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).