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Congruences for Fishburn numbers modulo prime powers
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The Fishburn numbers ξ(n) are defined by the formal power series [Formula: see text] Recently, Andrews and Sellers discovered congruences of the form ξ(pm + j) ≡ 0 modulo p, valid for all m ≥ 0. These congruences have then been complemented and generalized to the case of r-Fishburn numbers by Garvan. In this note, we answer a question of Andrews and Sellers regarding an extension of these congruences to the case of prime powers. We show that, under a certain condition, all these congruences indeed extend to hold modulo prime powers.
Title: Congruences for Fishburn numbers modulo prime powers
Description:
The Fishburn numbers ξ(n) are defined by the formal power series [Formula: see text] Recently, Andrews and Sellers discovered congruences of the form ξ(pm + j) ≡ 0 modulo p, valid for all m ≥ 0.
These congruences have then been complemented and generalized to the case of r-Fishburn numbers by Garvan.
In this note, we answer a question of Andrews and Sellers regarding an extension of these congruences to the case of prime powers.
We show that, under a certain condition, all these congruences indeed extend to hold modulo prime powers.
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