All Questions
5
questions
0
votes
0
answers
40
views
Finding particular solution using domain transformation
$$ φ(n)=5 φ\left(\frac{n}{2}\right)-6 φ\left(\frac{n}{4}\right)+n $$ where $$ \varphi (1) = 2 \\ \text{and} \\ \varphi (2) = 1 $$
With $n=2^x$, I have the following equation. Am I wrong in this ...
-1
votes
1
answer
39
views
Coefficient of an expansion
Find the coefficient of $x^k$ in $(x+a)(x+b)...(x+n)$ where $a$, $b$ and $n$ are integers.
I am not able to approach this problem.
- 524
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
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
$$ \begin{align}
\operatorname{S}_N(x) &= x^{N+1} \sum_{n=0}^{N} \binom{N+n}{n} \binom{2N+1}{N-n} (-x)^{n} \qquad N \in \mathbb{Z}...
11
votes
1
answer
202
views
Is the given binomial sum almost everywhere negative as $K\to\infty$?
The binomial sum is as follows:
$$\mathcal {L}^K(\theta)= \sum_{i=\lceil{K/2}\rceil}^K \binom{K}{i}\theta^i\left((1-\theta)^{K-i}-\frac{1}{2}(1-\theta)^{-K}(1-2\theta)^{K-i}\right)$$
which can also ...
- 7,892