Let \(\{x_{\nu}\}^ n_{\nu =1}\) be the roots of the polynomials \(\omega_ n(x)=-n(n-1)\int^{x}_{-1}P_{n-1}(t)dt=(1,x^ 2)P'_{n-1}(x),\quad\) \((n=2,3,...)\), where \(P_{n-1}(x)\) is the Legendre polynomial of degree \(n-1\) with the normalization \(P_{n-1}(1)=1\). Let also \(\{\xi_{\nu}\}^{n-1}_{\nu =1}\) be the roots of \(\omega'_ n(x)=-n(n-1)P_{n-1}(x)\). There exists a unique interpolation polynomial \(Q_{2n-1}(x;f)\) of degree \(\leq 2n-1\) interpolating the function \(f(x)\in C^{(r)}[-1,1]\) and \(r\geq 1\) such that \(Q_{2n-1}(x_{\nu};f)=f(x_{\nu})\), \((\nu =1,...,n\); \(n=2,3,...)\), \(Q_{2n-1}(\xi_{\nu};f)=f'(\xi_{\nu})\), \((\nu =1,...,n-1\); \(n=2,3,...)\), \(Q'_ n(-1;f)=f'(-1)\) \((n=2,3,...)\).