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On Some Generalisations of Gauss Sequences

Sathyanarayan Narayan, N. Uday Kiran

TL;DR

This work introduces Euler-Gauss sequences as a natural generalization bridging Gauss and Euler congruences for integer sequences and extends this framework to $q$-analogues. It proves structural containment (Gauss ⊂ Euler-Gauss ⊂ Euler) and identifies strong Euler-Gauss sequences, providing numerous examples and counterexamples. The SPF and GPF sequences emerge as key nontrivial instances, with new CSP conditions established for their $q$-analogues, enriching the combinatorial interpretation via Lambert-series characterizations. The paper also develops a robust CSP connection for $q$-Euler-Gauss sequences and outlines modified $q$-Gauss congruences, highlighting subtle differences between integer and polynomial settings.

Abstract

In this paper, we introduce integer sequences satisfying new congruence properties inspired by the Euler and Gauss congruences, which we call Euler-Gauss sequences. Noting that every Gauss sequence is an Euler-Gauss sequence, we compare them with certain generalisations of Gauss sequences and provide several counterexamples. In particular, the Smallest Prime Factor (SPF) and Greatest Prime Factor (GPF) sequences (suitably defined at 1) studied extensively by Erdos and Alladi arise as natural examples of Euler-Gauss sequences that are not Gauss sequences. We further extend these congruence-based integer sequences to a q-analog setting and establish characteristic properties that reveal their structure and fill gaps in the literature on q-Gauss sequences. In recent works, q-Gauss sequences have been shown to admit interesting combinatorial interpretations and to exhibit the Cyclic Sieving Phenomenon (CSP). Not only do our q-Euler-Gauss sequences satisfy the standard CSP with some restriction, but we also derive a new CSP condition for the SPF and GPF sequences, not hitherto known in the literature.

On Some Generalisations of Gauss Sequences

TL;DR

This work introduces Euler-Gauss sequences as a natural generalization bridging Gauss and Euler congruences for integer sequences and extends this framework to -analogues. It proves structural containment (Gauss ⊂ Euler-Gauss ⊂ Euler) and identifies strong Euler-Gauss sequences, providing numerous examples and counterexamples. The SPF and GPF sequences emerge as key nontrivial instances, with new CSP conditions established for their -analogues, enriching the combinatorial interpretation via Lambert-series characterizations. The paper also develops a robust CSP connection for -Euler-Gauss sequences and outlines modified -Gauss congruences, highlighting subtle differences between integer and polynomial settings.

Abstract

In this paper, we introduce integer sequences satisfying new congruence properties inspired by the Euler and Gauss congruences, which we call Euler-Gauss sequences. Noting that every Gauss sequence is an Euler-Gauss sequence, we compare them with certain generalisations of Gauss sequences and provide several counterexamples. In particular, the Smallest Prime Factor (SPF) and Greatest Prime Factor (GPF) sequences (suitably defined at 1) studied extensively by Erdos and Alladi arise as natural examples of Euler-Gauss sequences that are not Gauss sequences. We further extend these congruence-based integer sequences to a q-analog setting and establish characteristic properties that reveal their structure and fill gaps in the literature on q-Gauss sequences. In recent works, q-Gauss sequences have been shown to admit interesting combinatorial interpretations and to exhibit the Cyclic Sieving Phenomenon (CSP). Not only do our q-Euler-Gauss sequences satisfy the standard CSP with some restriction, but we also derive a new CSP condition for the SPF and GPF sequences, not hitherto known in the literature.

Paper Structure

This paper contains 9 sections, 4 theorems, 123 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Background
  3. q-Analogs
  4. Euler-Gauss Sequences
  5. Examples
  6. q-Euler-Gauss Sequences
  7. Examples
  8. Smallest Prime Factor and Greatest Prime Factor Sequences
  9. Cyclic Sieving Phenomenon

Key Result

Theorem 1

Let $(S_n(q))$ be the sequence of $q$-analogs of the SPF sequence terms. Define $X_{1}=\emptyset$. For an $n>1$, let $X_{n}$ be a set with $|X_{SPF(n)}|=SPF(n)$ and $\mathbb{Z}_n$ act on $X_n$ such that for all $i\in\mathbb{Z}_n$, where $SPF(n)$ is the smallest prime factor of $n$. Then $\left(X_n,\mathbb{Z}_n,S_n(q)\right)$ is a CSP triple.

Figures (2)

  • Figure 1: Gauss sequences and generalisations
  • Figure 2: $q$-Gauss sequences and generalisations

Theorems & Definitions (19)

  • Theorem : CSP for the $\mathbf{q}$-SPF sequence
  • Theorem
  • Theorem
  • proof
  • Theorem : goro_q-gauss
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 9 more