ \leq E ∣ U ∣$ (Linearity, monotonicity, and the modulus bound for expectation).

Proof

technique · direct

1.1givenF1F2F3F4algebra

Vanishing of odd moments: by [F1] the function $φ_{Z} (t) = e^{- t^{2} / 2}$ is even, and by induction with [F4] each derivative $φ_{Z}^{(j)}$ has parity $(- 1)^{j}$ (differentiating flips parity); hence $φ_{Z}^{(2 k + 1)}$ is odd and therefore $φ_{Z}^{(2 k + 1)} (0) = 0$ for every $k \geq 0$ . All absolute moments of $Z$ are finite, since by [F3] $E ∣ Z ∣^{2 m} = (2 m - 1)!!$ and [F3] gives $E ∣ Z ∣^{2 m + 1} \leq (E ∣ Z ∣^{4 m + 2})^{1 / 2} < \infty$ ; so [F2] applies to every order and gives $E [Z^{2 k + 1}] = i^{- (2 k + 1)} φ_{Z}^{(2 k + 1)} (0) = 0$ .

1.2givenF4algebra

Derivative relation: $H_{m}^{'} = m H_{m - 1}$ for every $m \geq 1$ . This holds for $m = 1$ , since $H_{1}^{'} = 1 = H_{0}$ , and for $m = 2$ , since $H_{2}^{'} = 2 x = 2 H_{1}$ ; for $m \geq 2$ , if it holds for all indices up to $m$ , then differentiating $H_{m + 1} = x H_{m} - m H_{m - 1}$ with [F4] gives $H_{m + 1}^{'} = H_{m} + x H_{m}^{'} - m H_{m - 1}^{'} = H_{m} + m x H_{m - 1} - m (m - 1) H_{m - 2}$ , and substituting $x H_{m - 1} = H_{m} + (m - 1) H_{m - 2}$ yields $H_{m + 1}^{'} = (m + 1) H_{m}$ .

1.3givenF5algebra

Monomial expansion: every $m \geq 0$ admits the expansion $x^{m} = \sum_{j = 0}^{⌊ m / 2 ⌋} c_{m, j} H_{m - 2 j}$ with $c_{m, j} = \frac{m!}{2^{j} j! (m - 2 j)!}$ . Indeed $x^{0} = H_{0}$ gives $m = 0$ ; if the expansion holds for $m - 1 \geq 0$ , then multiplying by $x$ and using $x H_{l} = H_{l + 1} + l H_{l - 1}$ gives coefficient of $H_{m - 2 j}$ equal to $c_{m - 1, j} + (m + 1 - 2 j) c_{m - 1, j - 1}$ (with $c_{r, l} : = 0$ whenever $l < 0$ or $2 l > r$ , and $x H_{0} = H_{1}$ supplying the boundary case), which equals $c_{m, j}$ : for $j \geq 1$ the common denominator $2^{j} j! (m - 2 j)!$ turns it into $\frac{(m - 1)! (m - 2 j) + 2 j (m - 1)!}{2^{j} j! (m - 2 j)!} = \frac{m!}{2^{j} j! (m - 2 j)!}$ (using [F5]), and for $j = 0$ it gives $c_{m - 1, 0} = 1 = c_{m, 0}$ . By [F5], $c_{m, j} = (\binom{m}{2 j}) (2 j - 1)!!$ is an integer and $c_{m, 0} = 1$ . Moreover the monicity and degree $\deg H_{l} = l$ make the matrix of coefficients of $x^{0}, \dots, x^{m}$ in the basis $H_{0}, \dots, H_{m}$ unitriangular with diagonal entries $c_{m, 0} = 1$ , so the expansion is the unique one and defines an invertible unitriangular change of basis in each finite degree.

2.1givenF1F3step 1.1algebraF6

Stein identity: for every real polynomial $p$ , $E [Z p (Z)] = E [p^{'} (Z)]$ . Write $p (x) = \sum_{j = 0}^{d} a_{j} x^{j}$ ; then $E [Z p (Z)] = \sum_{j = 0}^{d} a_{j} E [Z^{j + 1}]$ and $E [p^{'} (Z)] = \sum_{j = 1}^{d} j a_{j} E [Z^{j - 1}]$ , both finite sums. The constant term contributes $a_{0} E [Z] = 0$ on the left and nothing on the right, and for every $j \geq 1$ one has $E [Z^{j + 1}] = j E [Z^{j - 1}]$ : when $j$ is even both sides vanish by step 1.1; when $j$ is odd, [F3] gives $E [Z^{j + 1}] = (j)!!$ and $j E [Z^{j - 1}] = j (j - 2)!! = j!!$ ; here $(- 1)!! : = 1$ covers $j = 1$ , where both sides are $E [Z^{2}] = 1$ . Hence the two sums are equal.

2.2givenstep 1.3algebra

Multivariate expansion: let $N \geq 1$ and $m_{1}, \dots, m_{N} \geq 0$ . Expanding each factor by step 1.3 and multiplying out, $\prod_{k} x_{k}^{m_{k}} = \sum_{j_{1}, \dots, j_{N}} (\prod_{k} c_{m_{k}, j_{k}}) \prod_{k} H_{m_{k} - 2 j_{k}} (x_{k})$ ; each index $m_{k}^{'} : = m_{k} - 2 j_{k}$ satisfies $m_{k}^{'} \leq m_{k}$ and $m_{k}^{'} \equiv m_{k} (m o d 2)$ , the coefficients are integers by step 1.3, and the single tuple $(j_{1}, \dots, j_{N}) = (0, \dots, 0)$ contributes $\prod_{k} H_{m_{k}}$ with coefficient $1$ .

3.1givenstep 1.1step 1.2step 2.1algebraF6

Zero means: $E [H_{0} (Z)] = 1$ and $E [H_{m} (Z)] = 0$ for every $m \geq 1$ , by induction on $m$ : the case $m = 1$ is $E [Z] = 0$ from step 1.1, and for $m \geq 1$ the recurrence and step 2.1 give $E [H_{m + 1} (Z)] = E [Z H_{m} (Z)] - m E [H_{m - 1} (Z)] = E [H_{m}^{'} (Z)] - m E [H_{m - 1} (Z)] = m E [H_{m - 1} (Z)] - m E [H_{m - 1} (Z)] = 0$ , where step 1.2 identifies $H_{m}^{'} = m H_{m - 1}$ and the induction hypothesis handles $E [H_{m - 1}]$ .

4.1givenstep 1.2step 2.1step 3.1algebraF6

Orthogonality: put $a (M, n) : = E [H_{M} (Z) H_{n} (Z)]$ for $M, n \geq 0$ . We show $a (M, n) = n! δ_{M n}$ . First $a (0, n) = E [H_{n} (Z)] = δ_{0 n}$ by step 3.1. For $M = 1$ and $n \geq 1$ , step 2.1 gives $a (1, n) = E [Z H_{n}] = E [H_{n}'] = n a (0, n - 1)$ . For $M \geq 2$ and $n \geq 1$ , the recurrence $H_{M} = x H_{M - 1} - (M - 1) H_{M - 2}$ , the Stein identity of step 2.1 applied to $p = H_{M - 1} H_{n}$ and the derivative relation of step 1.2 give $a (M, n) = E [Z H_{M - 1} H_{n}] - (M - 1) a (M - 2, n) = E [(H_{M - 1} H_{n})^{'}] - (M - 1) a (M - 2, n) = (M - 1) a (M - 2, n) + n a (M - 1, n - 1) - (M - 1) a (M - 2, n) = n a (M - 1, n - 1)$ , while for $n = 0$ and $M \geq 1$ one has $a (M, 0) = E [H_{M} (Z)] = 0$ by step 3.1. If $k \leq M$ and $k \leq n$ , iteration gives $a (M, n) = n (n - 1) \dots (n - k + 1) a (M - k, n - k)$ ; taking $k = n$ when $n \leq M$ yields $a (M, n) = n! a (M - n, 0)$ , which is $0$ for $M > n$ and is $n! a (0, 0) = n!$ for $M = n$ by step 3.1, while taking $k = M$ when $M < n$ yields $a (M, n) = n (n - 1) \dots (n - M + 1) a (0, n - M) = 0$ because $n - M \geq 1$ . Hence $a (M, n) = n! δ_{M n}$ , which is claim (i) together with step 3.1.

5.1givenstep 1.3step 2.2step 3.1step 4.1∎

Conclusion: step 3.1 and step 4.1 prove (i); step 1.3 proves (ii) (including the unitriangularity clause); step 2.2 proves (iii). No step used anything beyond the published derivative, moment and characteristic-function facts listed above.

Depends on

Used by

Kerov's central limit theorem for normalized cycle characters Theorem

Dependency tree · two levels

88 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Vladimir Ivanov and Grigori Olshanski, Kerov's central limit theorem for the Plancherel measure on Young diagrams, arXiv:math/0304010; survey-paper version in Symmetric Functions 2001, NATO Science Series II 74 (2002), 93-151 (standard reference, not scraped)