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I am interested in polynomials $G_n(z)$ defined by the recurrence $$G_{n+1}(z) - 2G_n(z) + (1-nz)G_{n-1}(z)=0$$ for $n\ge1$ with the initial values $G_0(z) = 1$ and $G_1(z) = 1$. The next few values ...

For $\alpha,\beta\in\mathbb{C}$ and $\gamma\in\mathbb{C}\setminus\{0,-1,-2,\dotsc\}$, Gauss' hypergeometric function ${}_2F_1(\alpha,\beta;\gamma;z)$ can be defined by the series \begin{equation}\...

Let $f(z)$ be a generating function (so in particular, its power series coefficients are nonnegative). I am interested in conditions which would ensure that for every $n$, the coefficient of $z^n$ in $...

First, define a sequence $F_0,F_1,\dots$ of functions by $$F_0(x,z) = z,$$ $$F_k(x,z)=x\exp\left(F_{k-1}(x,z)\right) \quad \text{for }k\geq1.$$ So, for example, $$F_1(x,z) = x e^z, \quad F_2(x,z)=xe^{...

I am interested in using series of the form $\sum_{n=-\infty}^{\infty} a_nt^n$ (where $a_n\in\mathbb C$) as generating functions. In general, multiplication of such series goes against the "formal ...

Let $a_n$ be a sequence of non-negative real numbers, and $A(x) = \sum_{n=0}^{\infty} a_n \frac{x^n}{n!}$ its exponential generating function. Also, suppose $B(x) = \sum_{n=0}^{\infty} b_n \frac{x^n}{...