Abstract: In this paper, we show that if a problem of (0, 1,..,m-2, m) -interpolation on the zeros of (1 -x)P_m-1^(α,β)(x)(α>1,β ≥ -1) has an infinity of solutions then the general form of the solutions is f₀(x) +Cf(x) with an arbitrary constant C,where P_m-1^(α,β)(x) stands for the (n-1)th Jacobi polynomial, and f₀ (x) and f(x) are fixed polynomials of degree ≤ mn-1,and, meanwhile. the explicit form of f(x) is given. Moreover, a necessary and sufficient condition of quadrature regularity of the interpolation in a manageable form is established.