# A sequence of generating functions

I came on this sequence of generating functions on trying to count lambda terms (in lambda calculus)

$T^{\langle m+1\rangle} = \frac{T^{\langle m \rangle}(z)}{z} - (T^{\langle m\rangle}(z))^2$

with

$T^{\langle m\rangle}(0) = 0.$

What can be said about these functions?

I would be happy to have an expression for $T^{\langle 0\rangle}(z)$ and say something about the asymptotic behavior of the coefficients $T_{n,m}$ for

$T^{\langle m\rangle}(z) = \sum_{n=0}^{\infty}\ T_{n,m} z^n.$

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This appears to say nothing at all about $T^{0}.$ –  Igor Rivin Oct 20 '11 at 20:11
actually it seems to me that if you really want these $T^m(z)$ to be power series with vanishing zero degree coefficient, they have to be identically zero; I post the details as an answer (though I guess it is not the answer you want) in order to clarify what I mean. –  Pietro Majer Oct 20 '11 at 21:51
Something is wrong. If tdeg() denotes the tailing degree of a series (the smallest degree with nonzero coefficient), then the recurrent formula implies $$tdeg(T^{\angle m+1 \rangle}(z)) = tdeg(T^{\angle m \rangle}(z)) - 1$$ meaning that at certain point it will drop below 0. That is, for some $m$, $T^{\angle m \rangle}$ will have nonzero coefficients of negative degrees of $z$. –  Max Alekseyev Oct 20 '11 at 22:37
unless $tdeg=\infty$, that is, all $T^m$ are identically zero, as said. –  Pietro Majer Oct 21 '11 at 7:49
Actually my problem was underdetermined. In my problem, I also want to say that $[z]T^{\langle m\rangle}(z) = T^{\langle m\rangle}'(0) = m.$ In other word, the coefficient of $z$ in the power serie of $T^{\langle m\rangle}(z)$ is $m$. –  Pierre Lescanne Oct 28 '11 at 20:49

## 1 Answer

We want all the (formal) power series $T^m(z)$ to be elements of the ideal $z\mathbb{R}[[z]]$ (which rephrases $T_{0,m}=0$). Also, the relation $T^m(z)=zT^{m+1}(z)+zT^m(z)^2$ implies that if for some $n$ we have $T^m(z)\in z^n\mathbb{R}[[z]]$ for all $m$, then also $T^m(z)\in z^{n+1}\mathbb{R}[[z]]$ for all $m$. By induction, we should conclude that the $T^m(z)$ are identically zero for all $m$..

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Does this make some sort of philosophical statement about lambda calculus? –  Igor Rivin Oct 21 '11 at 7:55