On Turán's inequality for Legendre polynomials (Q996999)

Let \(P_n\) be the Legendre polynomial of degree \(n,\) then for all \(x\in[-1,1]\) and \(n\in \mathbb N\) the inequality \[ \alpha_n(1-x^2)\leq P_n(x)^2-P_{n-1}(x)P_{n+1}(x)\leq\beta_n(1-x^2) \] holds where \[ \alpha_n=\mu_{[n/2]}\,\mu_{[(n+1)/2]},\qquad \beta_n=\frac{1}{2} \] are the best possible constants and \(\mu_n=2^{-2n}\binom{2n}{n}\) is the normalized binomial mid-coefficient. This generalizes a classical inequality of Turán. It should be noted that the key inequality of the proof was proved by help of the Mathematica package SumCracker.