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

$$\sigma(s)y''_n + \tau(s)y'_n - n [\tau'+ {\frac{1}{2}}(n-1)\sigma''] y_n = 0, \qquad n \in \mathbb{N}_0$$

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 ^(m) _n (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

$$y_n^{(m)}(\theta_a)= m! \binom{n}{m} \frac{(\omega_a)_n(\omega_a +\omega_{-a}+n-1)_m} {(\omega_a)_m(\omega_a+\omega_{-a}+n-1)_n} \,(\theta_a-\theta_{-a})^{n-m},$$

in which ${a=\mp \Delta \ne 0}$ and ${\theta_{\mp a}}$ are the discriminant and the roots of σ(s), respectively, and${\omega_{\mp a}}$ denote a parameter depending on the coefficients of the differential equation. Furthermore, the interrelations that arise between ${y_n^{(m)}(\theta_a)}$ and ${y_n^{(m)}(\theta_{-a})}$ are also introduced. Finally, special values corresponding to the limiting and exceptional cases have been presented explicitly for completeness.