All Questions
4
questions
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
-1
votes
1
answer
39
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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
2
votes
1
answer
60
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Root of a plynomial in (0,1)
Define $$f_K(x)=\sum_{i=K+1}^{2K} \binom{2K}{i}x^{i-1}(1-x)^{2K-i}.$$ How to show that $qf_K(x)-f_K(1-x)$ has exactly one real root in $(0,1)$ for any $q > 0$ and $K \geq 1$. The proof for $q=1$ ...
- 29
1
vote
2
answers
125
views
Tartaglia triangle for real field?
Let's say i need to find the coefficient of the 2nd term in a binomia, we can use pascal's triangle. This works only if the power is a natural number.
What if we want to know it for a real power ...
- 613