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Polynomials Characterizing Hyper ***b*-ary Representations

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Karl Dilcher

Department of Mathematics and Statistics

Dalhousie University

Halifax, NS B3H 4R2

Canada

Larry Ericksen

P. O. Box 172

Millville, NJ 08332-0172

USA

**Abstract:**

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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