All Questions
18
questions
1
vote
0
answers
92
views
Example of a specific polynomial
I need four non-trivial polynomials $P(x)$, $Q(y)$, $R(z)$ and $S(w)$ such that $$P(x)Q(y)R(z)S(w)=\sum_{i}a_i b_i c_i d_i x^i y^i z^i w^i+ \sum_{j\neq k\neq l\neq m}e_j f_k g_l h_m x^j y^k z^l w^m $$ ...
- 910
0
votes
2
answers
259
views
Product of two finite sums $\left(\sum_{k=0}^{n}a_k\right) \left(\sum_{k=0}^{n}b_k\right)$
What is the product of the following summation with itself:
$$\left(\sum_{k=2}^{n}k(k-1)a_{k}t^{k-2} \right) \left(\sum_{k=2}^{n}k(k-1)a_{k}t^{k-2}\right) $$
Is the above equal to the double summation ...
- 85
1
vote
1
answer
70
views
Is there a closed form of $x_2x_3\cdots x_n + x_1x_3\cdots x_n + \dots + x_1x_2\cdots x_{n-1}$?
Given real numbers $x_1,\dots,x_n \in \mathbb{R}$, does there exist a closed form for the expression
$$A_n := x_2x_3\cdots x_n + x_1x_3\cdots x_n + \dots + x_1x_2\cdots x_{n-1} = \sum_{i=1}^n \prod_{j=...
- 2,740
2
votes
2
answers
55
views
Algebraic expansion of $1 + (1 + x) + ... + (1 + x)^{p-1}$
I was proving that $p(x) = 1 + x + \text{ ... } + x^{p-1}$, where $p$ is a prime number, is irreducible over the rationals, by using the translation $1 + (1 + x) + \text{ ... } + (1 + x)^{p-1}$. I ...
- 2,663
1
vote
0
answers
40
views
Prove $\sum_{n=0}^{N+1}P(n)(-1)^n\binom{N+1}{n}=0$ with degree at most $N$ of $P$ [duplicate]
If $P$ is a polynomial of degree at most $N$, we have
$$\sum_{n=0}^{N+1}P(n)(-1)^n\binom{N+1}{n}=0$$
I tried using pascal's identity $\binom{N+1}{n}=\binom{N}{n}+\binom{N}{n-1}$. I think that ...
- 11
1
vote
1
answer
64
views
Evaluation of a telescoping sum
I have come to a problem in a book on elementary mathematics that I don't understand the solution to. The problem has two parts :
a.) Factorize the expression $x^{4} + x^{2} + 1$
b.) Compute the ...
- 595
0
votes
1
answer
63
views
Expansion of $ \left(\sum_{i=1}^n x_i\right)^m$
What is the expansion of
$
\left(\sum_{i=1}^n x_i\right)^m?
$
For example,
\begin{align*}
\left(\sum_{i=1}^n x_i\right)^2&=\sum_{i=1}^n x_i^2+2\sum_{i < j}^n x_ix_j,\\
\left(\sum_{i=1}^n ...
- 2,398
1
vote
0
answers
48
views
Transforming a long sum of products for efficient computation (based on spanning trees in a complete graph)
Let's say there is a set of $n$ real coefficients: $a_1,...,a_n$. My task is to calculate the value of a rather simple sum of k products: ...
- 239
0
votes
1
answer
839
views
Sigma notation in the context of roots of polynomials?
Got these problems listed in a text book. I can't figure out what the meaning of the sigma notation is here. How did they obtain those expansions of $\sum\alpha^3\beta$ & $\sum \frac{\alpha}{\beta}...
- 21
1
vote
3
answers
325
views
A formula for $1^4+2^4+...+n^4$
I know that
$$\sum^n_{i=1}i^2=\frac{1}{6}n(n+1)(2n+1)$$
and
$$\sum^n_{i=1}i^3=\left(\sum^n_{i=1}i\right)^2.$$
Here is the question: is there a formula for
$$\sum^n_{i=1}i^4.$$
- 4,934
3
votes
1
answer
1k
views
Please help explain how "Any symmetric sum can be written as a Polynomial of the elementary symmetric sum functions"
I was browsing through the Art Of Problem Solving website and came across this:
"Any symmetric sum can be written as a Polynomial of the elementary symmetric sum functions, for example
$x^3 + y^3 + ...
- 637
4
votes
1
answer
687
views
$a \in (0,1]$ satisfies $a^{2008} -2a +1 = 0$ and we define $S$ as $S=1+a+a^2+a^3........a^{2007}$. The sum of all possible value(s) of $S$ is?
This is homework.
Let $a \in (0,1]$ satisfies the equation $$a^{2008} -2a +1 = 0$$
and we define $S$ as $$S=1+a+a^2+a^3........a^{2007}$$
The sum of all possible value(s) of $S$ is?
My Attempt
$...
- 2,944
1
vote
1
answer
56
views
Given variable $m$, how do I find zeros of a polynomial in terms of $m$?
This is a summation question about a finite series with sum $m$. I'm trying to write a computer program that takes in a given integer $m$ (which represents the sum of a series) and outputs the number ...
- 11
2
votes
1
answer
134
views
Trouble with understanding the definition of cyclic sums?
This is from my book:
Given an $n-$ variable expression $f(x_1, x_2, x_3, ..., x_n)$, we will denote the cyclic sum by
$\displaystyle \sum_{\sigma} f(x_1, x_2, x_3, ..., x_n) = $
$f(x_1, x_2, x_3, ......
- 23.8k
1
vote
1
answer
681
views
Sum of squared coefficients of polynomial
Let $P(x) = a_0 + a_1 x + \dots a_nx^n$ be a polynomial, and let $\mu$ be a constant.
Is there any way to write $$\sum_{i = 1}^n a_i^2 \mu^i$$ as a function of $P(x)$ or related polynomials? Clearly ...
- 21.2k