All Questions
20
questions
1
vote
1
answer
97
views
Sum with binomial coefficient using identity
I want to prove:
$\displaystyle \sum_{k=0}^n (-1)^k \binom{x}{k} = (-1)^n \binom{x-1}{n}$
using: $(1-z)^x \cdot \frac{1}{1-z} = (1-z)^{x-1}$
I know how to do it with induction but i somehow can't ...
- 121
0
votes
1
answer
51
views
Show with the help of binomial theorem that these two equations are equal?
Show with the help of binomial theorem that these two expression are equal for $n\ge 0$ then this $$ \sum_{k=0}^n \binom n k x^k (2+x)^k = \sum_{k=0}^{2n} \binom {2n} k x^k $$
I don’t know how to do ...
- 51
2
votes
1
answer
180
views
$p(q(t)) \neq q(p(t))$ for every $t$, find all degrees of monic polynomials
Find all the positive numbers $m, n$ for which there exist some monic polynomials $p$ and $q$, for which $\deg p = m$ and $\deg q = n$, while $f \circ g$ and $g \circ f$ have no intersection points.
...
0
votes
2
answers
193
views
AIME 1986 Problem - Polynomials
OBSERVATIONS
Let $$f(x)=1-x+x^2...+x^{16}-x^{17}$$
If we try to replace $x=y-1$ then we see that $y^2$ appears in every term after $1-x$ in $f(x)$.
By Binomial Expansion/observation Coefficient of $y^...
- 129
-1
votes
1
answer
77
views
What is the coefficient of $x^{11}$ in $(x+x^2+x^3+x^4+x^5)^4(1+x+x^2+...)^4$?
I got
$$(x+x^2+x^3+x^4+x^5)^4=x^4((1-x^5)/1-x)^4$$
and
$$(1+x+x^2+...)^4=1/(1-x)^4,$$
what I should do next? Please help me with it, thank you.
- 1
-1
votes
3
answers
55
views
I need to develop $P_n(X)=\frac{(X+i)^{2n+1} - (X - i)^{2n+1}}{2i}$ using the binomial coefficients formula and show a few properties [duplicate]
In the previous questions I've proved that $(1+i)^{2n+1}=a_n+ib_n$ where $a_n$ and $b_n$ are $\pm 2^n$ and that $(1-i)^{2n+1}=a_n - ib_n$ and that $|P_n(1)|=2^n$ using the previous statements. But now ...
- 143
1
vote
0
answers
71
views
Why do two ways of expanding the same formal polynomial lead to matching coefficients?
There are a few proofs in which the technique is to expand the product of some formal polynomials in $\mathbb{R}[x_1,x_2,\ldots,x_k]$ in more than one distinct way and then we can match up the ...
- 3,415
9
votes
3
answers
415
views
An interesting problem of polynomials
In the polynomial
$$
(x-1)(x^2-2)(x^3-3) \ldots (x^{11}-11)
$$
what is the coefficient of $x^{60}$?
I've been trying to solve this question since a long time but I couldn't. I don't know whether ...
- 1,135
0
votes
1
answer
110
views
Symmetric Polynomials: Binomial identity
Consider the following equality of symmetric polynomials of degree $n$:
$\sum\limits_{i=1}^{n-1}c_ix^iy^{n-i}+\sum\limits_{i=1}^{n-1}c_i(x+y)^iz^{n-i}=\sum\limits_{i=1}^{n-1}c_iy^iz^{n-i}+\sum\...
- 33
4
votes
2
answers
84
views
In how many ways a student can get $2m $ Marks
An examination contains four Question papers each paper carrying maximum marks as $m$. Find number of ways a student appearing for all the four papers gets a total of $2m$ Marks.
I used generating ...
- 13.2k
1
vote
4
answers
60
views
Computing the coefficient of the term of a certain degree in a polynomial
Given the polynomial
${1\over8}((1+z)^9 + 3(1-z)^4(1+z)^5 + (1-z)^6(1+z)^3)$
(which is the weight enumerator of a code)
how do I find out the coefficient of $z^2$?
The solution given is ${1 \over ...
- 293
2
votes
4
answers
134
views
Swapping the order of summation without writing a few terms and guessing the pattern
I am wondering if there is a general way to manipulate double summations, that does not involve writing a few terms and 'guessing' the pattern. For example, consider a polynomial of order $n$:
$$
p(x)=...
- 9,666
1
vote
2
answers
43
views
How the pattern emerging in the exponents of the terms of a binomial expansion is proven?
Every article I've read just shows that the exponents of a term in the expansion are of the form $$a^{n-k}b^{k}$$ but how to prove that this is really true?
- 1,201
0
votes
2
answers
296
views
Computing the coefficient in a large polynomial
What is the coefficient of $x^{10}$ in the expansion of $(5 x^2 + 3)^{14}$?
(I would prefer to know the answer as a mathematical expression rather than a number.)
May I know how to approach this ...
- 39
11
votes
3
answers
1k
views
Smoothstep sigmoid-like function: Can anyone prove this relation?
The Smoothstep sigmoid-like function is defined as the polynomial
$$ \begin{align}
\operatorname{S}_N(x) &= x^{N+1} \sum_{n=0}^{N} \binom{N+n}{n} \binom{2N+1}{N-n} (-x)^{n} \qquad N \in \mathbb{Z}...
