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

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

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

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

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

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

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

Inequalities for indices of coincidence and entropies

We consider a probability distribution depending on a real parameter $x$. As functions of $x$, the R\'enyi entropy and the Tsallis entropy can be expressed in terms of the associated index of coincidence $S(x)$. We establish recurrence relations and inequalities for $S(x),$ which can be used in order to get information concerning the two entropies

Elementary hypergeometric functions, Heun functions, and moments of MKZ operators

We consider some hypergeometric functions and prove that they are elementary functions. Consequently, the second order moments of Meyer-Konig and Zeller type operators are elementary functions. The higher order moments of these operators are expressed in terms of elementary functions and polylogarithms. Other applications are concerned with the expansion of certain Heun functions in series or finite sums of elementary hypergeometric functions

Discrete Operators associated with Linking Operators

We associate to an integral operator a discrete one which is conceptually simpler, and study the relations between them