Any Info on Generalized Binomial and Exponential Series?

Question

I was reading Concrete Mathematics by Knuth et al. and on p. 200, it references the generalized binomial series $\mathcal B_t(z)$ and the generalized exponential series $\mathcal E_t(z)$, defined as $$\mathcal B_t(z)=\sum_{k\ge0}(tk)^{\underline{k-1}}\frac{z^k}{k!}\quad\text{and}\quad\mathcal E_t(z)=\sum_{k\ge0}(tk+1)^{k-1}\frac{z^k}{k!},$$ where $x^{\underline k}$ denotes the falling factorial. I was interested in the generalized binomial series because it satisfied the identity $$\frac{\mathcal B_t(z)^r}{1-t+t\mathcal B_t(z)^{-1}}=\sum_{k\ge0}\binom{tk+r}{k}z^k.$$ I've been trying to find a way to compute the sum $\sum_{n\le k\le m}\binom{tk+r}{k}$, so this seemed related. It seems that $\mathcal E_1(z)$ is also somewhat related to the Lambert $W$ function, though I might be wrong. Searching them up online didn't seem to get anywhere, and I just want to find more sources where I can learn about these functions. If anyone can find some resources, I'd appreciate it!

Answer 1

There is a construction that generalizes both $\mathcal{B}_t$ and $\mathcal{E}_t$ (also given by D.E.Knuth in The Art of Computer Programming vol. 2, but definitely known before; I have to trace its history back yet).

We define, the usual way, $\big(F(z)\big)^t$ for $t\in\mathbb{C}$ and a formal power series $F(z)=\sum_{k\geq 0}F_k z^k$ over $\mathbb{C}$, with $F_0=1$ (or even for any $F_0\neq 0$ instead of $F_0=1$, provided that we fix the value of $(F_0)^t$). Then we can define $F^{\{t\}}(z)$ as the unique formal power series $G(z)$ that satisfies $G(z)=F\Big(z\big(G(z)\big)^t\Big)$. Our $\mathcal{B}_t$ and $\mathcal{E}_t$ are particular cases: $\mathcal{B}_t=B^{\{t\}}$ for $B(z)=1+z$, and $\mathcal{E}_t=E^{\{t\}}$ for $E(z)=e^z$.

This construction has many properties. An easy one is $(F^{\{t\}})^{\{s\}}=F^{\{t+s\}}$. A more important one, under the notation $[z^k]G(z)=G_k$ for $G(z)=\sum_{k\geq 0}G_k z^k$, and proven using Lagrange inversion theorem, is $$[z^k]\big(F^{\{t\}}(z)\big)^s=\frac{s}{s+kt}[z^k]\big(F(z)\big)^{s+kt}.$$ The power series for $\mathcal{B}_t$ and $\mathcal{E}_t$ are again easy particular cases of this formula. And $\mathcal{E}_1(z)=-W_0(-z)/z$ is indeed related to the Lambert W-function.