All Questions
Tagged with real-analysis co.combinatorics
133 questions
106
votes
5
answers
10k
views
integral of a "sin-omial" coefficients=binomial
I find the following averaged-integral amusing and intriguing, to say the least. Is there any proof?
For any pair of integers $n\geq k\geq0$, we have
$$\frac1{\pi}\int_0^{\pi}\frac{\sin^n(x)}{\...
- 43.2k
30
votes
2
answers
1k
views
Minimum number of $|\cdot|$ operations necessary to express $\max$
For two variables, their maximum
$\max\{x_1,x_2\}$ can be expressed using one $|\cdot|$ operation:
$$
\max\{x_1,x_2\} = \frac12(x_1+x_2+|x_1-x_2|).
$$
For $3$ variables, it seems fairly clear that ...
- 6,541
27
votes
3
answers
2k
views
Kasteleyn's formula for domino tilings generalized?
It seems a marvel when a bunch of irrational numbers "conspire" to become rational, even better an integer. An elementary example is $\prod_{j=1}^n4\cos^2\left(\pi j/(2n+1)\right)=1$.
Kasteleyn's ...
- 43.2k
26
votes
4
answers
2k
views
$\binom{x}{2}+\binom{x}{4}+\cdots+\binom{x}{2u}$ is a convex function on $[0,+\infty)$?
Let $f(x)=\binom{x}{2}+\binom{x}{4}+\cdots+\binom{x}{2u}$, where $u\in\mathbb{Z}^+$ and $\binom{x}{l}=\frac{x(x-1)\dots(x-l+1)}{l!}$ for all $l\in\mathbb{Z}^+$.
Then can we prove $f(x)$ is a convex ...
- 271
23
votes
4
answers
2k
views
Identity for an infinite product
Here is an experimental "result" exhibiting the difference of two (formal) infinite products that "almost factorizes".
QUESTION. Is this true?
$$\prod_{n\geq1}(1+x^{2n-1})^{24} - \...
- 43.2k
21
votes
2
answers
2k
views
Real rootedness of a polynomial
Let's consider $m$ and $n$ arbitrary positive integers, with $m\leq n$, and the polynomial given by:
$$ P_{m,n}(t) := \sum_{j=0}^m \binom{m}{j}\binom{n}{j} t^j$$
I've found with Sage that for every $...
- 1,889
20
votes
3
answers
1k
views
mixing convex and concave for convexity
Let $n\in\mathbb{N}$ and $0<x<1$ be a real number. Is the following a convex function of $x$?
$$G_n(x)=\log\left(\frac{(1+x^{4n+1})(1+x^{4n-1})(1+x^{2n})(1-x^{2n+1})}{(1+x^{2n+1})(1-x^{2n+2})}\...
- 43.2k
19
votes
0
answers
775
views
A Linear Order from AP Calculus
In teaching my calculus students about limits and function domination, we ran into the class of functions
$$\Theta=\{x^\alpha (\ln{x})^\beta\}_{(\alpha,\beta)\in\mathbb{R}^2}$$
Suppose we say that $...
- 433
16
votes
6
answers
2k
views
Alternative proofs sought after for a certain identity
Here is an identity for which I outlined two different arguments. I'm collecting further alternative proofs, so
QUESTION. can you provide another verification for the problem below?
Problem. Prove ...
- 43.2k
14
votes
2
answers
873
views
"sinc'n determinant"
The function $\text{sinc}(x)=\frac{\sin x}x$ permeates mathematics and physics in several aspects, and it carries multiple presentations/formulations. My interest is to inject yet another one of such.
...
- 43.2k
14
votes
1
answer
481
views
A question on a real sequence
Let $\{a_n\}_{n\ge1}$ be a real sequence that decays faster than any algebraic speed, that is, $\lim_{n\to \infty} n^pa_n = 0$ for every positive integer $p$. Assume that $$\sum_{n\ge 1}(n+1)^kn^ka_n =...
- 903
13
votes
2
answers
1k
views
On Hamkins' answer to a problem by Michael Hardy
Based on a post by Michael Hardy and Hamkins' answer to it Andreas Blass, Will Brian, Joel Hamkins, Michael Hardy and Paul Larson introduced a new cardinal characteristic of the continuum $\mathfrak{...
- 2,381
12
votes
1
answer
596
views
Equality of two $q$-series. Proof?
Recall the notation $(z;q)_n=(1-z)(1-zq)(1-zq^2)\cdots(1-zq^{n-1})$.
My earlier MO question did not find enough interest or yield an answer. Perhaps the modulo $2$ part might have thrown people off. ...
- 43.2k
12
votes
1
answer
525
views
An inequality about unit vector orthogonal to $(1,1,...,1)$
Does there exist a constant $\alpha>0$ such that the following holds?
$$\liminf_{n\to\infty}\inf_{x\in\mathbb{R}^n, \sum_{i=1}^nx_i^2=1, \sum_{i=1}^nx_i=0}\frac{\sum_{i<j, |i-j|\leq\frac{n}{4}}(...
- 1,495
12
votes
1
answer
933
views
Real-rootedness, interlacing, root-bounds of a sequence of polynomials
Problem: the number $a(n,k)$ is defined by the following recurrence
\begin{equation}
a(n,k)=(k+1)(k+2)\, a(n-1, k)+\frac{(k+1)(k+2)(k+3)}{k} \,a(n-1, k-1),
\end{equation}
with $a(1,1)=1$ and $a(n,k)=0$...
- 459
11
votes
6
answers
872
views
A question on the real root of a polynomial
For $n\geq 1$, given a polynomial
\begin{equation*}
\begin{aligned}
f(x)=&\frac{2+(x+3)\sqrt{-x}}{2(x+4)}(\sqrt{-x})^n+\frac{2-(x+3)\sqrt{-x}}{2(x+4)}(-\sqrt{-x})^n \\
&+\frac{x+2+\...
- 145
11
votes
2
answers
425
views
Maximization of a cubic form over the $14$-dimensional sphere
For any integers $i$ and $j$ such as $1\le i<j\le6$, let $x_{ij}$ be a nonnegative real number.
Is it true that, given the condition
$$\sum_{1\le i<j\le6}x_{ij}^2=1,$$
the sum
$$\sum_{1\le i<...
- 128k
11
votes
2
answers
539
views
Reference request: A multidimensional generalization of the fundamental theorem of calculus
$\newcommand\R{\mathbb R}$Let $f\colon\R^p\to\R$ be a continuous function. For $u=(u_1,\dots,u_p)$ and $v=(v_1,\dots,v_p)$ in $\R^p$, let
$[u,v]:=\prod_{r=1}^p[u_r,v_r]$;
$u\wedge v:=\big(\min(u_1,v_1)...
- 128k
10
votes
0
answers
287
views
Coefficients of polynomials vs trigonometric product
Let's consider the family of sequences of coefficients in the expansion
$$\prod_{i=0}^{n-1}(1+x^{3^i}+x^{3^{i+1}})=\sum_{k\geq0}a_n(k)\, x^k.$$
Remark. Evidently, the RHS is a finite sum.
Here is a ...
- 43.2k
9
votes
2
answers
354
views
Asymptotics of a quadratic recursion
Consider the sequence defined by
\begin{align}
c_0 &{}= 1 \\
c_n &{}= 2\,n\,c_{n-1}-\frac{1}{2}\sum_{m=1}^{n-1}c_m\,c_{n-m}.
\end{align}
How can you prove that it has the following asymptotics ...
9
votes
2
answers
440
views
How to prove this sum involving powers of cosec is an integer?
It is claimed that the following function produces only integer values for all integer $m \geq 1$, $N \geq 2$.
$F(m,N)=\frac{N^m}{2^m}\displaystyle \sum_{j=1}^{N-1} \operatorname{cosec} ^{2m}\left(\...
- 201
9
votes
0
answers
180
views
Infinite series identities in search of a proof
This comes in relation to the Fishburn numbers.
I stumbled on the following relation for which I ask a proof if true.
Let $Q_i(z):=1-(1-t)^{i-1}(1-zt)$. Then
$$\sum_{n=0}^{\infty}\frac{(n+1)zt}{...
- 43.2k
8
votes
1
answer
783
views
Real-rooted polynomials
I proposed this question at MO which was resolved neatly by Gerald Edgar in the form
$$
u_n(x) =
{2}^{n-1}\prod _{k=0}^{n-1}(2x+2k+1)
-{2\,n-1\choose n-1}\prod _{k=0}^{n-1}(x+k).$$
Now that we ...
- 43.2k
8
votes
1
answer
460
views
Real-rootedness of some polynomials
Denote the unsigned Stirling numbers of the first kind by $s(n,j)$.
Question. Is it true that the polynomials
$$P_n(x)=\sum_{j\geq0}s(n,j)\binom{x}j$$
have only real roots?
Note. Obviously, the ...
- 43.2k
8
votes
2
answers
1k
views
does this sum have a limit?
Define the sequence given by the finite sum
$$a_n:=\sum_{k=2}^{n+1}\binom{2k}k\binom{n+1}k\frac{k-1}{2^k\binom{4n}k}.$$
Questions.
(1) Is $0<a_n<1$?
(2) Does the limit $\lim_{n\rightarrow\infty}...
- 43.2k
8
votes
2
answers
323
views
Matrix rescaling increases lowest eigenvalue?
Consider the set $\mathbf{N}:=\left\{1,2,....,N \right\}$ and let $$\mathbf M:=\left\{ M_i; M_i \subset \mathbf N \text{ such that } \left\lvert M_i \right\rvert=2 \text{ or }\left\lvert M_i \right\...
- 225
7
votes
1
answer
226
views
Unusual isoperimetry and maximizing the measure of unions of translates of a set
Let me state a standard result first. Let a $A\subset \mathbb{R}^d$ be a set of fixed volume. Define $A_t$ to be the set of all points at distance at most $t$ from $A$. Then the volume of $A_t$ is ...
- 2,288
7
votes
4
answers
1k
views
The non-convergence of f(f(x))=exp(x)-1 and labeled rooted trees
This question is closely related to MO f(f(x))=exp(x)-1 and other functions “just in the middle” between linear and exponential. Consider $e^{e^x-1}$, this is the generating function of the Bell ...
user37691
7
votes
1
answer
321
views
Taylor's polynomials and loss of real roots
Real-rootedness, log-concavity, and unimodality are intertwined properties. It's in this light that I was prompted to ask the question below.
Suppose the roots of a polynomial $p(x)$ are all real and ...
- 43.2k
7
votes
1
answer
463
views
Boundedness of total current in electrical network
Consider the following symmetric matrix (adjacency matrix):
$$A=(a_{ij})_{1\leq i,j\leq n}$$
such that $a_{ij}=a_{ji}, a_{ii}=0$ and $a_{ij}=0$ for $|i-j|\geq k$ where $k\geq3$. We also have $1\leq a_{...
- 1,495
7
votes
0
answers
203
views
Permutations which change the value of a convergent series
I'm interested in the following combinatorial problem: What is a necessary and sufficent condition on a permutation $\sigma : \mathbb{N} \rightarrow \mathbb{N}$, so that there exist a summable ...
- 71
6
votes
2
answers
755
views
Prove positivity of a binomial sum
Some problems appear easy on the face of it, but perhaps they are not. Here is an instance of a certain calculation which is slightly reformulated from its original encounter in a current work. I have ...
- 43.2k
6
votes
3
answers
1k
views
Is there an entropy proof for bounding a weighted sum of binomial coefficients?
Given a probability $p \in (0,1)$ and parameter $\alpha \in (0,1)$, is there an entropy-based proof which yields a good upper bound for the sum
$$\sum_{\ell = 0}^{\alpha n} \binom{n}{\ell}p^\ell(1-p)^{...
- 557
6
votes
1
answer
331
views
A combinatorial identity involving increasing functions from $\{1, \dots, n\}$ to itself
This is related to the post An order statistics problem with some interesting geometry. The following identity arose in the context of the problem.
Fix an integer $N \geq 2$. Let $\mathcal S_N^+$ ...
- 6,155
6
votes
2
answers
2k
views
Distribution of $\max_{n \ge 0} S_n$, random walk
Say we have a random walk that is a nearest neighbor random walk on the integers where at each step the probability of moving one step to the right is $p$ and the probability of moving one step to the ...
- 61
6
votes
3
answers
554
views
Computing the volume of a simplex-like object with constraints
For any $n \geq 2$, let
$$D_n [r , (a_1, b_1 ) , \ldots , (a_n, b_n) ] =
\{ (x_1 , \ldots , x_n ) \in \mathbb R^n \mid
\sum_i x_i = r \mbox{ and } b_i \geq x_i \geq a_i \, \forall i \},$$
where $r \...
- 61
6
votes
1
answer
274
views
Limits (growth rates) of power series coefficients
Take two positive integers $m$ and $n$ and consider the rational function
$$G_{m,n}(x,t)=\frac{d}{dx}\left(\frac1{(1-x^m)(1-tx^n)}\right)$$
and the corresponding Taylor expansion as
$$G_{m,n}(x,t)=u_0(...
- 43.2k
6
votes
1
answer
411
views
Interesting behaviour of binomial coefficients
Let $\binom{n}{k}:=\frac{\Gamma(n+1)}{\Gamma(k+1)\Gamma(1-k+n)}$ be the generalized binomial coefficient then I noticed by playing around with Mathematica that the function $f:[0,n/2] \rightarrow \...
- 63
6
votes
1
answer
290
views
What is the growth rate of the products of binomial coefficients?
Question 1: Are the following empirically observed relationships true
$$
{n \choose 1^a}{n \choose 2^a}{n \choose 3^a}\cdots {n \choose m^a}
\sim \exp\bigg(\frac{2n^{1 + \frac{1}{a}}}{a+3}\bigg)
$$
...
- 5,470
6
votes
1
answer
308
views
Operation preserving log-concavity of sequences
Here a log-concave sequence $(a_0,a_1,a_2,\ldots)$ is a sequence of positive real numbers such that $a_i^2 \geq a_{i-1}a_{i+1}$ for each $i\geq 1$. These are pervasive within mathematics.
A polynomial ...
- 1,889
5
votes
3
answers
5k
views
Zeros of "exponential" function
Define ${f}_{i}(x) = \sum_{j=1}^{i} (-1)^{i-j}{i \choose j}j^x$, where $i=1,2,3,...$ and $x \in \mathbb{R}$.
For integer $x \geq i$, ${f}_{i}(x)$ reduces to ${f}_{i}(x)=i!S(x,i)$, where $S(x,i)$ is ...
- 2,619
5
votes
1
answer
258
views
Dimension reduction for non-negativity of elementary symmetric polynomials
Fix integers $1 \leq k \leq n$ and suppose $\mathbf{x} \in \mathbb{R}^n$ is such that $e_j(x_1,x_2,\ldots,x_n) \geq 0$ for all $1 \leq j \leq k$, where $e_j$ is the $j$-th elementary symmetric ...
- 6,351
5
votes
1
answer
333
views
Do monotone functions on the interval have an "Alexander duality" property?
Let $X,Y$ be two copies of the unit interval $[0,1]$. Consider functions $X\rightarrow Y$ and $Y\rightarrow X$ both as subsets of the cartesian product $X\times Y$. (More precisely: identify a ...
- 2,851
5
votes
1
answer
215
views
Smoothening a measure, II
There is an almost invisible, but significant difference between the question below and that recently answered by Boris Bukh.
Given a probability measure $\mu$ supported on a finite set $S\subset{\...
- 23k
5
votes
1
answer
425
views
"Interlacing property" of certain polynomials
I posted this question on MO which was quickly and decidedly answered by Noam D. Elkies.
Once more referring to the same set of polynomials
$$u_n(x) =
{2}^{n-1}\prod _{k=0}^{n-1}(2x+2k+1)
-{2\,n-1\...
- 43.2k
5
votes
1
answer
2k
views
Sum of multinomial coefficients (even distribution)
By multinomial expansion formula, we know that $$ \sum_{p_1 + \cdots + p_k = r} \binom{r}{p_1,\ldots,p_k} = k^r, $$ where the multinomial coefficient is defined by $ \binom{r}{p_1, \ldots, p_k} := \...
user83150
5
votes
0
answers
190
views
Number of discrete Lipschitz functions with given Lipschitz constant
Fix $T, K, N \in \mathbb Z_+$. How many distinct Lipschitz functions $f: \{0, \dots, T\} \to \mathbb Z$ are there with Lipschitz constant $K$, and supremum norm at most $N$ satisfying $f(0) = 0$?
In ...
- 6,155
5
votes
0
answers
167
views
Bounding elementary symmetric polynomials away from zero
Let $2 \leq m \leq n$ be integers and let $\mathbf{x} \in \mathbb{R}^n$ (importantly, I am not assuming that the entries of $\mathbf{x}$ are non-negative). The elementary symmetric polynomials are ...
- 6,351
5
votes
0
answers
107
views
Generalized Puiseux series for diagonal reflections of the curves $y = \frac{x}{(1-ax)(1-bx)^m}$
Reflection of the curve $y = f_m(x) = \frac{x}{(1-ax)(1-bx)^m}$ through the diagonal line $y=x$ in the $xy$-plane can be regarded as local compositional inversion of the curve $y=f_m(x)$. ($x,y,a,b$ ...
- 10.5k
5
votes
0
answers
170
views
operation on Ord., Exp., Dri. generating functions
The ordinary, exponential and Dirichlet generating functions for a sequence $\{a_n\}_{n\geq0}$ are given (at least on the formal side), respectively, by
$$F(x)=\sum_{n\geq0}a_nx^n, \qquad E(x)=\sum_{n\...
- 43.2k