Two questions about Bell polynomials

Question

In my research in Quantum Field Theory, I have encountered two questions that involve partial Bell polynomials:

1. Let $u$ and $x_i$ be indeterminates. I have checked that the following conjectured identity

$$ \sum _{m=0}^\infty u^m \frac{(m+1)!}{(2m+2)!} B_{2m+2,m+2}(x_1,\dots,x_{m+1})= \frac{1}{2} \left( \sum_{n=0}^\infty u^n \frac{n!}{(2n+1)!} B_{2n+1,n+1}(x_1,\dots, x_{n+1}) \right)^2 $$

holds for the first orders in $u$. Is this identity a known result? A proof or any reference would be very welcome.

2. I would like to find the radius of convergence of a similar infinite sum, in the particular case when $x_j = j^{j-2}$, or at least prove that it is different from zero. Namely, I would like to discuss the convergence of

$$ f(u) = \sum_{m=1}^\infty u^m \frac{(m-1)!}{(2m)!} B_{2m,m+1}(1^{1-2},2^{2-2},3^{3-2},\dots,m^{m-2}) $$

near $u=0$, but I don't know how to estimate the asymptotic growth of $B_{2m,m+1}(1^{1-2},2^{2-2},3^{3-2},\dots,m^{m-2})$ for large $m$. Again, any help or reference would be welcome.

Answer 1

With respect to the first question, I'm not sure about whether the given identity is known, but here is a proof. It also expresses the involved series in terms of reversion of the series $\frac{t}{g(t)}$ as defined below.

It is convenient to express the given identity in terms of ordinary Bell polynomials: $$\hat{B}_{n,k}(\frac{x_1}{1!},\frac{x_2}{2!},\ldots,\frac{x_{n-k+1}}{(n-k+1)!}) = \frac{k!}{n!}B_{n,k}(x_1,x_2,\ldots,x_{n-k+1})$$ and omitting the indeterminates $x_j$ and multiplying by $u^2$ as $$\sum_{m=0}^{\infty} \frac{u^{m+2}}{m+2} \hat{B}_{2m+2,m+2} = \frac{1}{2}\left(\sum_{n=0}^{\infty} \frac{u^{n+1}}{n+1} \hat{B}_{2n+1,n+1} \right)^2.$$

From the generating function for $\hat{B}_{n,k}$ it follows that $$\hat{B}_{2m+2,m+2} = [t^{2m+2}]\ \left(\sum_{j\geq 1} \frac{x_j}{j!}t^j\right)^{m+2} = [t^{m}]\ g(t)^{m+2}$$ and $$\hat{B}_{2n+1,n+1} = [t^{2n+1}]\ \left(\sum_{j\geq 1} \frac{x_j}{j!}t^j\right)^{n+1} = [t^n]\ g(t)^{n+1},$$ where $g(t):=\sum_{j\geq 1} x_j\frac{t^{j-1}}{j!}$.

Using Lagrange–Bürmann formula, we get $$\sum_{m\geq 0} \hat{B}_{2m+2,m+2} t^{m+1} = \frac{w(t)w'(t)}t,$$ $$\sum_{n\geq 0} \hat{B}_{2n+1,n+1} t^n = w'(t),$$ where function $w(t)$ satisfies the functional equation: $w(t)=tg(w(t))$.

Correspondingly, $$\sum_{m\geq 0} \hat{B}_{2m+2,m+2} \frac{u^{m+2}}{m+2} = \int_0^u w(t) w'(t)\ {\rm d}t = \frac12 w(u)^2,$$ $$\sum_{n\geq 0} \hat{B}_{2n+1,n+1} \frac{u^{n+1}}{n+1} = \int_0^u w'(t) {\rm d}t = w(u),$$ from where the required identity follows instantly.

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As for the second question, we first notice that $f'(u) = \frac12 w(u)^2$. Also, when $x_j=j^{j-2}$, we have $$g(t)=\sum_{j\geq 1} j^{j-2}\frac{t^{j-1}}{j!},$$ which can be expressed in terms of Lambert W function as $$g(t) = \frac{1}{2t}\left(1-(1+W_0(-t))^2\right).$$ Then convergence of $w(t)$ can be imposed from viewing it as a series reversion of $\frac{t}{g(t)}$.