Polynomials Characterizing Hyper b-ary Representations
Department of Mathematics and Statistics
Halifax, NS B3H 4R2
P. O. Box 172
Millville, NJ 08332-0172
Given an integer base b ≥ 2, a hyper b-ary
representation of a positive integer n is a representation of
n as a linear combination of nonnegative powers of b,
with integer coefficients between 0 and b. We use a system of
recurrence relations to define a sequence of polynomials in b
variables and with b parameters, and we show that all hyper
b-ary representations of n are characterized by the
polynomial with index n+1. This extends a recent result of
Defant on the number of hyper b-ary representations based on a
b-ary analogue of Stern's diatomic sequence. The polynomials
defined here extend this numerical sequence, and they can be seen as
generalized b-ary Stern polynomials.
Full version: pdf,
(Concerned with sequences
Received March 23 2018; revised versions received May 3 2018; May 7 2018.
Published in Journal of Integer Sequences, May 7 2018.
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