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For an elementary fact like this, which may have been reinvented a thousand times, it is hard to find the first paper where this appeared. However, let me give some missing context. There is a whole industry in constructive quantum field theory and statistical mechanics about related "smart" interpolation formulas or Taylor formulas with integral remainders. These are used to perform so-called cluster expansions. For the OP's identity, there is no loss of generality in taking $u=(0,0,\ldots,0)$ and $v=(1,1,\ldots,1)$. In this case, via Möbius inversion in the Boolean lattice, the formula comes from the following identity.

Let $L$ be a finite set. Let $f:\mathbb{R}^L\rightarrow \mathbb{R}$, $\mathbf{x}=(x_{\ell})_{\ell\in L}\mapsto f(\mathbf{x})$ be a sufficiently smooth function, and let $\mathbf{1}=(1,\ldots,1)\in\mathbb{R}^L$, then $$ f(\mathbf{1})=\sum_{A\subseteq L}\int_{[0,1]^A}d\mathbf{h} \left[\left(\prod_{\ell\in A}\frac{\partial}{\partial x_{\ell}}\right)f\right](\psi_A(\mathbf{h})) $$ where $\psi_A(\mathbf{h})$ is the element $\mathbf{x}=(x_{\ell})_{\ell\in L}$ of $\mathbb{R}^L$ defined from the element $\mathbf{h}=(h_{\ell})_{\ell\in A}$ in $[0,1]^A$ by the rule: $x_{\ell}=0$ if $\ell\notin A$ and $x_{\ell}=h_{\ell}$ if $\ell\in A$. Of course one needs to 1) apply this to all $L$'s which are subsets of $[p]$, 2) use MoebiusMöbius inversion in the Boolean lattice, and 3) specialize to $L=[p]$, and this gives the OP's identity.

The above formula is the most naive one of its kind used to do a "pair of cubes" cluster expansion. See formula III.1 in the article

A. Abdesselam and V. Rivasseau, "Trees, forests and jungles: a botanical garden for cluster expansions".

It is also explained in words on page 115 of the book

V. Rivasseau, "From Perturbative to Constructive Renormalization".

Now the formula is a particular case of a much more powerful one, namely, Lemma 1 in

A. Abdesselam and V. Rivasseau, "An explicit large versus small field multiscale cluster expansion",

where one sums over "allowed" sequences $(\ell_1,\ldots,\ell_k)$ of arbitrary length of elements of $L$, instead of subsets of $L$. The notion of allowed is based on an arbitrary stopping rule. The above identity corresponds to "allowed"$=$"without repeats", or the stopping rule that one should not tack on an $\ell$ at the end of a sequence where it already appeared. By playing with this kind of choice of stopping rule one can use Lemma 1 of my article with Rivasseau, to prove the Hermite-Genocchi formula, the anisotropic Taylor formula by Hairer in Appendix A of "A theory of regularity structures" and many other things. When $f$ is the exponential of a linear form for instance, one can obtain various algebraic identities as in the MO posts

rational function identity

Identity involving sum over permutations

I forgot to mention, one can use Lemma 1 to derive the Taylor formula from calculus 1. This corresponds to $L$ having one element and defining allowed sequences as the ones of length at most $n$. See

https://math.stackexchange.com/questions/3753212/is-there-any-geometrical-intuition-for-the-factorials-in-taylor-expansions/3753600#3753600

For an elementary fact like this, which may have been reinvented a thousand times, it is hard to find the first paper where this appeared. However, let me give some missing context. There is a whole industry in constructive quantum field theory and statistical mechanics about related "smart" interpolation formulas or Taylor formulas with integral remainders. These are used to perform so-called cluster expansions. For the OP's identity, there is no loss of generality in taking $u=(0,0,\ldots,0)$ and $v=(1,1,\ldots,1)$. In this case, via Möbius inversion in the Boolean lattice, the formula comes from the following identity.

Let $L$ be a finite set. Let $f:\mathbb{R}^L\rightarrow \mathbb{R}$, $\mathbf{x}=(x_{\ell})_{\ell\in L}\mapsto f(\mathbf{x})$ be a sufficiently smooth function, and let $\mathbf{1}=(1,\ldots,1)\in\mathbb{R}^L$, then $$ f(\mathbf{1})=\sum_{A\subseteq L}\int_{[0,1]^A}d\mathbf{h} \left[\left(\prod_{\ell\in A}\frac{\partial}{\partial x_{\ell}}\right)f\right](\psi_A(\mathbf{h})) $$ where $\psi_A(\mathbf{h})$ is the element $\mathbf{x}=(x_{\ell})_{\ell\in L}$ of $\mathbb{R}^L$ defined from the element $\mathbf{h}=(h_{\ell})_{\ell\in A}$ in $[0,1]^A$ by the rule: $x_{\ell}=0$ if $\ell\notin A$ and $x_{\ell}=h_{\ell}$ if $\ell\in A$. Of course one needs to 1) apply this to all $L$'s which are subsets of $[p]$, 2) use Moebius inversion in the Boolean lattice, and 3) specialize to $L=[p]$, and this gives the OP's identity.

The above formula is the most naive one of its kind used to do a "pair of cubes" cluster expansion. See formula III.1 in the article

A. Abdesselam and V. Rivasseau, "Trees, forests and jungles: a botanical garden for cluster expansions".

It is also explained in words on page 115 of the book

V. Rivasseau, "From Perturbative to Constructive Renormalization".

Now the formula is a particular case of a much more powerful one, namely, Lemma 1 in

A. Abdesselam and V. Rivasseau, "An explicit large versus small field multiscale cluster expansion",

where one sums over "allowed" sequences $(\ell_1,\ldots,\ell_k)$ of arbitrary length of elements of $L$, instead of subsets of $L$. The notion of allowed is based on an arbitrary stopping rule. The above identity corresponds to "allowed"$=$"without repeats", or the stopping rule that one should not tack on an $\ell$ at the end of a sequence where it already appeared. By playing with this kind of choice of stopping rule one can use Lemma 1 of my article with Rivasseau, to prove the Hermite-Genocchi formula, the anisotropic Taylor formula by Hairer in Appendix A of "A theory of regularity structures" and many other things. When $f$ is the exponential of a linear form for instance, one can obtain various algebraic identities as in the MO posts

rational function identity

Identity involving sum over permutations

I forgot to mention, one can use Lemma 1 to derive the Taylor formula from calculus 1. This corresponds to $L$ having one element and defining allowed sequences as the ones of length at most $n$. See

https://math.stackexchange.com/questions/3753212/is-there-any-geometrical-intuition-for-the-factorials-in-taylor-expansions/3753600#3753600

For an elementary fact like this, which may have been reinvented a thousand times, it is hard to find the first paper where this appeared. However, let me give some missing context. There is a whole industry in constructive quantum field theory and statistical mechanics about related "smart" interpolation formulas or Taylor formulas with integral remainders. These are used to perform so-called cluster expansions. For the OP's identity, there is no loss of generality in taking $u=(0,0,\ldots,0)$ and $v=(1,1,\ldots,1)$. In this case, via Möbius inversion in the Boolean lattice, the formula comes from the following identity.

Let $L$ be a finite set. Let $f:\mathbb{R}^L\rightarrow \mathbb{R}$, $\mathbf{x}=(x_{\ell})_{\ell\in L}\mapsto f(\mathbf{x})$ be a sufficiently smooth function, and let $\mathbf{1}=(1,\ldots,1)\in\mathbb{R}^L$, then $$ f(\mathbf{1})=\sum_{A\subseteq L}\int_{[0,1]^A}d\mathbf{h} \left[\left(\prod_{\ell\in A}\frac{\partial}{\partial x_{\ell}}\right)f\right](\psi_A(\mathbf{h})) $$ where $\psi_A(\mathbf{h})$ is the element $\mathbf{x}=(x_{\ell})_{\ell\in L}$ of $\mathbb{R}^L$ defined from the element $\mathbf{h}=(h_{\ell})_{\ell\in A}$ in $[0,1]^A$ by the rule: $x_{\ell}=0$ if $\ell\notin A$ and $x_{\ell}=h_{\ell}$ if $\ell\in A$. Of course one needs to 1) apply this to all $L$'s which are subsets of $[p]$, 2) use Möbius inversion in the Boolean lattice, and 3) specialize to $L=[p]$, and this gives the OP's identity.

The above formula is the most naive one of its kind used to do a "pair of cubes" cluster expansion. See formula III.1 in the article

A. Abdesselam and V. Rivasseau, "Trees, forests and jungles: a botanical garden for cluster expansions".

It is also explained in words on page 115 of the book

V. Rivasseau, "From Perturbative to Constructive Renormalization".

Now the formula is a particular case of a much more powerful one, namely, Lemma 1 in

A. Abdesselam and V. Rivasseau, "An explicit large versus small field multiscale cluster expansion",

where one sums over "allowed" sequences $(\ell_1,\ldots,\ell_k)$ of arbitrary length of elements of $L$, instead of subsets of $L$. The notion of allowed is based on an arbitrary stopping rule. The above identity corresponds to "allowed"$=$"without repeats", or the stopping rule that one should not tack on an $\ell$ at the end of a sequence where it already appeared. By playing with this kind of choice of stopping rule one can use Lemma 1 of my article with Rivasseau, to prove the Hermite-Genocchi formula, the anisotropic Taylor formula by Hairer in Appendix A of "A theory of regularity structures" and many other things. When $f$ is the exponential of a linear form for instance, one can obtain various algebraic identities as in the MO posts

rational function identity

Identity involving sum over permutations

I forgot to mention, one can use Lemma 1 to derive the Taylor formula from calculus 1. This corresponds to $L$ having one element and defining allowed sequences as the ones of length at most $n$. See

https://math.stackexchange.com/questions/3753212/is-there-any-geometrical-intuition-for-the-factorials-in-taylor-expansions/3753600#3753600

added 344 characters in body
Source Link

For an elementary fact like this, which may have been reinvented a thousand times, it is hard to find the first paper where this appeared. However, let me give some missing context. There is a whole industry in constructive quantum field theory and statistical mechanics about related "smart" interpolation formulas or Taylor formulas with integral remainders. These are used to perform so-called cluster expansions. For the OP's identity, there is no loss of generality in taking $u=(0,0,\ldots,0)$ and $v=(1,1,\ldots,1)$. In this case, via Möbius inversion in the Boolean lattice, the formula comes from the following identity.

Let $L$ be a finite set. Let $f:\mathbb{R}^L\rightarrow \mathbb{R}$, $\mathbf{x}=(x_{\ell})_{\ell\in L}\mapsto f(\mathbf{x})$ be a sufficiently smooth function, and let $\mathbf{1}=(1,\ldots,1)\in\mathbb{R}^L$, then $$ f(\mathbf{1})=\sum_{A\subseteq L}\int_{[0,1]^A}d\mathbf{h} \left[\left(\prod_{\ell\in A}\frac{\partial}{\partial x_{\ell}}\right)f\right](\psi_A(\mathbf{h})) $$ where $\psi_A(\mathbf{h})$ is the element $\mathbf{x}=(x_{\ell})_{\ell\in L}$ of $\mathbb{R}^L$ defined from the element $\mathbf{h}=(h_{\ell})_{\ell\in A}$ in $[0,1]^A$ by the rule: $x_{\ell}=0$ if $\ell\notin A$ and $x_{\ell}=h_{\ell}$ if $\ell\in A$. Of course one needs to 1) apply this to all $L$'s which are subsets of $[p]$, 2) use Moebius inversion in the Boolean lattice, and 3) specialize to $L=[p]$, and this gives the OP's identity.

The above formula is the most naive one of its kind used to do a "pair of cubes" cluster expansion. See formula III.1 in the article

A. Abdesselam and V. Rivasseau, "Trees, forests and jungles: a botanical garden for cluster expansions".

It is also explained in words on page 115 of the book

V. Rivasseau, "From Perturbative to Constructive Renormalization".

Now the formula is a particular case of a much more powerful one, namely, Lemma 1 in

A. Abdesselam and V. Rivasseau, "An explicit large versus small field multiscale cluster expansion",

where one sums over "allowed" sequences $(\ell_1,\ldots,\ell_k)$ of arbitrary length of elements of $L$, instead of subsets of $L$. The notion of allowed is based on an arbitrary stopping rule. The above identity corresponds to "allowed"$=$"without repeats", or the stopping rule that one should not tack on an $\ell$ at the end of a sequence where it already appeared. By playing with this kind of choice of stopping rule one can use Lemma 1 of my article with Rivasseau, to prove the Hermite-Genocchi formula, the anisotropic Taylor formula by Hairer in Appendix A of "A theory of regularity structures" and many other things. When $f$ is the exponential of a linear form for instance, one can obtain various algebraic identities as in the MO posts

rational function identity

Identity involving sum over permutations

I forgot to mention, one can use Lemma 1 to derive the Taylor formula from calculus 1. This corresponds to $L$ having one element and defining allowed sequences as the ones of length at most $n$. See

https://math.stackexchange.com/questions/3753212/is-there-any-geometrical-intuition-for-the-factorials-in-taylor-expansions/3753600#3753600

For an elementary fact like this, which may have been reinvented a thousand times, it is hard to find the first paper where this appeared. However, let me give some missing context. There is a whole industry in constructive quantum field theory and statistical mechanics about related "smart" interpolation formulas or Taylor formulas with integral remainders. These are used to perform so-called cluster expansions. For the OP's identity, there is no loss of generality in taking $u=(0,0,\ldots,0)$ and $v=(1,1,\ldots,1)$. In this case, via Möbius inversion in the Boolean lattice, the formula comes from the following identity.

Let $L$ be a finite set. Let $f:\mathbb{R}^L\rightarrow \mathbb{R}$, $\mathbf{x}=(x_{\ell})_{\ell\in L}\mapsto f(\mathbf{x})$ be a sufficiently smooth function, and let $\mathbf{1}=(1,\ldots,1)\in\mathbb{R}^L$, then $$ f(\mathbf{1})=\sum_{A\subseteq L}\int_{[0,1]^A}d\mathbf{h} \left[\left(\prod_{\ell\in A}\frac{\partial}{\partial x_{\ell}}\right)f\right](\psi_A(\mathbf{h})) $$ where $\psi_A(\mathbf{h})$ is the element $\mathbf{x}=(x_{\ell})_{\ell\in L}$ of $\mathbb{R}^L$ defined from the element $\mathbf{h}=(h_{\ell})_{\ell\in A}$ in $[0,1]^A$ by the rule: $x_{\ell}=0$ if $\ell\notin A$ and $x_{\ell}=h_{\ell}$ if $\ell\in A$. Of course one needs to 1) apply this to all $L$'s which are subsets of $[p]$, 2) use Moebius inversion in the Boolean lattice, and 3) specialize to $L=[p]$, and this gives the OP's identity.

The above formula is the most naive one of its kind used to do a "pair of cubes" cluster expansion. See formula III.1 in the article

A. Abdesselam and V. Rivasseau, "Trees, forests and jungles: a botanical garden for cluster expansions".

It is also explained in words on page 115 of the book

V. Rivasseau, "From Perturbative to Constructive Renormalization".

Now the formula is a particular case of a much more powerful one, namely, Lemma 1 in

A. Abdesselam and V. Rivasseau, "An explicit large versus small field multiscale cluster expansion",

where one sums over "allowed" sequences $(\ell_1,\ldots,\ell_k)$ of arbitrary length of elements of $L$, instead of subsets of $L$. The notion of allowed is based on an arbitrary stopping rule. The above identity corresponds to "allowed"$=$"without repeats", or the stopping rule that one should not tack on an $\ell$ at the end of a sequence where it already appeared. By playing with this kind of choice of stopping rule one can use Lemma 1 of my article with Rivasseau, to prove the Hermite-Genocchi formula, the anisotropic Taylor formula by Hairer in Appendix A of "A theory of regularity structures" and many other things. When $f$ is the exponential of a linear form for instance, one can obtain various algebraic identities as in the MO posts

rational function identity

Identity involving sum over permutations

For an elementary fact like this, which may have been reinvented a thousand times, it is hard to find the first paper where this appeared. However, let me give some missing context. There is a whole industry in constructive quantum field theory and statistical mechanics about related "smart" interpolation formulas or Taylor formulas with integral remainders. These are used to perform so-called cluster expansions. For the OP's identity, there is no loss of generality in taking $u=(0,0,\ldots,0)$ and $v=(1,1,\ldots,1)$. In this case, via Möbius inversion in the Boolean lattice, the formula comes from the following identity.

Let $L$ be a finite set. Let $f:\mathbb{R}^L\rightarrow \mathbb{R}$, $\mathbf{x}=(x_{\ell})_{\ell\in L}\mapsto f(\mathbf{x})$ be a sufficiently smooth function, and let $\mathbf{1}=(1,\ldots,1)\in\mathbb{R}^L$, then $$ f(\mathbf{1})=\sum_{A\subseteq L}\int_{[0,1]^A}d\mathbf{h} \left[\left(\prod_{\ell\in A}\frac{\partial}{\partial x_{\ell}}\right)f\right](\psi_A(\mathbf{h})) $$ where $\psi_A(\mathbf{h})$ is the element $\mathbf{x}=(x_{\ell})_{\ell\in L}$ of $\mathbb{R}^L$ defined from the element $\mathbf{h}=(h_{\ell})_{\ell\in A}$ in $[0,1]^A$ by the rule: $x_{\ell}=0$ if $\ell\notin A$ and $x_{\ell}=h_{\ell}$ if $\ell\in A$. Of course one needs to 1) apply this to all $L$'s which are subsets of $[p]$, 2) use Moebius inversion in the Boolean lattice, and 3) specialize to $L=[p]$, and this gives the OP's identity.

The above formula is the most naive one of its kind used to do a "pair of cubes" cluster expansion. See formula III.1 in the article

A. Abdesselam and V. Rivasseau, "Trees, forests and jungles: a botanical garden for cluster expansions".

It is also explained in words on page 115 of the book

V. Rivasseau, "From Perturbative to Constructive Renormalization".

Now the formula is a particular case of a much more powerful one, namely, Lemma 1 in

A. Abdesselam and V. Rivasseau, "An explicit large versus small field multiscale cluster expansion",

where one sums over "allowed" sequences $(\ell_1,\ldots,\ell_k)$ of arbitrary length of elements of $L$, instead of subsets of $L$. The notion of allowed is based on an arbitrary stopping rule. The above identity corresponds to "allowed"$=$"without repeats", or the stopping rule that one should not tack on an $\ell$ at the end of a sequence where it already appeared. By playing with this kind of choice of stopping rule one can use Lemma 1 of my article with Rivasseau, to prove the Hermite-Genocchi formula, the anisotropic Taylor formula by Hairer in Appendix A of "A theory of regularity structures" and many other things. When $f$ is the exponential of a linear form for instance, one can obtain various algebraic identities as in the MO posts

rational function identity

Identity involving sum over permutations

I forgot to mention, one can use Lemma 1 to derive the Taylor formula from calculus 1. This corresponds to $L$ having one element and defining allowed sequences as the ones of length at most $n$. See

https://math.stackexchange.com/questions/3753212/is-there-any-geometrical-intuition-for-the-factorials-in-taylor-expansions/3753600#3753600

added 685 characters in body
Source Link

For an elementary fact like this, which may have been reinvented a thousand times, it is hard to find the first paper where this appeared. However, let me give some missing context. There is a whole industry in constructive quantum field theory and statistical mechanics about related "smart" interpolation formulas or Taylor formulas with integral remainders. These are used to perform so-called cluster expansioncluster expansions. For the OP's identity, there is no loss of generality in taking $u=(0,0,\ldots,0)$ and $v=(1,1,\ldots,1)$. In this case, via Moebius inversion in the Boolean latticeMöbius inversion in the Boolean lattice, the formula comes from the following identity.

Let $L$ be a finite set. Let $f:\mathbb{R}^L\rightarrow \mathbb{R}$, $\mathbf{x}=(x_{\ell})_{\ell\in L}\mapsto f(\mathbf{x})$ be a sufficiently smooth function, and let $\mathbf{1}=(1,\ldots,1)\in\mathbb{R}^L$, then $$ f(\mathbf{1})=\sum_{A\subset L}\int_{[0,1]^A}d\mathbf{h} \left[\left(\prod_{\ell\in A}\frac{\partial}{\partial x_{\ell}}\right)f\right](\psi_A(\mathbf{h})) $$$$ f(\mathbf{1})=\sum_{A\subseteq L}\int_{[0,1]^A}d\mathbf{h} \left[\left(\prod_{\ell\in A}\frac{\partial}{\partial x_{\ell}}\right)f\right](\psi_A(\mathbf{h})) $$ where $\psi_A(\mathbf{h})$ is the element $\mathbf{x}=(x_{\ell})_{\ell\in L}$ of $\mathbb{R}^L$ defined from the element $\mathbf{h}=(h_{\ell})_{\ell\in A}$ in $[0,1]^A$ by the rule: $x_{\ell}=0$ if $\ell\notin A$ and $x_{\ell}=h_{\ell}$ if $\ell\in A$. Of course one needs to 1) apply this to all $L$'s which are subsets of $[p]$, 2) use Moebius inversion in the Boolean lattice, and 3) specialize to $L=[p]$, and this gives the OP's identity.

The above formula is the most naive one of its kind used to do a "pair of cubes" cluster expansion. See formula III.1 in the article

A. Abdesselam and V. Rivasseau, https"Trees, forests and jungles://arxiv.org/abs/hep-th/9409094 a botanical garden for cluster expansions".

It is also explained in words on page 115 of the book

V. Rivasseau, http://www.rivasseau.com/resources/book.pdf"From Perturbative to Constructive Renormalization".

Now the formula is a particular case of a much more powerful one, namely, Lemma 1 in

A. Abdesselam and V. Rivasseau, https://arxiv.org/abs/hep-th/9605094"An explicit large versus small field multiscale cluster expansion",

where one sums over "allowed" sequences $(\ell_1,\ldots,\ell_k)$ of arbitrary length of elements of $L$, instead of subsets of $L$. The notion of allowed is based on an arbitrary stopping rule. The above identity corresponds to "allowed"$=$"without repeats", or the stopping rule that one should not tack on an $\ell$ at the end of a sequence where it already appeared. By playing with this kind of choice of stopping rule one can use Lemma 1 of my article with Rivasseau, to prove the Hermite-Genocchi formula, the anisotropic Taylor formula by Hairer in Appendix A of https://arxiv.org/abs/1303.5113"A theory of regularity structures" and many other things. When $f$ is the exponential of a linear form for instance, one can obtain various algebraic identities as in the MO posts

rational function identity

Identity involving sum over permutations

For an elementary fact like this, which may have been reinvented a thousand times, it is hard to find the first paper where this appeared. However, let me give some missing context. There is a whole industry in constructive quantum field theory and statistical mechanics about related "smart" interpolation formulas or Taylor formulas with integral remainders. These are used to perform so-called cluster expansion. For the OP's identity, there is no loss of generality in taking $u=(0,0,\ldots,0)$ and $v=(1,1,\ldots,1)$. In this case, via Moebius inversion in the Boolean lattice, the formula comes from the following identity.

Let $L$ be a finite set. Let $f:\mathbb{R}^L\rightarrow \mathbb{R}$, $\mathbf{x}=(x_{\ell})_{\ell\in L}\mapsto f(\mathbf{x})$ be a sufficiently smooth function, and let $\mathbf{1}=(1,\ldots,1)\in\mathbb{R}^L$, then $$ f(\mathbf{1})=\sum_{A\subset L}\int_{[0,1]^A}d\mathbf{h} \left[\left(\prod_{\ell\in A}\frac{\partial}{\partial x_{\ell}}\right)f\right](\psi_A(\mathbf{h})) $$ where $\psi_A(\mathbf{h})$ is the element $\mathbf{x}=(x_{\ell})_{\ell\in L}$ of $\mathbb{R}^L$ defined from the element $\mathbf{h}=(h_{\ell})_{\ell\in A}$ in $[0,1]^A$ by the rule: $x_{\ell}=0$ if $\ell\notin A$ and $x_{\ell}=h_{\ell}$ if $\ell\in A$. Of course one needs to 1) apply this to all $L$'s which are subsets of $[p]$, 2) use Moebius inversion in the Boolean lattice, and 3) specialize to $L=[p]$, and this gives the OP's identity.

The above formula is the most naive one of its kind used to do a "pair of cubes" cluster expansion. See formula III.1 in the article

https://arxiv.org/abs/hep-th/9409094

It is also explained in words on page 115 of the book

http://www.rivasseau.com/resources/book.pdf

Now the formula is a particular case of a much more powerful one, namely, Lemma 1 in

https://arxiv.org/abs/hep-th/9605094

where one sums over "allowed" sequences $(\ell_1,\ldots,\ell_k)$ of arbitrary length of elements of $L$, instead of subsets of $L$. The notion of allowed is based on an arbitrary stopping rule. The above identity corresponds to "allowed"$=$"without repeats", or the stopping rule that one should not tack on an $\ell$ at the end of a sequence where it already appeared. By playing with this kind of choice of stopping rule one can use Lemma 1 of my article with Rivasseau, to prove the Hermite-Genocchi formula, the anisotropic Taylor formula by Hairer in Appendix A of https://arxiv.org/abs/1303.5113 and many other things.

For an elementary fact like this, which may have been reinvented a thousand times, it is hard to find the first paper where this appeared. However, let me give some missing context. There is a whole industry in constructive quantum field theory and statistical mechanics about related "smart" interpolation formulas or Taylor formulas with integral remainders. These are used to perform so-called cluster expansions. For the OP's identity, there is no loss of generality in taking $u=(0,0,\ldots,0)$ and $v=(1,1,\ldots,1)$. In this case, via Möbius inversion in the Boolean lattice, the formula comes from the following identity.

Let $L$ be a finite set. Let $f:\mathbb{R}^L\rightarrow \mathbb{R}$, $\mathbf{x}=(x_{\ell})_{\ell\in L}\mapsto f(\mathbf{x})$ be a sufficiently smooth function, and let $\mathbf{1}=(1,\ldots,1)\in\mathbb{R}^L$, then $$ f(\mathbf{1})=\sum_{A\subseteq L}\int_{[0,1]^A}d\mathbf{h} \left[\left(\prod_{\ell\in A}\frac{\partial}{\partial x_{\ell}}\right)f\right](\psi_A(\mathbf{h})) $$ where $\psi_A(\mathbf{h})$ is the element $\mathbf{x}=(x_{\ell})_{\ell\in L}$ of $\mathbb{R}^L$ defined from the element $\mathbf{h}=(h_{\ell})_{\ell\in A}$ in $[0,1]^A$ by the rule: $x_{\ell}=0$ if $\ell\notin A$ and $x_{\ell}=h_{\ell}$ if $\ell\in A$. Of course one needs to 1) apply this to all $L$'s which are subsets of $[p]$, 2) use Moebius inversion in the Boolean lattice, and 3) specialize to $L=[p]$, and this gives the OP's identity.

The above formula is the most naive one of its kind used to do a "pair of cubes" cluster expansion. See formula III.1 in the article

A. Abdesselam and V. Rivasseau, "Trees, forests and jungles: a botanical garden for cluster expansions".

It is also explained in words on page 115 of the book

V. Rivasseau, "From Perturbative to Constructive Renormalization".

Now the formula is a particular case of a much more powerful one, namely, Lemma 1 in

A. Abdesselam and V. Rivasseau, "An explicit large versus small field multiscale cluster expansion",

where one sums over "allowed" sequences $(\ell_1,\ldots,\ell_k)$ of arbitrary length of elements of $L$, instead of subsets of $L$. The notion of allowed is based on an arbitrary stopping rule. The above identity corresponds to "allowed"$=$"without repeats", or the stopping rule that one should not tack on an $\ell$ at the end of a sequence where it already appeared. By playing with this kind of choice of stopping rule one can use Lemma 1 of my article with Rivasseau, to prove the Hermite-Genocchi formula, the anisotropic Taylor formula by Hairer in Appendix A of "A theory of regularity structures" and many other things. When $f$ is the exponential of a linear form for instance, one can obtain various algebraic identities as in the MO posts

rational function identity

Identity involving sum over permutations

Source Link
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