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7 votes
0 answers
218 views

Analytic continuation of Dixon's identity

Many well-known combinatorial identities has an analytic version. For example, the following identities $$ 2^n = \sum_{k=0}^n \binom{n}{k} $$ $$ \binom{2n}{n} = \sum_{k=1}^n \binom{n}{k}^2 $$ can be ...
Pluviophile's user avatar
  • 1,608
12 votes
3 answers
784 views

Can the equation $1+z+z^2=z^n$ for natural $n$ have multiple complex roots $z$?

The question is stated in the title of this post. It is easy to see that, if $z$ is a multiple root of $p_n(z):=1+z+z^2-z^n$, then $(n-2)z^2+(n-1)z+n=0$, so that we can successively express $z^2,\dots,...
Iosif Pinelis's user avatar
0 votes
1 answer
129 views

Seeking an integral formulation for an algebraic function

While working with a generating function for the Catalan numbers, I came across the integral representation $$\frac1{1+\sqrt{1-4x}}=\frac1{2\pi}\int_0^{\infty}\frac{\sqrt{t}}{(t+\frac14)(t-x+\frac14)}\...
T. Amdeberhan's user avatar
3 votes
0 answers
185 views

"Circulant-Vandermonde" matrix: in search of a formula

An $n\times n$ circulant matrix $\mathbf{X}_n$ has the form \begin{align} \mathbf{X}_n= \begin{bmatrix} x_1 & x_2 & \cdots & x_{n-1} & x_n \\ x_2 & x_3 & \cdots & x_n&...
T. Amdeberhan's user avatar
2 votes
1 answer
177 views

Another combinatorial identity

Is it true that $$\sum_{r=0}^p \sum_{i=0}^r a_{n,p,r,i}=0$$ for all natural $n$ and all natural $p\ge2n$, where $$a_{n,p,r,i}:=\frac{(-1)^r (n+p-r-1)! (n p-i (r-i))}{i!(r-i)! (n-i)! (p-r+i)! (n-r+i)! ...
Iosif Pinelis's user avatar
12 votes
2 answers
1k views

Short research articles

I am a masters student. I am interested in short articles which have counter examples and very few references. I want to write a short and interesting article. For example; One of the best known ...
1 vote
1 answer
218 views

Generating series of rational$\times \exp($rational$)$

It is known that rational functions $f\in \mathbb C(x)$, $0$ not a pole, are the sum of generating series $\sum_{n\geq 0} a_nx^n$ where $(a_n)_n$ is solution of a linear recurrence with constant ...
Loïc Teyssier's user avatar
4 votes
1 answer
230 views

Poles of an integral of a meromorphic function with toric poles

Suppose I have a meromorphic function in several variables $f(x_1,\ldots,x_k,y_1,\ldots,y_m)$ and I want to integrate along the torus $T^m$ given by $|y_1|=\cdots=|y_m|=1$. It is not true in general ...
Anton Mellit's user avatar
  • 3,772
27 votes
3 answers
2k views

Kasteleyn's formula for domino tilings generalized?

It seems a marvel when a bunch of irrational numbers "conspire" to become rational, even better an integer. An elementary example is $\prod_{j=1}^n4\cos^2\left(\pi j/(2n+1)\right)=1$. Kasteleyn's ...
T. Amdeberhan's user avatar