A new kind of orthogonal polynomial -- \(A_{n,r}(x)\) (Q2751004)

It is a classical result that the Legendre polynomials \(P_n(x)\) are orthogonal over the interval \([-1,-1]\). The author considers the polynomial defined by \(A_{n,r}(x)=(x^2-1)^r P_n^{(2r)}(x)\) and shows that the \(\{A_{n,r}\}_n^\infty\) is also orthogonal over \([-1,-1]\). Recurrence relations are given. It is commented that \(\{A_{n,r}\}_n^\infty\) already includes the Legendre polynomials and Perkai polynomial \(Q_n(x):=n(n-1)P_n(x)=2P_{(n-1)}^\prime (x)\) [\textit{P. Kh. Derkack} [Ukr. Mat. Zh. 12, 466-471 (1960; Zbl 0116.04902)] by choosing \(r=0\) and \(1\) respectively.