# Questions tagged [generating-functions]

A generating function is a way of encoding an infinite sequence of numbers by treating them as the coefficients of a formal power series. Tag questions involving generating functions in this.

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### Combinatorial consequences of de Branges's Theorem?

I'm usually not a proponent of the mentality “here is a tool, what results can we prove with it?” (as I prefer to start at the other end with a well-motivated problem), but this famously entertaining ...
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### Transcendental functions with two prescribed values

Let $\alpha$ and $\beta$ two algebraic numbers lying in unit ball. Let $T:=(t_k)_k$ be an increasing sequence of positive integers such that $t_{k+1}/t_k$ tends to $1$ as $k\to \infty$. I would like ...
1 vote
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### Something (which might be called multi-continued fraction) for the A112487

Let $a(n)$ be A112487 i.e. an integer sequence with exponential generating function $$A(x)=\exp\left(\int (A(x)+A(x)^2)\,dx\right), \\ A(0)=1$$ However, the definition in the name of the sequence is ...
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### Recursion for the A266328 by analogy with non-standard recursion for factorials

Let $a(n)$ be A266328 i.e. an integer sequence with exponential generating function $$A(x)=\exp\int B(x) \,dx$$ such that $$B(x)=\exp(-x)\exp\int A(x) \,dx$$ where the constant of integration is ...
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### Products involving exponents of tribonacci numbers

The Fibonacci numbers $F_n$ can be given by $$\sum_{k\geq0}F_kx^k=\frac{x}{1-x-x^2}.$$ Among many many properties of this sequence, consider the following two results: (1) the coefficients of the ...
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### Sequences that sum up to possible generalization of Euler or up/down numbers (A000111)

Let $a(n,m,k)$ be an integer sequence with e.g.f. $$A(x)=\operatorname{exp}\left(x + m\int\int (A(x))^k \, dx \, dx\right)$$ I don't know much about integrals, so here's a concrete example: $a(n,1,3)$...
1 vote
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### Closed formula for Hermite polynomials

Hermite polynomials $H_k(x), x \in \mathbb{R}, k \in \mathbb{N}$ are defined by the formula $$H_k(x)=(-1)^k e^{x^2} \frac{d^k}{d x^k}\left(e^{-x^2}\right) .$$ Each $H_k(x)$ is a polynomial of exact ...
1 vote
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### Arithmetic triangles and unimodality of its rows

Let's consider the sequence of coefficients of $\prod_{i}\frac {1-x^{d_i}} {1-x}$, where $d_i$ is a monotonically increasing nonnegative integer sequence. How to prove that the coefficients form an ...
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### Usefulness of total algebras and exotic generating series

In his first Algebra volume, Bourbaki  defines the structure of a “total algebra” i.e. the space of functions on a monoid $M$ (to a ring $k$) with the convolution product ( a function $f:\ M\to k$ ...
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### Two dice yielding uniform distribution, part 2

Since this question is on the front page again, a generalization. Let $p$ be prime, and let $a$ and $b$ be positive integers with $a+b=p-1$. Is it possible to have two loaded dice, one with sides ...
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### A multi-variable "Fibonacci polynomial"?

There is a tremendous literature on the Fibonacci sequence, including its polynomial analogue $F_{-1}=0, F_0=1$ and $$F_n(x)=xF_{n-1}(x)+F_{n-2} \qquad \text{for n\geq1}.$$ What I have found is the ...
1 vote