The tag has no wiki summary.

learn more… | top users | synonyms

8
votes
1answer
131 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 ...
39
votes
2answers
4k views

Hirzebruch's motivation of the Todd class

In Prospects in Mathematics (AM-70), Hirzebruch gives a nice discussion of why the formal power series $f(x) = 1 + b_1 x + b_2 x^2 + \dots$ defining the Todd class must be what it is. In particular, ...
0
votes
0answers
73 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 ...
0
votes
1answer
344 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 ...
1
vote
1answer
116 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 ...
18
votes
4answers
1k views

Are the q-Catalan numbers q-holonomic?

The generating function $f(z)$ of the Catalan numbers which is characterized by $f(z)=1+zf(z)^2$ is D-finite, or holonomic, i.e. it satisfies a linear differential equation with polynomial ...
1
vote
1answer
171 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
2answers
321 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 ...
4
votes
0answers
80 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
0answers
27 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 ...
19
votes
1answer
634 views

Monomer-Dimer tatami tilings need better relationships with other math. Summary of results.

A monomer-dimer tiling of a rectangular grid with $r$ rows and $c$ columns satisfies the \emph{tatami} condition if no four tiles meet at any point. (or you can think of it as the removal of a ...
7
votes
2answers
307 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 ...
5
votes
2answers
166 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 ...
1
vote
0answers
118 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
4answers
850 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 ...
41
votes
10answers
6k views

The functional equation $f(f(x))=x+f(x)^2$

I'd like to gather information and references on the following functional equation for power series $$f(f(x))=x+f(x)^2,$$$$f(x)=\sum_{k=1}^\infty c_k x^k$$ (so $c_0=0$ is imposed). First things that ...
3
votes
1answer
249 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 ...
2
votes
2answers
327 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 ...
4
votes
1answer
245 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
1answer
58 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
0answers
91 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
1answer
245 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
0answers
125 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 ...
2
votes
1answer
460 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 ...
0
votes
0answers
97 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 ...
9
votes
3answers
1k views

Solving a general two-term combinatorial recurrence relation

What is known about explicit (not necessarily closed-form) solutions to the recurrence $$R^n_k= (\alpha n) R^{n-1}_k + (\alpha' n + \beta' k) R^{n-1} _{k-1},$$ with initial condition $R_0^0 = 1$ and ...
3
votes
0answers
117 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
0answers
111 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
2answers
98 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
1answer
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 ...
5
votes
1answer
162 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} ...
2
votes
2answers
906 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 ...
-1
votes
1answer
248 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
0answers
95 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" ...
17
votes
3answers
2k views

Need help proving that $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0$

Hello. I have been trying very hard to show that $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0$ and could not quite get anywhere. This inequality has been verified by computer ...
2
votes
0answers
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
1answer
122 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. ...
7
votes
1answer
747 views

Generating function for Random Walk Hitting Time, taking the wrong root

In a calculation of the hitting time for a Bernoulli random walk we have to calculate the hitting time $\tau(1)=\inf\{n\ge 0:S_n=1\}$ to reach $+1$ and the generating function has the recursion ...
3
votes
0answers
214 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: ...
6
votes
2answers
694 views

Closed form or/and asymptotics of a hypergeometric sum

Dear mathematicians, I am a computer scientist wandering in the deep sea of combinatorics and asymptotics to pursue a recent interest in average case analysis of algorithms. In doing so, I designed ...
2
votes
0answers
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
1answer
241 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 + ...
2
votes
2answers
276 views

Alternating sum of binomial coefficients times logarithm

Trying to find a closed form expression for the following sum, or an asymptotic expression in terms of well known functions (like the Gamma function, for instance). Let $m,n$ be positive integers ...
9
votes
2answers
837 views

Can you find linear recurrence relation for dimensions of invariant tensors?

Let $V$ be a finite dimensional highest weight representation of a (semi)-simple Lie algebra. For each $n\ge 0$ take $a_n$ to be the dimension of the space of invariant tensors in $\otimes^n V$. In ...
8
votes
2answers
1k views

Expressions involving Eulerian numbers of the second kind: trying to show $\sum_{m=0}^{n} (-1)^m(m)m!(2n-m-2)!\left\langle\left\langle n\atop m\right\rangle\right\rangle\neq0$ for even $n$.

Considering the success of a previous question involving Eulerian numbers, I thought I might throw this question into the mix. It comes from some localization computations in GW theory, but in this ...
0
votes
1answer
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
2answers
175 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
1answer
307 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
0answers
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
0answers
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 ...