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<p>We present a method for obtaining congruences modulo powers of a prime number&#xa0;<i>p</i> for combinatorial sequences whose generating function satisfies an algebraic differential equation. This method generalises the one by Kauers and the authors [<i>Electron. J. Combin.</i> <b>18</b>(2) (2012), Art.&#xa0;P37] from <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(p=2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> to arbitrary primes. Our applications include congruences for numbers of non-crossing graphs and numbers of Kreweras walks modulo powers of&#xa0;3, as well as congruences for Fuß–Catalan numbers and blossom tree numbers modulo powers of arbitrary primes.</p>

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A method for determining the mod-\(p^k\) behaviour of recursive sequences

  • 期刊论文

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  • 发表日期：

    2025年9月8日

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  • C. Krattenthaler,
  • T. W. Müller

摘要

We present a method for obtaining congruences modulo powers of a prime number p for combinatorial sequences whose generating function satisfies an algebraic differential equation. This method generalises the one by Kauers and the authors [Electron. J. Combin. 18(2) (2012), Art. P37] from \(p=2\) p = 2 to arbitrary primes. Our applications include congruences for numbers of non-crossing graphs and numbers of Kreweras walks modulo powers of 3, as well as congruences for Fuß–Catalan numbers and blossom tree numbers modulo powers of arbitrary primes.