SOME CLASSES OF GENERALIZED MinMax POLYNOMIALS

Abstract

The so-called MinMax numbers {M_n} and their subsidiary numbers {N_n} for the Pell numbers {P_n} were studied by (for example) A.F. Horadam [3].  These MinMax numbers {M_n} are positive integers which are the minimal and maximal representations by means of the Pell numbers.  Analogous results for the MinMax numbers {D_n} and their subsidiary numbers {R_n}, and for the modified Pell numbers {q_n}, are obtained in [3], Q_n := 2q_n being the Pell-Lucas numbers. A.F.  Horadam [4], on the other hand, expanded this MinMax number system to the algebraic polynomials {M_n(x)}, {N_n(x)}, {D_n(x)} and {R_n(x)}.  Our aim in this paper is to investigate results, which are similar to those in [4], but which hold true instead for the following generalized sequences of polynomials:

     {P_n,m(x)}   and   {Q_n,m(x)}   (m ∈ \(\mathbb{N}\) ; n ∈ \(\mathbb{N}\) ∪ {0}),

N being the set of positive integers.