All Questions
Tagged with binomial-coefficients polynomials
31
questions with no upvoted or accepted answers
12
votes
0
answers
393
views
How to prove this $p^{j-\left\lfloor\frac{k}{p}\right\rfloor}\mid c_{j}$
Let $p$ be a prime number and $g\in \mathbb{Z}[x]$.
Let $$\binom{x}{k}=\dfrac{x(x-1)(x-2)\cdots(x-k+1)}{k!} \in \mathbb{Q}[x]$$ for every $k \geq 0$.
Fix an integer $k$. Write the integer-valued ...
- 459
10
votes
0
answers
331
views
Expanding a product of linear combinations with coefficients $1$ and $-1$
For any odd natural number $n$, denote $t \equiv \frac{n-1}{2}$. Let $K$ be a field such that $\operatorname{char} K \neq 2$. Working over the polynomial ring $K\left[x_1,x_2,...,x_{n} \right]$, ...
- 124
7
votes
0
answers
460
views
Best representation of a polynomial as a linear combination of binomial coefficients
Newton's interpolation formula
shows that every polynomial is a linear combination of binomial coefficients. For instance,
$$
\frac{n^3}{3}+\frac{n^2}{2}+\frac{n}{6}
= 0 \binom{n}{0}+1\binom{n}{1}+3\...
- 218k
4
votes
0
answers
145
views
A rational function with hidden symmetry and alternating poles and zeros.
Upon answering a question about an equivalence of two binomial sums I have noted that a naturally appearing function has some interesting properties.
Consider the function:
$$
f(m,n_1,n_2;z)=\frac{1}{...
- 26.7k
3
votes
1
answer
75
views
How to prove that this binomial sum remains positive for $x>1$?
Let's say you have this function for real numbers $x>1$, for some positive integer $n \geq 1$
$$
\sum_{k=0}^{\left \lfloor n/2 \right \rfloor} {x \choose 2k+\frac{1-(-1)^n}{2}}
$$
How would you ...
- 633
3
votes
0
answers
108
views
Is there a simple formula for this polynomial sum?
The following seemingly arbitrary polynomial has popped up in my research on period polynomials of modular forms, and I would like to try to understand it a little better. I hope it is OK to ask about ...
- 3,559
3
votes
0
answers
49
views
Algebra problem related to calculus and binomial formula
This might look like a trivial calculus question, but I did tons of calculation
and research to turn the property I am interested into this simple form (actually it would be a sufficient condition ...
- 166
3
votes
0
answers
68
views
Can a certain polynomial have all its coefficients in some basis divisible by a prime $p$?
I fix $n\in\mathbf{N}^{*}$ and $n$ elements $\alpha_1,\ldots,\alpha_n$ in $\mathbf{N}^{*}$. Consider the polynomial $$Q(T)=\prod\limits_{1\leq i \leq n} \prod\limits_{0\leq j \leq \alpha_i -1} (\...
- 369
2
votes
0
answers
77
views
Closed expression for a combinatorial sum
The following equality is true for every positive integer $n$ :
$$\sum_{k=0}^n {n \choose k} = 2^n $$
It is a special case ($p = 0$) of the sequence :
$$S_{p, n}=\sum_{k=0}^n k^p {n \choose k} $$
For ...
- 389
2
votes
0
answers
56
views
$f_n(ab) = f_n(a)f_n(b) + f_n(a-1)f_n(b-1) + f_n(a-2)f_n(b-2)+ ...+f_n(a-n)f_n(b-n)$
I was toying around when I noticed for $a,b > 0$:
$$f(ab) = f(a)f(b) + f(a-1)f(b-1)$$
is satisfied by $f(n) = T_n$ ; the triangular numbers $n(n+1)/2$.
This equation is not an addition formula, not ...
- 16.4k
2
votes
0
answers
70
views
Expectation of a certain polynomial expression in Rademacher random variables.
Let $N_1,k \ge 1$ be integers and let $N = N_1 k$. Let $G_1,...,G_k$ be an equi-partition of $[N] := \{1,2,\ldots,N\}$. Thus, $|G_j| = N_1$ for all $i$. Let $\mathcal S$ be the transversal of this ...
- 9,575
2
votes
0
answers
168
views
Polynomial Division of a "Special" Polynomial
Let $$f(m)=(2n+1)((2n+1)^2-1^2)((2n+1)^2-3^2)\ldots((2n+1)^2-(2m-3)^2)/(2m-1)!$$
for some Positive integers $n,m$
we have to find the coefficients of $t^{1-k}$ quotient on polynomial Division of
$...
- 39
2
votes
0
answers
39
views
Coefficients of a particular polynomial
Let $f$ be the polynomial obtained by taking the terms $1$, $x^3$, $x^6$, $x^9$, ... $x^k$,... for $k\equiv 0,3,6$ (mod 9) in the expansion of $(1+x)^n$ with the corresponding coefficients. That is,
$$...
- 546
1
vote
0
answers
67
views
Closed formula for $\sum_{k=0}^{\lfloor n/2\rfloor} x^k\binom{n-k}{k}$
Let $$P_n(x) = \sum_{k=0}^{\lfloor n/2\rfloor} x^k\binom{n-k}{k}.$$ It's known that $P_n(1) = F_{n+1}$, the $(n+1)$th Fibonacci number, see for example here. Can we find a closed form for this ...
- 10.5k
1
vote
0
answers
50
views
Nonnegative integer combinations of binomial coefficients
A classical result of Polya and Ostrowski states that integral linear combinations of binomial coefficients ${x\choose k}$ is exactly the set of all polynomials $f(x)\in\mathbb{Q}[x]$ such that $f(\...
- 427