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

Using a simple factorization scheme we obtain the recurrence-shift relations of the polynoll1ial functions of Aldaya, llisquert amI Navarro-Salas (ABNS), F~"(~JN x), i. e., one-step first-orocr differential rclations refcrring to N, as follows. Firstly, we apply thc 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 Gegenhauer polynomials to get the recurrcncc relations with respect to their parameter. Next, we make use of Nagel's result, showing thc connection between Gegenbauer polynomials and the ABNS functions, to writc down the recurrence-shift relations for the latter ones. Such relations may be used in the study of the spatial structure of pair-creation proccsses in an Anti-de Sitter gravitational background.