We consider the indices of coincidence for the binomial, Poisson, and negative binomial distributions. They are related in a simple manner to the Rényi 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.

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 Renyi and Tsallis entropies of order 2.

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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 Rényi and Tsallis entropies of order 2.

Heun functions are important for many applications in mathematics, physics and in thus in interdisciplinary phenomena modelling. They satisfy second order differential equations and are usually represented by power series. Closed forms and simpler polynomial representations are useful. Therefore, we study and derive 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.