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Looking for references for the following theorem.

Given the formal Taylor series/exponential generating function

$$T(z) = \sum_{n \ge 1} a_n \; \frac{z^n}{n!},$$

for which the indeterminates $a_n$ and the independent variable $z$ may be complex, its formal compositional inverse

$$T^{(-1)}(z) = \sum_{n \ge 1} b_n \; \frac{z^n}{n!}=\sum_{n \ge 1} Prt_n(a_1,a_2,...,a_n) \; \frac{z^n}{n!} $$

satisfies the differential equation

$$-\frac{\partial}{\partial a_{n}}\;T^{(-1)}(z)= \frac{\partial}{\partial z}\;\frac{(T^{(-1)}(z))^{n+1}}{(n+1)!}.$$

Edit: Don't quite follow Pietro's derivation (I already had two other derivations of this theorem), but Pietro's inspires the following third derivation;

With $S(\omega,a_n) = T^{(-1)}(\omega,a_n) = z$ and $\omega = T(z,a_n)$, then $z = S(T(z,a_n),a_n)$,

so, suppressing the $a_n$ in the arguments,

$$\partial_{a_n} z = 0 = \partial_T \;S(T(z))\; \partial_{a_n} \;T(z) + \partial_{a_n} \;S(T(z))$$

$$= \partial_T \;S(T(z))\; \frac{z^n}{n!} + \partial_{a_n} \;S(T(z))$$

$$=\partial_T \;S(T(z))\; \frac{(S(T(z)))^n}{n!} + \partial_{a_n} \;S(T(z))$$

$$=\partial_T \; \frac{(S(T(z)))^{n+1}}{(n+1)!} + \partial_{a_n} \;S(T(z)),$$

implying

$$\partial_{\omega} \; \frac{(S(\omega))^{n+1}}{(n+1)!} + \partial_{a_n} \;S(\omega)=0.$$

Instances I had already found of this integrable hierarchy of diff ids / conservation laws are in the section "5.4 Deformation of flow equations" on p. 57 and in Theorem 5.4 on pp. 66 of "On Emergent Geometry of the Gromov-Witten Theory of Quintic Calabi-Yau Threefold" by Jian Zhou.

One reason I ask for more references is that this diff id implies some interesting relations w.r.t. free probability theory, associahedra, and the inviscid Burgers-Hopf differential equation, so I'm hoping to see some perspectives from others on these and will give due credit on such in any of my postings in the future.

Edit (5/28/22): The derivation is more accurately expressed as

$$\partial_{a_n} \; z=0 =\partial_{a_n} \; S(T(z,a_n),a_n) =\partial_{a_n} \; S(\beta_1,\beta_2)$$

$$= (\partial_{\beta_1} \; S(\beta_1,\beta_2)) \;\partial_{a_n}\beta_1 +(\partial_{\beta_2} \; S(\beta_1,\beta_2)) \; \partial_{a_n}\beta_2$$

$$ = (\partial_{\beta_1} \; S(\beta_1,\beta_2)) \;\frac{z^n}{n!} +\partial_{\beta_2} \; S(\beta_1,\beta_2) $$

$$ = (\partial_{\beta_1} \; S(\beta_1,\beta_2)) \;\frac{( S(\beta_1,\beta_2))^n}{n!} +\partial_{\beta_2} \; S(\beta_1,\beta_2) $$

$$ = \partial_{\beta_1} \; \;\frac{(S(\beta_1,\beta_2))^{n+1}}{(n+1)!} +\partial_{\beta_2} \; S(\beta_1,\beta_2) $$

which may be expressed as

$$0 = \partial_{u} \; \;\frac{(S(u,a_n))^{n+1}}{(n+1)!} +\partial_{a_n} \; S(u,a_n).$$

Tom Copeland
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  • One implication of the diff id is given in "Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory" (https://mathoverflow.net/questions/412573/combinatorics-for-the-action-of-virasoro-kac-schwarz-operators-partition-poly). – Tom Copeland May 27 '22 at 06:52
  • I wonder if this is also connected with Lagrange Inversion Theorem in the form $$[z^m]T^{(-1)}(z)=\frac1m[z^{-1}]T(z)^{-m}$$ – Pietro Majer May 27 '22 at 09:13
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    @PietroMajer, connection to the LIT is in my posts "The Laplace Transform and Compositional Inversion" (https://tcjpn.wordpress.com/2022/05/23/the-laplace-transform-and-compositional-inversion/) and the earlier "Ruling the inverse universe, the inviscid Hopf-Burgers evolution equation: Compositional inversion, free probability, associahedra, diff ids, integrable hierarchies, and translation" (https://tcjpn.wordpress.com/2022/01/27/ruling-the-inverse-universe-the-inviscid-hopf-burgers-evolution-equation-compositional-inversion-associahedra-diff-ids-integrable-hierarchies-and-translation/). – Tom Copeland May 27 '22 at 16:12
  • Very interesting thank you – Pietro Majer May 28 '22 at 09:23
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    See also the Riemann hierarchy on p. 4 of "Remarks on intersection numbers and integrable hierarchies. I. Quasi-triviality" by Dubrovin and Yang (https://arxiv.org/pdf/1905.08106.pdf) and p. 93 of "Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants" by Dubrovin and Zhang (https://arxiv.org/pdf/math/0108160.pdf). – Tom Copeland May 28 '22 at 22:11
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    See eqn 1.28 on p. 7 of "Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs" by Dubrovin, Liu, Yang, and Zhang (https://arxiv.org/pdf/1409.4616.pdf). – Tom Copeland May 28 '22 at 22:39
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    And the Riemann-Hopf hierarchy on p. 5 of "On cubic Hodge integrals and random matrices" by Dubrovin and Yang (https://arxiv.org/pdf/1606.03720.pdf). – Tom Copeland May 28 '22 at 23:21
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    p. 2 of "Deformations of the Monge/Riemann hierarchy and approximately integrable systems" by Strachan (https://arxiv.org/pdf/nlin/0205051.pdf). – Tom Copeland May 28 '22 at 23:34
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    As noted in "Ruling the inverse universe . . . ", the paper "Quantum Deformation Theory of the Airy Curve and Mirror Symmetry of a Point" by Zhou (https://arxiv.org/pdf/1405.5296.pdf) contains the diff id on p. 11 and discusses its relation to the inviscid Burgers-Hopf equation (IBHE) and thence to Lagrange inversion. The association between Lagrange inversion of the generating function for the signed coarse face polynomials (refined are A133437) of the associahedra and the IBHE was noted earlier in http://oeis.org/A086810, suggested by notes by Buchstaber and Panov since generalized. – Tom Copeland May 29 '22 at 01:11

1 Answers1

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I do not have a reference, but I’d say it is a plain instance of the Implicit Function Theorem for formal power series. I add the computation below, in case you had a different one.

We choose our index $n\ge1$ and see $T$ as a power series in $z$ and $a:=a_{n}$, that is, as an element of $\mathbb{C}[[z, a]]$. The corresponding compositional inverse of $T(z, a)$, is the formal series $S(z, a)$ well defined as implicit function by $$z=T(S(z, a), a) $$ (with $a_1\neq0$). Differentiating w.r.to $a $ and multiplying by $\partial_{1 }S(z,a) $ we get

$$0=\Big[\partial_1 T\big(S(z, a ), a \big)\partial_2S(z,a)+\partial_2 T\big(S(z,a),a\big)\big] \partial_1S(z,a)=$$

$$=\partial_1 T\big(S(z, a ), a \big)\partial_1S(z,a) \partial_2S(z,a) +\partial_2T\big(S(z,a),a\big) \partial_1S(z,a)=$$

$$= \partial_1\big[T (S(z, a ), a)\big]\partial_2S(z,a) + \frac{S(z,a)^{n}}{n!} \, \partial_1S(z,a)=$$

$$=\partial_2S(z,a) + \partial_1\bigg(\frac{S(z,a)^{n+1}}{(n+1)!}\bigg).$$

Pietro Majer
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