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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 parameterized probability distribution p(x)=(p0(x),p1(x),…) and denote by S(x) the squared l2-norm of p(x). The properties of S(x) are useful in studying the Rényi entropy, the Tsallis entropy, and the positive linear operator associated with p(x). We show that for a family of distributions (including the binomial and the negative binomial distributions), S(x) is a Heun function reducible...
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