2
votes
1answer
31 views

Integer valued polynomials in two variables

The ring of integer valued polynomials, $\{ f \in \mathbb{Q}[x] : f(\mathbb{Z}) \subseteq \mathbb{Z} \}$ is fairly well-known to be generated as Abelian group by the binomial coefficients, $f_k(n) = ...
0
votes
2answers
24 views

Get polynomial function from 3 points

I need to understand how to define a polynomial function from 3 given points. Everything I found on the web so far is either too complicated or the reversed way around. (how to get points with a given ...
5
votes
7answers
293 views

If $x+\dfrac{1}{x}=5$, find the value of $x^5+\dfrac{1}{x^5}$.

If $x>0$ and $x+\dfrac{1}{x}=5$, find the value of $x^5+\dfrac{1}{x^5}$. Is there some other way to do find it? $$ \left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=23\cdot 110. $$ ...
0
votes
1answer
58 views

General term of a series

I am trying to find the general term of the series: $$( 1 + x + ... + x^{m-1} )^k$$ I am trying to implement the KZ filter and it requires the coefficients of the above series. Here, k and m are ...
3
votes
1answer
34 views

Binomial coefficient difference

I have the following difference of binomial coefficients: $$f(m)={m+n\choose n}-{m-d+n\choose n}$$ I believe the following two things should hold true: For $m$ large enough, $f(m)$ is a polynomial ...
2
votes
2answers
39 views

Expansion Coefficient needed

This is probably something very easy, but wth... my mind is totally stuck right now. I need to find the coefficient of $x^{11}$ of the expansion $(x^2 + 2\frac yx)^{10}$ Well I know that the answer ...
2
votes
0answers
115 views

Getting K heads out of N biased coins problem (formula generation ).

Problem- Given a set of coins n with each coin i having Pi probability to give heads. Find the probability of getting k heads, when all coins are tossed together. hi i have solved this problem ...
1
vote
2answers
119 views

How to perform a binomial expansion on $m*v^2$?

I have been told by a couple of folks in passing, one of whom was a mathematician, that through binomial expansion of $m*v^2$ (where v is used in place of c), that all 5 Major Forces (Strong Force, ...
-1
votes
1answer
78 views

Revisited: Binomial Theorem: An Inductive Proof

I'm asked to use the fact that $\begin{pmatrix}n\\r\end{pmatrix}+\begin{pmatrix}n\\r-1\end{pmatrix}=\begin{pmatrix}n+1\\r\end{pmatrix}$ to show, by induction, that ...
2
votes
2answers
87 views

Stirling Binomial Polynomial

Let $\{\cdot\}$ denote Stirling Numbers of the second kind. Let $(\cdot)$ denote the usual binomial coefficients. It is known that $$\sum_{j=k}^n {n\choose j} \left\{\begin{matrix} j \\ k ...
7
votes
1answer
162 views

Vandermonde identity in a ring

Let $R$ be a commutative $\mathbb{Q}$-algebra. For $r \in R$ and $n \in \mathbb{N}$ we can define the binomial coefficient $\binom{r}{n}$ as usual by $\binom{r}{0}=1$ and ...
3
votes
1answer
105 views

Equivalence of two binomial type equations

Given that $$A=\sum_{i=k}^{2k-1}\binom {2k-1} ix^i(1-x)^{2k-1-i}$$ and $$B=\sum_{i=k+1}^{2k}\binom {2k} i x^i(1-x)^{2k-i}+\frac{1}{2}\binom {2k} k x^k(1-x)^k$$ I would like to prove that $A=B$ ...
0
votes
2answers
46 views

Polynomial of degree 2: what happens when variable triples?

Let p(x,y,z) be a homogeneous polynomial of degree 2: if p(2,3,4) = 10, what is p(6,9,12)?
1
vote
1answer
260 views

Using the Multinomial Theorem to Calculate a Finite Sum raised to an exponent

I know it's a simple question, but I keep getting different general formulas for the coefficients when I am trying to use the multinomial theorem for the following: $$ ...
2
votes
2answers
175 views

Recurrence equation for central trinomial coefficients

I've come across the following exercise: Give a recurrence equation for the central coefficients $(a_n)$, where for all $n$, $a_n$ is the coefficient of $X^n$ in $(1+X+X^2)^n$. Here's what I've ...
3
votes
4answers
1k views

Derivation of binomial coefficient in binomial theorem.

How was the binomial coefficient of the binomial theorem derived? $$\frac{n!}{k!(n-k)!}$$
6
votes
3answers
275 views

Intuitive explanation for a polynomial expansion?

Is there an ituitive explanation for the formula: $$ \frac{1}{\left(1-x\right)^{k+1}}=\sum_{n=0}^{\infty}\left(\begin{array}{c} n+k\\ n \end{array}\right)x^{n} $$ ? Taylor expansion around x=0 ...
8
votes
3answers
293 views

Polynomial in $\mathbb{Q}[x]$ sending integers to integers?

We can view the binomial coefficient $\binom{x}{k}$ has a polynomial in $x$ with degree $k$. So taking some $f\in\mathbb{Q}[x]$, why is $f(n)\in\mathbb{Z}$ for all $n\in\mathbb{Z}$, precisely when the ...
4
votes
3answers
120 views

Property of a polynomial $f\in\mathbb{Q}[X]$ such that $f(n)\in\mathbb{Z}$ for all $n\in\mathbb{Z}$?

We can always view $\binom{x}{k}$ as a polynomial in $x$ of degree $k$. With this in mind, why is it so that a polynomial $f\in\mathbb{Q}[x]$ is such that $f(n)\in\mathbb{Z}$ for all $n\in\mathbb{Z}$ ...
0
votes
3answers
132 views

Evaluating $\binom{100}{i}a^i(1-a)^{(100-i)}$ in GMP-GNU [closed]

I want to calculate $\binom{100}{i}a^i(1-a)^{(100-i)}$ for different $i$ with $a=0.001$ using GMP-GNU. How can this be done?
7
votes
2answers
332 views

A Curious Binomial Sum Identity without Calculus of Finite Differences

Let $f$ be a polynomial of degree $m$ in $t$. The following curious identity holds for $n \geq m$, \begin{align} \binom{t}{n+1} \sum_{j = 0}^{n} (-1)^{j} \binom{n}{j} \frac{f(j)}{t - j} = (-1)^{n} ...
4
votes
1answer
118 views

How to calculate this efficiently?

If in the expansion of $(1 + x)^m \cdot (1 – x)^n $, the coefficients of $ x $ and $ x^2 $are 3 and -6 respectively, then m is ? I solved it in the following way : Expanding we get, the coefficient ...