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**3**

votes

**1**answer

177 views

### Generating function of alternating even terms in the Vandermonde Convolution

I have
\begin{equation}
G(x) = \sum_{i = 0}^{\infty} \sum_{r = 0}^{\infty} (-1)^i \binom{k}{2i} \binom{n-k}{r} x^{2i} x^{r} = \frac{1}{2} \left( (1 + x{\iota})^{k} + (1 - x \iota)^{k} \right)(1 + ...

**9**

votes

**0**answers

111 views

### types of generating functions (ordinary, exponential, ???) closed under substitution

A nice feature of ordinary and exponential generating functions is that they are closed under substitution: if $F(z)$ and $G(z)$ both have integer coefficients, then $F(G(z))$ also has integer ...

**9**

votes

**0**answers

651 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) = ...

**5**

votes

**0**answers

181 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, ...

**5**

votes

**0**answers

271 views

### When does a triangle of numbers have a zero row sum?

Suppose we have a triangle of numbers defined by the recurrence relation
$$\left| n \atop k \right| = f(n,k) \left| n-1 \atop k \right| +g(n,k) \left| n-1 \atop k-1 \right| + [n=k=0],$$
for some ...

**4**

votes

**0**answers

155 views

### Puzzling behaviour of a recursively defined sequence of functions

This question arose in connection with another problem that I described earlier in Constructing a generating function using a series with all negative and positive powers of a variable. I had certain ...

**4**

votes

**0**answers

70 views

### Is there a nice way to invert this expression?

Let us first define the Euler polynomials to be the polynomials $P_n(q)$ that satisfy
$$
\frac{qP_n(q)}{(1 - q)^{n+1}} = \Big(q\frac{d}{dq}\Big)^n\frac{q}{1 - q}.
$$
For example, $P_0(q) = P_1(q) = ...

**4**

votes

**0**answers

101 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) ( ...

**4**

votes

**0**answers

183 views

### probabilistic terminology for polynomials with positive coefficients

Given a polynomial $P(x) = p_0 + p_1 x + p_2 x^2 + ... + p_n x^n$ with non-negative coefficients, is there a standard name for (the function of $p_1,...,p_n$ equal to) the variance of an ...

**4**

votes

**0**answers

270 views

### Generating function related to 2-residues of partitions

Question
Express the following power series in two commuting variables $x,y$ as an infinite product, or a short sum of infinite products:
$$
...

**3**

votes

**0**answers

157 views

### Combination of Generating Functions

Suppose I have the following generating functions:
$$\frac{x^ke^{\left(z-\frac{1}{N}\right)x}}{N^{k-1}k!\sum_{j=0}^{N-1}w_N^{-jk}e^{\frac{w_N^jx}{N}}}=\sum_{j=0}^\infty H_{N,k,j}(z)\frac{x^j}{j!}$$
...

**3**

votes

**0**answers

124 views

### Generating function of a sequence involving reciprocals of binomial coefficients

Question: Is there a closed-form expression for the following sum
$$
F(z,k,r)=\sum_{n=0}^{r} \frac{z^n}{{n+k} \choose {k}}\label{sum}\tag{1}
$$
where $z\in\mathbb{C}$, and $r$, $k$ are non-negative ...

**3**

votes

**0**answers

125 views

### Involutions on $[0,1]$ given by power series (related to probability generating functions)

Let $A$ be a function from $[0,1]$ to $[0,1]$. $A$ is an involution if $A(A(x))=x$ for all $x\in[0,1]$.
Which involutions $A$ exist such that $A(x)=\sum_{k=0}^\infty a_k x^k$ with $a_0=1$ and ...

**3**

votes

**0**answers

59 views

### Using Mellin transform for a certain function

In short, I want to use the Mellin transform to obtain the asymptotic behavior of the sequence $D_n = \frac{ [z^n] D(z)} {C_n}$ where
$$
D(z) = \frac 1{2z}\sum_{p \ge 1}C_p \left( ...

**3**

votes

**0**answers

272 views

### Solving a doubly exponential generating function

I am analyzing the average time complexity of some algorithm on some probabilistic model, and I've come to a doubly exponential sequence for which I cannot find corresponding generating function. I ...

**3**

votes

**0**answers

318 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:
...

**3**

votes

**0**answers

185 views

### closure properties of q-differential equations

I am interested in q-differential equations of the form
$p(f(z), f(qz),\dots,f(q^kz))=0$
where $p$ is a polynomial and $k$ an nonnegative integer. I wonder about the closure properties of the class ...

**3**

votes

**0**answers

921 views

### Which Hadamard Products of Generating Functions Are Known?

The Hadamard product, Schur product, or entrywise product of two generating functions is computed as follows:
The Hadamard Product, H(x), given two generating functions f(x) and g(x) where
$$ f(x) = ...

**2**

votes

**0**answers

163 views

### Hilbert series of the weight 0 sub-algebra of the algebra of functions on GL(N)

Let $A$ be the algebra of polynomials in $N^2$ variables $x^i_j$, $i,j=1,\dots,N$. It is $\mathbb{Z}^N$ graded, with $\text{weight}(x^i_j)=e_i-e_j$. Here $(e_1,\dots,e_n)$ is the standard basis of ...

**2**

votes

**0**answers

66 views

### Inverses of probability generating functions: positivity of derivatives

Let $\mathcal{G}$ be the set of probability generating functions of random variables taking positive integer values, considered as functions on $[0,1]$.
So $G\in\mathcal{G}$ can be written ...

**2**

votes

**0**answers

87 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) = ...

**2**

votes

**0**answers

212 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 ...

**2**

votes

**0**answers

229 views

### Algebraic Dirichlet series and beyond

I wonder what the "right" notion of "algebraic Dirichlet series" might be. Here I'm thinking of formal Dirichlet series $D(s)=\sum_{n\geq 1} a_n/n^s$, say with $a_n$ being rational numbers.
I'm ...

**1**

vote

**0**answers

155 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 & = & ...

**1**

vote

**0**answers

131 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 $X$
$$\varphi (\theta ) = \sum\limits_{n = 0}^\infty {{p_n}} {e^{in\theta }}$$
...

**1**

vote

**0**answers

108 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, ...

**1**

vote

**0**answers

86 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 ...

**1**

vote

**0**answers

209 views

### An equation about generating functions and subfactorial

As I promised, I clone the problem from Math.SE to here, in order to find a solution.
Suppose $G_n(w)$ is a formal power series (really a probability generating function, see the following ...

**1**

vote

**0**answers

311 views

### Transfinite Sums Related to a Sequence

Hello,
Given a sequence $S$ indexed by the finite ordinals, a limit ordinal $\alpha$, and $k \in \mathbb{N}$, define $S_{\alpha+k}$(the extension of $S$ to $\alpha+k$) to be the sum over the products ...

**0**

votes

**0**answers

90 views

### Multidimensional recurrence relations

There are many methods of solving one-dimensional homogeneous linear recurrence relations, i.e. such of the form
$$ a_n = \sum_{k=1}^{m}\alpha_ka_{n-k}.$$
Most widespread use linear algebra or ...

**0**

votes

**0**answers

55 views

### Direct proof that a certain generating function is D-finite

Consider the set $T^{2,3}_n$ of all non-planar rooted trees with $n$ leaves labelled by $1,2,\ldots,n$ where each internal vertex can have two or three children. If we think of the binary / ternary ...

**0**

votes

**0**answers

119 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

**0**answers

110 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} = ...

**0**

votes

**0**answers

131 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" ...

**0**

votes

**0**answers

126 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 ...

**0**

votes

**0**answers

208 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 ...

**0**

votes

**0**answers

101 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} - ...

**0**

votes

**0**answers

663 views

### Any tips on finding generating functions from recurrence relations involving minimization and maximization?

Any general tips on or examples of finding interesting generating functions from recurrence relations involving minimization and maximization?
I'd imagine the case with one term of a minimization or ...