Related papers: On Tur\'an inequality for ultraspherical polynomia…
We investigate Riemannian manifolds $(M^n,g)$ whose curvature operator of the second kind $\mathring{R}$ satisfies the condition \begin{equation*} \alpha^{-1} (\lambda_1 +\cdots +\lambda_{\alpha}) > - \theta \bar{\lambda}, \end{equation*}…
For a convex domain $K$ in the complex plane, the well-known general Bernstein-Markov inequality holds asserting that a polynomial $p$ of degree $n$ must have $||p'|| < c(K) n^2 ||p||$. On the other hand for polynomials in general, $||p'||$…
In this note our aim is to point out that certain inequalities for modified Bessel functions of the first and second kind, deduced recently by Laforgia and Natalini, are in fact equivalent to the corresponding Tur\'an type inequalities for…
Ensembles of complex symmetric, and complex self dual random matrices are known to exhibit local statistical properties distinct from those of the non-Hermitian Ginibre ensembles. On the other hand, in distinction to the latter, the joint…
For $k \ge 2$ even, and $ \alpha \ge -(2k+1)/4 $, we provide a uniform approximation of the ultraspherical polynomials $ P_k^{(\alpha,\, \alpha)}(x) $ in the oscillatory region with a very explicit error term. In fact, our result covers all…
In this paper, we study the Tur\'{a}n problem for $C_{2k+1}^{-}$. Suppose that $\dot{G}$ is an unbalanced signed graph of order $n$ with $e(\dot{G})$ edges. Let $\lambda_{1} (\dot{G})$ be the largest eigenvalue of $\dot{G}$, and…
The first part of this paper is devoted to the study of the orthogonal polynomial on the circle, with respect of a weight of type $f_\alpha (\theta) = (2\cos \theta- 2\cos \theta_0)^{2\alpha} c_1$ with $\theta_0 \in ]0,\pi[$, -1/2…
We study solutions of three-term recurrence relations whose $N$-step transfer matrices belong to the uniform Stolz class. In particular, we derive the first order of their uniform asymptotics. For orthonormal polynomials we show more.…
Tuenter [Fibonacci Quarterly 40 (2002), 175-180] and other authors have considered centred binomial sums of the form \[S_r(n) = \sum_k \binom{2n}{k}|n-k|^r,\] where $r$ and $n$ are non-negative integers. We consider sums of the form…
Motivated by a Gan-Loh-Sudakov-type problem, we introduce the regular Tur\'an numbers, a natural variation on the classical Tur\'an numbers for which the host graph is required to be regular. Among other results, we prove a striking…
The Tur\'{a}n inequalities and the higher order Tur\'{a}n inequalities arise in the study of Maclaurin coefficients of an entire function in the Laguerre-P\'{o}lya class. A real sequence $\{a_{n}\}$ is said to satisfy the Tur\'{a}n…
In 1939 P. Tur\'an started to derive lower estimations on the norm of the derivatives of polynomials of (maximum) norm 1 on $I:=[-1,1]$ (interval) and $D:=\{z\in\mathbb{C}~:~|z|\le 1\}$ (disk), under the normalization condition that the…
Newton's inequalities $c_n^2 \ge c_{n-1}c_{n+1}$ are shown to hold for the normalized coefficients $c_n$ of the characteristic polynomial of any $M$- or inverse $M$-matrix. They are derived by establishing first an auxiliary set of…
Tur\'an, Mitrinovi\'c-Adamovi\'c and Wilker type inequalities are deduced for regular Coulomb wave functions. The proofs are based on a Mittag-Leffler expansion for the regular Coulomb wave function, which may be of independent interest.…
In this study we introduce a second type of higher order generalised geometric polynomials. This we achieve by examining the generalised stirling numbers $S(n; k;\alpha;\beta;\gamma)$ [Hsu & Shiue,1998] for some negative arguments. We study…
Tur\'an problems in extremal combinatorics ask to find asymptotic bounds on the edge densities of graphs and hypergraphs that avoid specified subgraphs. The theory of flag algebras proposed by Razborov provides powerful methods based on…
A one-parameter generalisation R_{\lambda}(X;b) of the symmetric Macdonald polynomials and interpolations Macdonald polynomials is studied from the point of view of branching rules. We establish a Pieri formula, evaluation symmetry,…
We consider differences between $\log \Gamma(x)$ and truncations of certain classical asymptotic expansions in inverse powers of $x-\lambda$ whose coefficients are expressed in terms of Bernoulli polynomials $B_n(\lambda)$, and we obtain…
Let \( \mathcal{F} \) be a family of graphs. The generalized Tur\'an number \( \operatorname{ex}(n, K_r, \mathcal{F}) \) is the maximum number of $K_r$ in an \( n \)-vertex graph that does not contain any member of \( \mathcal{F} \) as a…
Orthogonal polynomials of degree $n$ with respect to the weight function $W_\mu(x) = (1-\|x\|^2)^\mu$ on the unit ball in $\RR^d$ are known to satisfy the partial differential equation $$ [ \Delta - \la x, \nabla \ra^2 - (2 \mu +d) \la x,…