The moments of the density of zeros for the relativistic Hermite and Laguerre polynomials
Sum rules for zeros of two (closely related) types of orthogonal polynomials are given, using methods due to \textit{K. M. Case} [J. Math. Physics, 21, 702-714 (1980; Zbl 0453.33007, and Zbl 0453.33008)] and of the first author, \textit{B. Germano} and \textit{P. E. Ricci} [Comput. Math. Appl. 30, 3-6, 69-81 (1995; Zbl 0846.65005)]. The authors assert that their polynomials are new, but this is not true. The polynomials are (not only in limiting cases, but always) special Jacobi polynomials, already known since more than a century, and consequently the recent investigations are in the scope of the 19-th century \textit{B. Nagel}, J. Math. Phys. 35, 1549-1554 (1994; reviewed above). Every hypergeometric polynomial \(F(-n,b;c;x)\) is a Jacobi polynomial \(P_n^{(c-1, b-c-n)} (1-2x)\); the authors should read formulae 22.5.42-45 in the ``Handbook'' of M. Abramowitz and I. A. Stegun, cited by them! Hence, the facts treated here, as differential equations, recurrence relations, explicit representations, orthogonality properties, location of zeros (more effective to be treated with the help of Rodrigues formula!) are known. Further: the differential equation (1.1), (2.1), (3.1) is not ``a generalized hypergeometric-type'' equation, it is the good old Gaussian hypergeometric equation, with a simple linear transform of the independent variable, and even a computer system of symbolic integration as Maple offers the general solution (among it the polynomials discussed here) for all these equations -- without any difficulty through a single statement.
scientific article; zbMATH DE number 878535
scientific article