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Results tagged with polynomials
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user 48831
Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please, use at least one of the top-level tags, such as nt.number-theory, co.combinatorics, ac.commutative-algebra, in addition to it. Also, note the more specific tags for some special types of polynomials, e.g., orthogonal-polynomials, symmetric-polynomials.
6
votes
$\binom{x}{2}+\binom{x}{4}+\cdots+\binom{x}{2u}$ is a convex function on $[0,+\infty)$?
Here's an alternative proof based on probabilistic arguments (showing different aspects). Let
$$f_n(x):=\sum_{j=0}^n { x \choose j}=[t^n]\,\frac{(1+t)^x}{1-t}\;\;,$$
and let $^\prime$ denote deriv …
- 3,255
5
votes
Accepted
Under what conditions does the Mittag-Leffler function ${E_{\alpha ,1}}(z),(0 < \alpha < 1)$...
None of these functions have real zeroes, because they can be written as moment generating
functions of certain random variables $Y_\alpha$. More precisely, for $0<\alpha <1$
$$E_{\alpha,1}(z)=\mathb …
- 3,255
4
votes
$\prod_k(x\pm k)$ in binomial basis?
Here is an argument for the leading coefficient (and more).
We use (formal) generating functions.
Let $f_{n,x}(t):= \sum_{m=0}^n {n-x \choose m } {n+x \choose n-m} e^{(x+2m-n)t}$, we are interested
i …
- 3,255
3
votes
Accepted
A polynomial identity related to Catalan numbers
These assertions can be proved using (formal) generating functions.
Using that for $j\geq 0, k\geq 1$
\begin{align*}
\sum_{n\geq 0} {n-j+kj \choose kj} t^n &=\frac{t^j}{(1-t)^{kj+1} }\;\;\mbox{ …
- 3,255
2
votes
Absolute convergence of logarithm of polynomial with positive coefficient ($\ln G(z) = \sum\...
Not a complete solution, only a remark: assume $G(z)$ has no zeroes in the closed unit disk.
If the series for $G(z)$ has radius of convergence >1 the series for $\log(G(z)$ will converge absolutely i …