Recurrence-shift relations for the polynomial functions of Aldaya, Bisquert, and Navarro-Salas

Abstract

Using a simple factorization scheme we obtain the recurrence-shift relations of the polynomial functions of Aldaya, Bisquert and Navarro-Salas (ABNS), F_n^N(\frac\omega c\sqrtN x), i.e., one-step first-order differential relations referring to N, as follows. Firstly, we apply the scheme to the polynomial degree confirming the recurrence relations of Aldaya, Bisquert and Navarro-Salas, but also obtaining another slightly modified pair. Secondly, the factorization scheme is applied to the Gegenbauer polynomials to get the recurrence relations with respect to their parameter. Next, we make use of Nagel's result, showing the connection between Gegenbauer polynomials and the ABNS functions, to write down the recurrence-shift relations for the latter ones. Such relations may be used in the study of the spatial structure of pair-creation processes in an Anti-de Sitter gravitational background