Modified (0,2)-interpolation on the roots of Jacobi polynomials.: I (explicit formulae)

Let the set of knots -1 = x(n+1) < x(n) <... < x(1) < x(0) =1 (n greater than or equal to 1) be given on the interval [-1, 1]. Find a polynomial Q(m)(x) of minimal degree satisfying (0, 2)-interpolational conditions at the inner knots and boundary conditions at the endpoints, that is Q(m)((s))(x(i))=y(i)((s)) (s=0,2) fbr i=1,...,n, and Q(m)((j))(x(0)) = alpha(0)((j)) for j=0,...,k, Q(m)((j))(x(n+1)) = alpha(n+1)((j)) for j=0,...,l, where y(i)((s)), alpha(0)((i)), alpha(n+1)((j)), are arbitrarily given real numbers, and k, I are arbitrary fixed non-negative integers. In this paper the existence and uniqueness of the polynomial Q(m)(x) is proved if the inner nodal points are the zeros of Jacobi polynomials P-n((2k+1,2l-1))(x) or P-n((2k-1,2l+1))(x). Explicit formulae for the fundamental polynomials of interpolation are also given.