On the special values of monic polynomials of hypergeometric type

Special values of monic polynomials y _n (s), with leading coefficients of unity, satisfying the equation of hypergeometric type have been examined in its full generality by means of a unified approach, where σ(s) and τ(s) are at most quadratic and a linear polynomial in the complex variable s, respectively, both independent of n. It is shown, without actually determining the polynomials y _n (s), that the use of particular solutions of a second order difference equation related to the derivatives y _n ^(m)(z) is sufficient to deduce special values for some appropriate s = z points. Hence the special values of almost all polynomials and their derivatives can be generated by the universal formula in which and are the discriminant and the roots of σ(s), respectively, and denote a parameter depending on the coefficients of the differential equation. Furthermore, the interrelations that arise between and are also introduced. Finally, special values corresponding to the limiting and exceptional cases have been presented explicitly for completeness.