[Paper Review] A method for determining the mod-$p^k$ behaviour of recursive sequences

Christian Krattenthaler, Thomas W. Müller|arXiv (Cornell University)|Aug 11, 2015

Advanced Combinatorial Mathematics15 references9 citations

TL;DR

This paper presents a general method for computing congruences modulo prime powers $p^k$ for combinatorial sequences defined by algebraic differential equations. By expressing solutions as polynomials in the $p$-adic series $\Phi_p(z) = \sum_{n \geq 0} z^{p^n}$, and leveraging linear independence and coefficient extraction techniques for auxiliary series $H_{b_1,\dots,b_r}(z)$, the authors derive explicit congruences for Fuß–Catalan numbers, Kreweras walks, non-crossing graphs, and blossom trees modulo arbitrary powers of primes.

ABSTRACT

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. 8(2) (2012), Art. P37; arXiv:1107.2015] from $p=2$ 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.

Motivation & Objective

• To generalize the method for computing mod-$2^k$ congruences to arbitrary prime powers $p^k$.
• To develop a systematic framework for deriving congruences modulo $p^k$ for sequences satisfying algebraic differential equations.
• To enable efficient coefficient extraction from powers of $\Phi_p(z)$ via decomposition into linearly independent auxiliary series $H_{b_1,\dots,b_r}(z)$.
• To apply the method to concrete combinatorial sequences, including non-crossing graphs, Kreweras walks, and blossom trees, yielding new congruences modulo $p^k$.

Proposed method

• Represent solutions of differentially algebraic power series over $\mathbb{Z}$ as polynomials in $\Phi_p(z)$ with coefficients in $\mathbb{Z}[z,z^{-1}]$.
• Introduce auxiliary series $H_{b_1,\dots,b_r}(z) = \sum_{n_1 > \cdots > n_r \geq 0} z^{b_1 p^{n_1} + \cdots + b_r p^{n_r}}$ to decompose powers of $\Phi_p(z)$.
• Prove linear independence of $1$ and $H_{b_1,\dots,b_r}(z)$ over $\mathbb{Z}[z,z^{-1}]$ and $\mathbb{Z}/p^k\mathbb{Z}[z,z^{-1}]$ when parameters are coprime to $p$.
• Show that any $H_{b_1,\dots,b_r}(z)$ can be expressed as a $\mathbb{Z}[z,z^{-1}]$-linear combination of the independent basis series.
• Establish an effective algorithm for coefficient extraction from $H_{b_1,\dots,b_r}(z)$ using hypergeometric identities and the Pfaff–Saalschütz formula.
• Apply the framework to derive explicit congruences modulo $p^k$ for sequences like Fuß–Catalan and blossom tree numbers.

Experimental results

Research questions

• RQ1 Can the method for computing mod-$2^k$ congruences be generalized to arbitrary prime powers $p^k$?
• RQ2 What is the minimal degree of polynomial identities satisfied by $\Phi_p(z)$ modulo $p^k$, and how can it be bounded?
• RQ3 How can coefficients of powers of $\Phi_p(z)$ be extracted efficiently modulo $p^k$ for combinatorial applications?
• RQ4 What new congruences can be derived for combinatorial sequences like non-crossing graphs and Kreweras walks modulo $3^k$?
• RQ5 Can the method yield explicit congruences for Fuß–Catalan and blossom tree numbers modulo $p^k$ when $k$ is a power of $p$?

Key findings

• The paper establishes that $\Phi_p(z)$ is transcendental over $\mathbb{Z}[z]$ but algebraic modulo $p^k$, enabling a systematic approach to $p^k$-congruences.
• A conjectured minimal degree for $\Phi_p(z)$ modulo $p^k$ is shown to be a lower bound, though not necessarily tight.
• Congruences modulo $3^k$ are derived for the number of non-crossing graphs with $n$ vertices, extending prior results for $p=2$.
• Congruences modulo $3^k$ are obtained for the number of Kreweras walks of length $n$ in the plane, using the new method.
• For $k = p^m$, the paper proves explicit congruences modulo $p^k$ for Fuß–Catalan numbers $F(n;k) = \frac{1}{n}\binom{kn}{n-1}$.
• Congruences modulo $p^k$ are established for the number of $p$-ary blossom trees with $n$ white nodes, given by $B(n;p) = \frac{p+1}{n((p-1)n+2)}\binom{pn}{n-1}$.