The generating-functions tag has no wiki summary.

**1**

vote

**1**answer

47 views

### Proof of closed walk generating function identity

In 'Spectral Conditions for the Reconstructibility of a Graph' Godsil and McKay give a short proof of an identity (Lemma 2.1) that relates the generating function for the number of closed walks ...

**8**

votes

**1**answer

173 views

### Higher moments of information and Renyi entropy

For a given discrete probability distribution, Shannon entropy can be though as an expectation value $\langle - \log p \rangle$ (see also: What is entropy, really?, What is the role of the logarithm ...

**0**

votes

**0**answers

80 views

### Help solving a recurrence relation

For solving a related probability problem, I need to solve the following recurrence relation:-
$q(r,b,L)=\frac{b}{r+b}q(r,b-1,L)+\sum_{k=1}^{L}\frac{r}{r+b}\times\cdots \times ...

**1**

vote

**1**answer

176 views

### Name for series $\sum f_n x^n / (n! (n+k)!)$

Let $(f_n)_{n\ge0}$ be a real sequence. Then $\sum f_n {x^n \over n!}$ is called the exponential generating function of $(f_n)$.
Let $k\ge0$ be a nonnegative integer. If we add another factorial ...

**4**

votes

**0**answers

82 views

### Prove a complicated function (in epidemic spreading search) to be convex

When analyse epidemic spreading, I came across to prove that a complicated function $f(x)$ is convex when $0 \leq x \leq 1$.
\begin{equation}
f(x)=\frac{b_1g'(x) f_1(x)^{n-2}+g'(1) \gamma}{g'(1) ( ...

**0**

votes

**0**answers

30 views

### Coupled recurrence relations for generating functions, involving squared arguments

I'm trying to find a solution for the following system of three equations in terms of three bivariate generating functions.
$G(x,y)=b(x,y) \cdot \left( G(x,y^2) + I(x,y^2) \right) +y ;$
$H(x,y)=x ...

**4**

votes

**2**answers

326 views

### An interesting calculation of derivative

I was trying to get the probability distribution $p(n)$ from a generating function $G(s)$ like this:
$G(s) = e^{a(s-1)^2}=\sum s^np(n)$
I need first to do Maclaurin expansion of the exponential and ...

**7**

votes

**2**answers

311 views

### What is the number of noncrossing acyclic digraphs?

A noncrossing graph on $n$ vertices is a graph drawn on $n$ points numbered from $1$ to $n$ in counter-clockwise order on a circle such that the edges lie entirely within the circle and do not cross ...

**1**

vote

**0**answers

120 views

### Solving a recurrence (with the form of a convolution) involving binomial coefficients

While dealing with a problem related to intersection of hyperplanes I have come across the following recurrence to obtain the values of $K_{j}$
\begin{array}{cccccccccc}
1 & = & ...

**9**

votes

**4**answers

875 views

### Ordinary Generating Function for Bell Numbers

In the OEIS entry for Bell numbers, there appears a generating function
$$\sum_{k=0}^\infty B_k t^k = \sum_{r=0}^\infty \prod_{i=1}^r \frac{t}{1-it}$$
However, I could not locate any proof of ...

**3**

votes

**1**answer

254 views

### Is there a nice way to write the generating function obtained by taking the quadratic coefficients of another one?

Suppose that you have a generating function
$$
f(q) = \sum_{k=0}^\infty a_k q^k
$$
It's not too hard to obtain the generating function
$$
f_{n,m}(q) = \sum_{k=0}^\infty a_{nk + m}q^k
$$
by taking a ...

**4**

votes

**1**answer

246 views

### Examples of specializations of elementary symmetric polynomials

Let $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ be a set of $n$
indeterminates. The $h^{th}$elementary symmetric polynomial is the
sum of all monomials with $h$ factors
\begin{eqnarray*}
...

**1**

vote

**1**answer

122 views

### Matrix Generator for M/M/1 Queue Waiting Time Distribution

I "believe" that generator, $\bf W$, of the waiting time distribution for the M/M/1 queue is given by the following (I'm not sure if this is even correct):
${\bf W} =\left( \begin{array}{ccccc}
0 ...

**1**

vote

**1**answer

60 views

### Approaches to implicitly defining generating function

First,every language in Chomsky hierarchy(or c.e.language) corresponds to a generating function,the set of the functions is GF,now,a question : is every generating function with integral coefficient ...

**0**

votes

**0**answers

94 views

### Estimating when does a certain binomial sum exceed an upper bound

Given a fixed integer $n > 0$ and $0 \le m \le n$ let us define the numbers
$$f_{n,m} = \sum_{i=\lfloor m/2 \rfloor}^m {n-2i \choose n - m -i}{i+1 \choose m - i +1}.$$
For example $f_{n,0} = ...

**10**

votes

**1**answer

249 views

### Laurent polynomials associated to partitions and a $Q$-deformation of $\sigma(d)$

Let $\alpha \vdash d$ be a partition of $d$, i.e. $\alpha = (\alpha_1 \geq \alpha_2 \geq …\geq \alpha_l)$, where $\sum_k \alpha_k = d$. Define a Laurent polynomial in $Q$ as follows:
$$
P_\alpha(Q) = ...

**1**

vote

**0**answers

127 views

### How to prove this identity? (Perhaps related to partition) [closed]

How to prove this identity?
$ \sum_{n\ge 0} \frac{x^{n^2}}{(1-x)(1-x^2)\cdots(1-x^n)}= \frac{1}{\prod_{k \ge 0}(1-x^{5k+1})(1-x^{5k+4})}$
I will appreciate it a lot if a solution using method ...

**0**

votes

**0**answers

99 views

### Are there generating functions of rational or integral solutions of Diophantine equation that

As we know,there are generating functions for c.e.languages which are some retricted rational or algebraic or transcendental functions dependent on the class of languages like regular ...

**3**

votes

**0**answers

122 views

### Asymptotics and combinatorics

Wright's expansion of
$$
(1-z)^b\exp[A/(1-z)^c], \text{for } A > 0, 0<c<1\tag{1}
$$
is, in the words of the late, great Mark Kac "well known to those that know it well".
(See, for example, ...

**1**

vote

**0**answers

113 views

### Is there inverse FFT algorithm for Fourier transform of a integer-valued random variable?

In many applications, it is possible to derive an explicit expression for the
Fourier transform of a random variable $X$
$$\varphi (\theta ) = \sum\limits_{n = 0}^\infty {{p_n}} {e^{in\theta }}$$
...

**0**

votes

**1**answer

412 views

### Probability generating function zero implies random variable is infinite

Let $V$ be a random variable supported on the nonnegative integers (including $\infty$) and $f(x) = \mathbf E x^V$ be the probability generating function. In our model $V$ is the number of visits to ...

**0**

votes

**2**answers

106 views

### Hypergeometric sum specific value

How to show?
$${}_2F_1(1,1;\frac{1}{2}, \frac{1}{2}) = 2 + \frac{\pi}{2} $$
It numerically is very close, came up when evaluating:
$$ \frac{1}{1} + \frac{1 \times 2}{1 \times 3} + \frac{1 \times 2 ...

**2**

votes

**1**answer

254 views

### An infinite product: combinatorial interpretation

It is an undergraduate exercise to show that the generating function for the sequence of unrestricted integer partitions $p(n)$ is the celebrated infinite product
...

**2**

votes

**2**answers

916 views

### Number of 1 in binary representation of n

Let $1(n)$ be the number of digits $1$ in binary representation of number $n$.
For example, $13=1101_2$ so $1(13)=3\\$
Is there explicit form of $\,\,\sum{1(i)x^i} $?
I checked OEIS and didn't find ...

**5**

votes

**1**answer

165 views

### The number of partitions between two fixed partitions

Given two partitions M and N, with $M_i \leq N_i$ for all $1\leq i\leq \max\{l(M),l(N)\}$. Is there a formula for the generating function: $$\sum_{\lambda: M_i\leq \lambda_i\leq N_i} ...

**-1**

votes

**1**answer

249 views

### Does anyone recognize this generating function [closed]

$a_1=1, a_2=1, a_3=3, a_4=15, a_5=105$
Reccurence formula is
$a_{k+1}=\sum\limits_{\lambda_1+\lambda_2+\ldots+\lambda_s=k,\ \lambda_i\geq1} a_{\lambda_1}a_{\lambda_2}...a_{\lambda_s}{k \choose ...

**0**

votes

**0**answers

97 views

### What are the properties of this linear operator?

Suppose $f(x)$ is a function which satisfies the following condition:
$$f(x)=\sum_{k=0}^\infty G(2k)\frac{x^{2k}}{(2k)!}$$
Where the generating function $G(x)$ is a "natural" or "discrete-analytic" ...

**5**

votes

**2**answers

168 views

### Quantities whose generating functions are symmetric

This is inspired by an old Putnam problem from 2005, and a solution given by Professor Greg Martin (a Professor of Mathematics at the University of British Columbia, also a user on MO). The question ...

**2**

votes

**0**answers

76 views

### Lagrangean equations for the generating function of quadrangulations

Let $M(z)$ be the generating function of edge-rooted connected quadrangulations, with $z$ marking the number of edges. I derived the following Lagrangean equations for $M(z)$:
$$M(z) = ...

**1**

vote

**1**answer

124 views

### Truncated sums of symmetric polynomials; reference request for an algebraic derivation

EDIT: This is a case of being too wrapped up in a formulation
($e_j,p_i,$ and the like) to try something simple. It did not
occur to me to pull exp to the outside in the weeks I have
stared at this. ...

**2**

votes

**2**answers

329 views

### Asymptotic behaviour of sequence

I am interested in the sequence
$$a(n)=\sum_{k=0}^n {p(n-k) \choose k}$$
where $p(n)$ is a polynomial equation.
When $p(n)=n$ this reduces to the Fibonacci sequence, but what about when $p(n)$ is ...

**3**

votes

**0**answers

218 views

### Generating function for the characteristic function of prime numbers

What do we know about the generating function of $\chi(n)$ (A010051)
$$
f(x) = \sum_{n=0}^\infty \chi(n)x^n = \sum_{p\text{ prime}} x^p
$$
for $\chi(n)$ the characteristic function of the primes:
...

**2**

votes

**0**answers

64 views

### Rank generating functions on a graded poset and a linear extension of it

Let $A$ be a possibly infinite ensemble and let $\leq$ be a partial order between elements of $A$. We denote $P_{\leq}=(A,\leq)$ the corresponding poset. Furthermore, we suppose that $P$ is graded, ...

**6**

votes

**1**answer

242 views

### Does the following operation on modular forms yield something modular?

Let $f(z) = \sum_{n=0}^\infty a_nq^n$ be the fourier expansion of a (quasi-)modular form (with $q = e^{2\pi i z}$). Consider the following related functions:
$$f_{m,k}(z) = \sum_{n=0}^\infty a_{mn + ...

**0**

votes

**1**answer

165 views

### Given a generating function with “zeros”, can one derive the function for ONLY the “zeros”?

If I have an generating function (GF) --- ordinary or exponential --- defining a series with at least one coefficient equal to zero, is there a general method to find the "inverse GF", i.e., the GF ...

**2**

votes

**1**answer

465 views

### Recurrence relation for coefficients of product of generating functions for partition numbers

It is well known that
$$Z(x,q) = \prod_{n=1}^\infty\frac{1}{(1-xq^n)} = \sum_{m=0}^\infty\sum_{k=0}^m p_{m,k}x^kq^m$$
is the generating function for the number $p_{m,k}$ of partitions of $m$ in ...

**2**

votes

**2**answers

176 views

### Generating function of factorable binary words

A word $w$ on the alphabet $A := \{0, 1\}$ is factorable if
\begin{equation}
w = u^k \mbox{ where } u \in A^* \mbox{ and } k \geq 2.
\end{equation}
Let $L$ be the language of the set of ...

**6**

votes

**1**answer

309 views

### Generating functions with all non-zero coefficients equal to one

Inspired by this question, I have been wondering if there are any useful generating functions with all non-zero coefficients equal to one. Obviously, the trivial generating function $\frac{1}{1-x}$ ...

**1**

vote

**0**answers

82 views

### Generating series of free PROs

Let
\begin{equation}
G := \biguplus_{p \geq 0} \: \biguplus_{q \geq 0} G(p, q)
\end{equation}
be a bigraded set of generators and $\mathcal{F}(G)$ be the free PRO generated by $G$ (see [1] for a net ...

**0**

votes

**0**answers

109 views

### Using extended group rings for combinatorial generating functions

In work of mine recently, I have come to investigate generalised recurrence relations. The generalisation I have in mind is where, instead of natural numbers or integers, the recurrence is over some ...

**1**

vote

**1**answer

98 views

### Approximating rational generating functions

Suppose we have a initial segment $x_1,\ldots,x_N$ (for reasonably large $N$) of a sequence of natural numbers $(x_i)$. We have reason to believe the generating function $\sum_{i=0}^\infty x_iX^i$ is ...

**8**

votes

**4**answers

523 views

### Partitions-Sum of divisors identity

A few years ago I first read about the marvelous Euler identity:
$\sum_{n\in\mathbb{N}}p(n)z^n=\prod_{k\geq1}\frac{1}{1-z^k}$,
where $p(n)$ is the number of partitions of $n$ ($p(0)=1$ by ...

**0**

votes

**0**answers

192 views

### functional equation, how to solve

Suppose $x_i, y_i \in \mathbb{R}^n$, and $F,G: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$.
$$F(x, y) = \frac{x\circ Ay}{x^TAy}$$
$$G(x, y) = \frac{x\circ By}{x^TBy}$$
where $A$ and $B$ are ...

**9**

votes

**2**answers

446 views

### Series defined by a fixed-point functional equation

Description
I my work, I met fixed-point functional equations of a very specific form. Since I am not expert in the domain of functional equations, I have not pushed the study of these very far. Here ...

**3**

votes

**1**answer

237 views

### Repertory of the different sorts of operads

Many different types of operads have emerged in recent years (symmetric, shuffle, cyclic, anticyclic, coloured, etc.).
I would like, for any of these, list the following data:
Description of the ...

**0**

votes

**0**answers

94 views

### Theorem Leads for tied-down random walk

scribd.com/doc/87930409/18/Leads-for-tied-down-random-walk [Theorem 3.7, page 38]
Could you explain me the last equality in the proof? I mean this:
$$\frac{2[\sqrt{1 - s^2t^2} - ...

**1**

vote

**0**answers

186 views

### Generalization of Lagrange inversion with “skewed” formal parameter

I am interested in obtaining an analog of the Lagrange inversion formula, starting from a generalization of the implicit equation. Ordinary Lagrange reversion, as I am familiar with it, starts with ...

**8**

votes

**0**answers

364 views

### Generalization of Cauchy's identity

Let $ s_{\lambda} $ be the schur function associated to the partition $ \lambda $.
Cauchy's identity (as in Macdonald) states that
$$
\sum_{\lambda} s_{\lambda}(X)s_{\lambda}(Y) = ...

**14**

votes

**2**answers

432 views

### Combinatorial meaning of the functional equation for logarithm

If we set $\exp(x)=\sum x^k/k!$, then $\exp(x+y)=\exp(x)\cdot \exp(y)$. In terms of coefficients it means that $(x+y)^n=\sum \frac{n!}{k!(n-k)!} x^ky^{n-k}$, i.e. just binomial expansion.
Now ...

**10**

votes

**1**answer

345 views

### Elephant populations (and Dyck words)

Hello,
I'm relatively new to this forum so apologies if I have tagged my
question incorrectly.
I have been in contact with a wildlife biologist recently concerning
counting elephant populations and ...