1
vote
2answers
132 views

Under what conditions does the Mittag-Leffler function ${E_{\alpha ,1}}(z),(0 < \alpha < 1)$ has no real zero?

I want to know under what conditions does the Mittag-Leffler function ${E_{\alpha ,1}}(z),(0 < \alpha < 1)$ has no real zero, where ${E_{\alpha ,1}}(z) = \sum\limits_{k = 0}^\infty ...
6
votes
1answer
151 views

Inequality for Laguerre polynomials

Let $L_n$ be the $n$-th Laguerre polynomial defined by $\quad L_n (x)=\frac{e^x}{n!}\frac{d^n}{dx^n}(x^n e^{-x}).\quad $ I want to prove that $$ \forall n\in \mathbb N,\forall x\ge 0,\quad \sum_{0\le ...
7
votes
4answers
205 views

Is there an oscillating analog of the Gaussian distribution?

It frequently happens that, in some famillies of polynomials with positive coefficients, the coefficients of large polynomials look like a bell curve and tend to the distribution function of the ...
0
votes
0answers
177 views

Convert 2F1 to polynomial

Is there any transformation to convert each of the following versions of ${}_2F_1$ to a polynomial? The first one is $${}_2F_1\left(\frac{1-a}{2}, -\frac{a}{2}; b;\frac{4z}{(1+z)^2} \right), \quad ...
4
votes
1answer
825 views

minimal polynomials of trig functions of ($k \pi/p$) and divisibility of coefficients by p

Take an odd prime $p$ and put $x_0:=\sum\limits_{j=0}^{p-1}\left(a_{j}\sqrt{p}\cos\dfrac{j\pi}p+b_{j}\sin\dfrac{j\pi}p +c_{j}\tan\dfrac{j\pi}p\right)$, where the $a_{ij}$ are integers. If $f$ denotes ...
0
votes
1answer
283 views

Upper bounds on generalized Laguerre polynomials

I evaluated an integral and obtained an expression with a Laguerre polynomial. I'd like something more explicit and useable. Are there any known simple (e.g. exponential) upper bounds on the ...
2
votes
2answers
369 views

About Turan`s problem(inequality) in multivariable

Hi. I have a question related to Turan`s problem, that is Find a sequence of polynomial $P_n(x)$ satisfying $P_{n+1}(x)P_{n-1}(x) < P_{n}^2(x)$. I am considering the generalized question for ...
8
votes
1answer
775 views

Is this sequence of polynomials well-known?

While working on a problem in p-adic Hodge theory, and needing to write down a solution to a certain equation involving p-adic power series, I stumbled across a certain sequence of polynomials. Define ...
0
votes
1answer
451 views

modular arithmetic of Hermite polynomials

I wonder if there is anything known (formula, asymptotics, etc) of computing the remainder $R_{k,m} \equiv H_{k} ~ \mod H_m$ for $k > m$, where $H_m$ denotes the $m$th Hermite polynomial ...
8
votes
3answers
802 views

No simple duplication formula for factorials?

Many special functions including the gamma function have a duplication formula of some sorts. In the case of the gamma function it reads: Gamma(2z) = Gamma(z) Gamma(z+1/2) 22z-1/Gamma(1/2) On ...