# Recurrence Identities of b-ary Partitions

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 220)

## Abstract

Solving the b-ary partition problem, counting the number $$p_b(n)$$ of partitions of n into powers of b, is a pursuit which dates back to Euler. The function $$p_b(n)$$ satisfies a recurrence, and this note examines a family of identities which can be deduced by iterating the recurrence in a suitable way. These identities can then be used to calculate $$p_b(n)$$ for large values of n. Further, these identities correspond to generating function identities involving a sequence of polynomials which have suggestive connections to Eulerian polynomials.

### Keywords

• Integer partitions
• Partition functions
• Recurrence
• Congruences
• Generating functions
• Eulerian polynomials

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## Notes

1. 1.

See Table 1 for values of $$p_b(bn)$$ for small values of b and n. The expression $$p_b(bn)$$ is chosen because by Theorem 3.1 the value of $$p_b(n)$$ is constant on runs of b.

2. 2.

Note that the Stirling numbers $${{n}\brack {k}}$$ and $${{n}\brace {k}}$$ are defined on page 56.

3. 3.

Note that the Eulerian numbers $${\left\langle {n \atop k}\right\rangle }$$ are defined on page 56.

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## Acknowledgements

This research was supported, in part, under National Science Foundation Grants CNS-0958379, CNS-0855217, ACI-1126113 and the City University of New York High Performance Computing Center at the College of Staten Island.

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Correspondence to Dakota Blair .

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