44 research outputs found
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 and , and show
that the sum of the squared probabilities is a log-convex function of . 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 .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