Solution of the Szili type conjugate problem on the roots of the Laguerre polynomials (Q5928563)

The author considers the problem of constructing an interpolation polynomial of degree \(2n-1\) which takes given values at the roots of the generalized \(n\)-th Laguerre polynom \[ L_n^{(\alpha)}(x)=3D{1 \over n!} x^{-\alpha}e^x {d^n \over{dx^n}}(x^{\alpha +n}e^{-x}) \] and at the roots of \((L_n^{(\alpha)})'\) their derivatives are also given. The existence and the canonical form of such construction is proved for the values \(\alpha \in (-1,0]\).