Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.3

Polynomials Characterizing Hyper b-ary Representations

Karl Dilcher
Department of Mathematics and Statistics
Dalhousie University
Halifax, NS B3H 4R2

Larry Ericksen
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.

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(Concerned with sequences A002487 A054390.)

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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