Tsallis and Rényi divergences of generalized Jacobi polynomials

•We consider a sequence of generalized Jacobi polynomials.•With the help of these polynomials we define a discrete probability distribution.•Consider Tsallis (resp. Rényi) divergence for this discrete probability distribution.•We study the asymptotic behavior of Tsallis (resp. Rényi) divergence defined above.•For the quadratic Tsallis (resp. Rényi) divergence, an explicit formula is given.

We consider some discrete probability distribution ψn(x)=(ψn,1(x),ψn,2(x),…,ψn,n(x))ψn(x)=(ψn,1(x),ψn,2(x),…,ψn,n(x)). We show that, under suitable conditions, the sequence of Tsallis divergence (DT(ψn(x)))n(DT(ψn(x)))n and the sequence of Rényi divergence (DR(ψn(x)))n(DR(ψn(x)))n are convergent for any x∈(−1,1)x∈(−1,1).