  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
