Turán's inequality for ultraspherical polynomials revisited

Ultraspherical multipliers revisited

Sufficient ultraspherical multiplier criteria are refined in such a way that they are comparable with necessary multiplier conditions. Also new necessary conditions for Jacobi multipliers are deduced which, in particular, imply known Cohen type inequalities. Muckenhoupt's transplantation theorem is used in an essential way.

关于超球多项式的Turán不等式

On Turán inequality for ultraspherical polynomials

本文证明了归一化超球多项式满足带权重因子 $|x|^θ$ 的更强 Turán 不等式，并将类似推广结果扩展到由三项递推定义的对称正交多项式。

Inequalities for ultraspherical polynomials. Proof of a conjecture of I. Raşa

A recent conjecture by I. Raşa asserts that the sum of the squared Bernstein basis polynomials is a convex function in $[0,1]$. This conjecture turns out to be equivalent to a certain upper pointwise estimate of the ratio $P_n^{\prime}(x)/P_n(x)$ for $x\geq 1$, where $P_n$ is the $n$-th Legendre polynomial. Here, we prove both upper and lower pointwise estimates for the ratios $\big(P_n^{(λ)}(x)\big)^{\prime}/P_n^{(λ)}(x)$, $~x\geq 1$, where $P_n^{(λ)}$ is the $n$-th ultraspherical polynomial. In particular, we validate Raşa's conjecture.

Dual addition formulas associated with dual product formulas

We observe that the linearization coefficients for ultraspherical polynomials are the orthogonality weights for Racah polynomials with special parameters. Then it turns out that the linearization sum with such a Racah polynomial as extra factor inserted, can also be evaluated. The corresponding Fourier--Racah expansion is an addition type formula which is dual to the well-known addition formula for ultraspherical polynomials. The limit to the case of Hermite polynomials of this dual addition formula is also considered. Similar results as for ultraspherical polynomials, although only formal, ar…

Positivity of Turán determinants for orthogonal polynomials

The orthogonal polynomials $p_n$ satisfy Turán's inequality if $p_n^2(x)-p_{n-1}(x)p_{n+1}(x)\ge 0$ for $n\ge 1$ and for all $x$ in the interval of orthogonality. We give general criteria for orthogonal polynomials to satisfy Turán's inequality. This yields the known results for classical orthogonal polynomials as well as new results, for example, for the $q$--ultraspherical polynomials.

Turrittin定理再探：实情形

Turrittin's Theorem revisited. The real case

本文建立了Turrittin定理的实版本，证明具有亚纯系数的线性常微分方程组可化为实系数规范型，并回顾了复情形证明以适配实情形。