Special functions associated with positive linear operators

Many well-known positive linear operators (like Bernstein, Baskakov, Sz\'{a}sz-Mirakjan) are constructed by using specific fundamental functions. The sums of the squared fundamental functions have been objects of study in some recent papers. We investigate the relationship between these sums and some special functions. Consequently, we get integral representations and upper bounds for the sums. Moreover, we show that they are solutions to suitable second order differential equations. In particular, we provide polynomial or rational solutions to some Heun equations

The index of coincidence for the binomial distribution is log-convex

We consider the binomial distribution with parameters $n$ and $x$, and show that the sum of the squared probabilities is a log-convex function of $x$. This completes the proof of a conjecture formulated in 2014. Applications to R\'{e}nyi and Tsallis entropies are given

Complete monotonicity of some entropies

It is well-known that the Shannon entropies of some parameterized probability distributions are concave functions with respect to the parameter. In this paper we consider a family of such distributions (including the binomial, Poisson, and negative binomial distributions) and investigate the Shannon, R\'{e}nyi, and Tsallis entropies of them with respect to the complete monotonicity

Concavity of some entropies

It is well-known that the Shannon entropies of some parameterized probability distributions are concave functions with respect to the parameter. In this paper we consider a family of such distributions (including the binomial, Poisson, and negative binomial distributions) and investigate the concavity of the Shannon, R\'enyi, and Tsallis entropies of them.Comment: 8 pages; an oral presentation based on this work was delivered at ICMA 2015 (International Conference on Mathematics and its Applications

Entropies and the derivatives of some Heun functions

This short note contains a list of new results concerning the R\'{e}nyi entropy, the Tsallis entropy, and the Heun functions associated with positive linear operators.Comment: 7 page

A sharpening of a problem on Bernstein polynomials and convex functions

We present an elementary proof of a conjecture proposed by I. Rasa in 2017 which is an inequality involving Bernstein basis polynomials and convex functions. It was affirmed in positive by A. Komisarski and T. Rajba very recently by the use of stochastic convex orderings

Estimates for the differences of positive linear operators and their derivatives

The present paper deals with the estimate of the differences of certain positive linear operators and their derivatives. Our approach involves operators defined on bounded intervals, as Bernstein operators, Kantorovich operators, genuine Bernstein-Durrmeyer operators, Durrmeyer operators with Jacobi weights. The estimates in quantitative form are given in terms of first modulus of continuity. In order to analyze the theoretical results in the last section we consider some numerical examples

Heun functions and combinatorial identities

We give closed forms for several families of Heun functions related to classical entropies. By comparing two expressions of the same Heun function, we get several combinatorial identities generalizing some classical ones.Comment: Submitted to AAD

Bounds for some entropies and special functions

We consider a family of probability distributions depending on a real parameter and including the binomial, Poisson and negative binomial distributions. The corresponding index of coincidence satisfies a Heun differential equation and is a logarithmically convex function. Combining these facts we get bounds for the index of coincidence, and consequently for R\'{e}nyi and Tsallis entropies of order $2$.Comment: Accepted to be published in Carpathian Journal of Mathematics, 1/201

Heun functions related to entropies

We consider the indices of coincidence for the binomial, Poisson, and negative binomial distributions. They are related in a simple manner to the R\'{e}nyi entropy and Tsallis entropy. We investigate some families of Heun functions containing these indices of coincidence. For the involved Heun functions we obtain closed forms, explicit expressions, or representations in terms of hypergeometric functions.Comment: Submitted to RACSAM Series