# Questions tagged [generating-functions]

The generating-functions tag has no usage guidance.

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### The asymptotics of a vector sequence defined by a recursion relation

The sequence of vectors $(\mathbf{v}_0,\mathbf{v}_1,\mathbf{v}_2,\dots)$ obeys the recursion relation that
$A\mathbf{v}_j-\mathbf{v}_{j-1}=\sum_{k=0}^j diag(\mathbf{v}_k)B\mathbf{v}_{j-k}$,
where A ...

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71 views

### One-point partition

Let following be the partition function in infinitely many variables $x_i$ the linear coordinates on the vector space $V$.
$$
\mathcal{Z}=exp\Big(\sum_{\substack{g\geq 0\\n\geq 1}}\frac{h^{g-1}}{n!}\...

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103 views

### Exponential generating function for k^n / n

The ordinary generating function for $k^n/n$ is $-\log(1-kx)$. Is there a closed-form exponential generating function for this sequence?

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99 views

### Relate log concavity of sequence to convexity of 1/generating function

Let $Y$ be a random variable on the natural numbers, with $p_k = \mathbb{P}(Y = k)$ for $k \geq 0$.
Say that $Y$ is log-concave if $p_k^2 \geq p_{k-1} p_{k+1}$ for $k \geq 1$, and that $Y$ is log-...

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153 views

### Collapsed partitions and generating functions

Given $n\in\Bbb{N}$, the number of (unrestricted) integer partitions of $n$ are given by
$$\sum_{n\geq0}p(n)x^n=\prod_{j\geq1}\frac1{1-x^j}.$$
Define the collapsed partitions of $n$ to be the ...

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**1**answer

120 views

### Integer partitions with subset sums “not divisible” by p

I have the following questions: Let $N \in \mathbb{N}$ and
\begin{equation}
\sum_{i=1}^k n_i = N,
\end{equation}
with $n_i \in \mathbb{N}$ for $1 \le i \le k$ and some $k \in \mathbb{N}$, be an ...

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292 views

### Explicit formula for a generating function

In an enumeration problem the sequence of number of Dyck paths semilength n having no UUDD's starting at level 0 with generating function $$\frac{2}{(1+2z^2+\sqrt{1-4z})}$$ showed up, see also https://...

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43 views

### About basis of function spaces under composition

Say $L^2(M_1,\mathbb{R})$ is the space of square-integrable functions on $M_1$ which is a compact manifold with a measure defined on it. Let $L^2(\mathbb{R}^n,\mathbb{R})$ be the space of all square-...

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55 views

### Generating function for number of r-disjoint subsets each of size k

Fix $n, k$. Let
$$
C^{n,k}_r =\frac{1}{r!} \binom{n}{\underbrace{k, \ldots, k}_{\text{r times}}, n-rk} = \frac{n!}{r!(k!)^r(n - kr)!}
$$
be the number of ways to form $r$ disjoint subsets each of ...

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636 views

### Congruences Ramanujan-style

Let $t\in\Bbb{N}$ and consider the sequences $p_t(n)$ defined by
$$\sum_{n\geq0}p_t(n)x^n=\prod_{i\geq1}\frac1{(1-x^i)^t}=(x;x)_{\infty}^{-t}.$$
The numbers $p_t(n)$ can be regarded as enumerating ...

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267 views

### Seeking proof to an asymptotics of a recursion or functional equation

My question on math.stackexchange.com and the continuation by an answer to it gives the two summation expressions for the recursion
$$a_n = 1+\frac1{2^n}\sum_{k=0}^n {n\choose k}a_k,\, \forall n\in\...

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163 views

### Second order recurrence relation for third order polynomial root

Consider this recurrence relation:
$$
\begin{eqnarray*}
f_0&=&1\\
f_n&=&
\sum_{m=0}^{n-1} \frac{\left(\frac{m+3}{2}\right)_{m-1}}{\left(\frac{m+2}{2}\right)_m} f_{n-m-1} f_m\ \ \ \...

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85 views

### How to use this generalised 'generating function' for the Gegenbauer polynomials

Cohl (2011) gives a generalisation of the standard generating function for the Gegenbauer polynomials $C_n^\mu(x)$:
$(1 -2tx + t^2)^{-\nu} = A_{\mu,\nu} \frac{(1-t^2)^{-\nu+\mu+1/2}}{t^{\mu+1/2}} \...

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291 views

### Wherefore art thou a Borcherds Product?

This question essentially asks how can one recognize (or rule out) that a generating function of combinatorial origin may be given as a Borcherds type product. I'll start with a motivational example: ...

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144 views

### Is this a Borel summable $ S = \sum_{k=0}^\infty (-1)^k (k!)a_k $ with $ a_k$ alternating sequence?

let $ S = \sum_{k=0}^\infty (-1)^k (k!)a_k $ a divergent series such that $b_k=(-1)^k (k!)a_k >0 $ for $k>1$ , and $b_k$ signed this from $k=1$ to $20$ ,The asymptotic of the titled series ...

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**1**answer

133 views

### What can we know about “the half” of the generating series of Bessel function

I am interested in the series
$$\sum_{n\geq 1}I_n(x)\lambda^n$$
which is not the full generating series of the modified Bessel function of the first kind because it starts from $n=1$ and not at $-\...

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172 views

### What are the patterns of the sequence of polynomials? [closed]

In my research, I obtained a sequence of polynomials (I am only able to compute the first 4 of them):
\begin{align}
& f(2) = 1+t, \\
& f(3) = 1+4t+3t^2, \\
& f(4) = 1+6t+12t^2+7t^3, \\
&...

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2k views

### Generating function in graph theory

I am looking for a simple illustration of generating functions in graph theory.
So far, the matching polynomial seems to be the best. But I want something bit richer; at least a derivative should ...

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334 views

### A generalization of binary Krawtchouk polynomials

I am looking for orthogonal polynomials $P_{d,m,n}$ that have their values at integers $i$ specified by the following generating function
$(1-z)^i (1+z+z^2+ \ldots + z^d)^{n-i} = \sum_{m=0}^{i+d(n-i)}...

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196 views

### Asymptotic behavior of coefficient of a complicated generating function

Given a generating function
$$H(z)=\sum_{m=0}^\infty h_m z^m =\frac{P F(z)}{1-(1-P)F(z)}$$
where $0<P\leq 1$ and
$$F(z)=1-\frac{\pi z\sqrt{c}}{6\boldsymbol{\mathrm{K}}(k')} $$
in which $\...

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146 views

### Inverse of partial sums of general harmonic series

I would like to understand better the scaling of the following summation as a function of $r$ and $p > 1$:
$$
S_r(p) := \sum_{m=1}^{r} \left( \sum_{k=r-m+1}^{r} \left( \frac{k^q}{\sum_{k'=1}^{r}...

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766 views

### Proofs of some combinatorial identities

Just wondering if anyone knows any references in the literature to bijections corresponding to the following simple generating function identities. Let $B(z)=\dfrac{1}{\sqrt{1-4z}}$ and $C(z)=\dfrac{1-...

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711 views

### Ramanujan's Lost Notebook page 1 first equation and OEIS sequence A260195

In the 1988 Narosa edition of Ramanujan's The Lost Notebook and Other Unpublished Papers, on the first line of page 1 is the following:
$$ \Big(1+\frac1a\Big) \Bigg\{\frac{1}{(1-aq)(1-q/a)}+\frac{q(1+...

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112 views

### Holonomic generating function

Let $\lambda$ denote a hook of size $d$ and $c(\Box)$ the content of $ \Box \in \lambda $. Let $ \text{Hooks}(d) $ be the set of hooks with $d$ boxes. Define
\begin{align}
B(d)&=
\frac{1}{d!h^{d-1}...

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320 views

### Infinite sum and product associated with the Weierstrass elliptic function [closed]

Can anyone help me figure out how the identity below was obtained?
$ \frac{1}{\sqrt{(e_1-e_3)(e_2-e_3)}} = R \prod \limits_{n=1}^{\infty} \left(1 - \frac{1}{R^{4n}} \right)^{-4}\left(1 + \frac{1}{R^{...

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66 views

### Construct generating functions of series of palindromic polynomials

I have a problem that is generating a series ($d=2,4,\ldots,20,\ldots$) of pairs of $4 d$-degree palindromic (self-reciprocal) polynomials.
The first three members ($d=2,4,6$) of the first pair are:
\...

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177 views

### Asymptotics of Generating Functions

Given a generating function $A(x)$, are there any general techniques for finding the asymptotics of the associated sequence? For example, given the generating function satisfying $A(x) = 1 + x\cdot A\...

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352 views

### What is this sequence?

This is again a question that I asked at Stack Exchange, but got no answer so far, so I am trying here.
Let:
$$ a_n=\sum_{k\ge0}(k+1) {n+2\brack k+2}(n+2)^kB_k$$
$B_k$ is the Bernoulli number. ${n\...

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92 views

### Find bivariate generating function for two-dimensional sequence

How to find generating function for triangle of squares of elements in this sequence? I. e. for
$1 + (1 + 4x)y + (1 + 9x + 16x^2)y^2 + ...$ ? It seems that ordinary approach with arithmetic ...

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92 views

### Number of occurrences of certain generators in expressions in Coxeter groups

Let $W$ be a Coxeter group (finite or infinite) with (finite) set $S$ of Coxeter generators, and let $I \subseteq S$ be some subset. If $w\in W$ then I call $m_I(w)$ the minimum total number of ...

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355 views

### Expression for infinite product

can anyone show me how
$$\displaystyle\frac{4}{R}\displaystyle\Pi_{n=1}^{\infty} \left(\frac{1+R^{-4n}}{1+R^{-4n+2}}\right)^4= \frac{1}{R}\left(1+2 \sum_{n=1}^ {\infty} \frac{1}{R^{2n(n+1)}}\...

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248 views

### Counting some binary trees with lots of extra stucture

While working on some computations on Hilbert schemes, I came across the following combinatorial problem.
Let $D(k,n)$ be the weighted number of binary trees (children are left/right) with $n$ ...

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121 views

### asymptotics for coefficients of generating functions involving logarithms

I have a generating function that has a closed form like $1/(\log(z-a)+b)$ and I would like to get asymptotics for the size of the coefficients of it.
I was going to use the methods in Chapter 5 of ...

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218 views

### Multivariate Generating Function Related to Lambert $W$ Function and Counting Trees with a Certain Property

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^{...

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252 views

### Independent families of functions on $\omega$ of size continuum

In Hausdorff's article "Über zwei Sätze von G. Fichtenholz und L. Kantorovich''(1935) one can find the (simplified) proofs of the following two theorems:
1) There are continuum many essentially ...

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101 views

### D-finiteness of Hilbert series of non-commutative invariant ring under reductive group

Let $G$ be reductive group over a field of characteristic $0$ ($GL_n$ fine for this question). Let $V$ be a linear representation of $G$. Then $G$ acts on the tensor algebra $T(V) = \bigoplus_{n \ge 0}...

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63 views

### Is there an anti-commutator analog of Zassenhaus formula?

Is anyone familiar with an anti-commutator analog Zassenhaus formula? I have been able to find the anti-commutator analog of the BCH formula
$$e^ABe^A= B + \{B,A\}+\frac{1}{2!}\{\{B,A\},A\}+ \frac{1}{...

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174 views

### Reference request: Reduced reflection length in Coxeter groups

I recently read this paper, where the authors define on page 26 what they call the reduced reflection length. For that we take a Coxeter group $G$ with Coxeter generators $S$ and transpositions $T$. ...

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98 views

### Recovering a distribution from sample averages?

I'm working on a problem where I have $n^2$ real numbers $x_{11},...,x_{nn}$, all drawn i.i.d. from the same distribution $F$. I don't observe each $x_{ij}$, but I do observe the $n$ means:
$$\bar{x}...

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211 views

### Bijective proof of formula for rooted binary forests

For $n\ge 1$, let $f(n)$ be the number of rooted complete (unordered) binary trees with $n$ leaves labeled from $1$ to $n$ ("complete binary" means that every vertex has either $0$ or $2$ children and ...

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207 views

### partition theory: meet the COP

Recall that $(a;q)_0:=1,\,(a;q)_n=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1})$ and
$(a;q)_{\infty}=(1-a)(1-aq)(1-aq^2)\cdots$. Let's introduce the following (generalized) concept.
A colored overpartition (...

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95 views

### Unrestricting The Parameters of a Functional Equation

Good evening. I am looking into methods of generalization of Bernoulli polynomials. First, define
$$\Phi_{N,k}(x)=\frac{1}{N}\sum_{j=0}^{N-1}\omega_N^{-jk}\exp\left(\omega_N^jx\right)$$
where $\...

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110 views

### Recurrence relation asymptotics

A continuation from my two previous posts:
I have got the following recurrence which describes polynomials:
$$
C_n(a) = \sum\limits_{t=0}^{n-1} \binom{n-1}{t} a^{t(n-t)} C_t(a)
$$
where $C_1(a)=C_0(...

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920 views

### A mystery sequence

This question arose from the recent one, roots of a polynomial linked to mock theta function?. Let
$$
g(x):=\sum_{k=0}^\infty x^k\prod_{j=1}^{k-1}(1 + x^j)^2\\=1+x+x^2+3 x^3+4 x^4+6 x^5+10 x^6+15 x^7+...

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176 views

### A question about integer representation as a sum of two coprime integers

It is easy to see that every natural number $n$ can be written in a unique way $n = a+b$ where $gcd(a,b)=1$, $b>a$ and $b-a$ is minimal with this property. For instance if $n$ is odd the ...

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287 views

### generalizing Wilf's conjecture: Uppuluri-Carpenter numbers

The complementary Bell numbers have the exponential generating function
$$\sum_{n\geq0}\tilde{B}_nx^n=e^{1-e^x}.$$
Herb Wilf conjectured that $\tilde{B}_n=0$ only for $n=2$. By now, there are a few ...

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160 views

### an algebra generated by some known series

Denote the e.g.f. for the number of (unordered) rooted labeled trees on $n$ nodes by
$$\Phi(x)=\sum_{n\geq1}\frac{n^{n-1}}{n!}x^n.$$
And, the related series $\Psi(x)=\sum_{n\geq1}\frac{n^n}{n!}x^n$. ...

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**1**answer

183 views

### Product of polynomial coefficients of a recurrence

A recurrence is given by
$f[0]=2x$, $f[1]=3x^3-x^2+x+1$,
$$
f[n]=(x^{2^n}+1)f[n-1]+(x^{2^n}+1)(x^{2^n-1}+1)
$$
How does the PRODUCT of the nonzero coefficients of $f[n]$ scale with $n$?

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458 views

### Total progeny of a Galton-Watson branching process - standard textbook question

While analyzing some parallel-computing related algorithm, I came across a probability distribution with a particularly nice property (at least to me), but I am unable to write it down explicitly.
...

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234 views

### Generating function for products of complex Hermite polynomials

By making use of the generating function
$$\sum_{m=0}^\infty \frac{H_m(x)}{m!} t^m=e^{-t^2 + 2xt} $$ for the real Hermite polynomials $H_m$, we get easily that
$$(*)\quad \sum_{m,n=0}^\infty \frac{u^...