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Polynomial Analogues of Restricted ***b*-ary Partition Functions

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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 *b* ≥ 2, a well studied concept of a *b*-ary
partition function of a positive integer *n* counts the number of
representations of *n* as sums of powers of *b*, with each
power occurring up to λ times, for a fixed λ ≥ 1.
In this paper we
introduce and study a multivariable polynomial sequence that reduces to
the restricted *b*-ary partition function when all variables are
taken to be 1. In particular, we show that this polynomial sequence
characterizes all restricted *b*-ary partitions for each *n*,
generalizing previous results on hyperbinary and hyper *b*-ary
representations. All this will follow from considering more general
concepts of restricted *b*-ary partition functions.

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(Concerned with sequences
A002487
A018819.)

Received December 2 2018;
revised version received May 8 2019.
Published in *Journal of Integer Sequences*,
May 17 2019.

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