Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the determination of $p$-Frobenius and related numbers using the $p$-Apéry set

Takao Komatsu

TL;DR

This work generalizes the Frobenius problem by introducing the p-Apéry set and defining the p-Frobenius, p-Sylvester, and related numerical semigroup invariants through counting representations up to multiplicity $p$. It provides a unified, explicit framework: a central formula expresses sums of representable numbers $\sum_{d(n)\le p} n^{\mu}$ in terms of the $p$-Apéry set and Bernoulli numbers, from which $g_p$, $n_p$, and $s_p$ follow. The authors extend the theory to weighted sums with weight $\lambda^n$, deriving closed forms involving Eulerian numbers and Bernoulli numbers, and discuss degenerate and root-of-unity cases. The results enable systematic, direct computation of p-Frobenius and related quantities and are demonstrated with concrete examples, including new explicit formulas for several special sequences.

Abstract

In this paper, we give convenient formulas in order to obtain explicit expressions of a generalized Frobenius number called the $p$-Frobenius number as well as its related values. Here, for a non-negative integer $p$, the $p$-Frobenius number is the largest integer whose number of solutions of the linear diophantine equation in terms of positive integers $a_1,a_2,\dots,a_k$ with $\gcd(a_1,a_2,\dots,a_k)=1$ is at most $p$. When $p=0$, the problem is reduced to the famous and classical linear Diophantine problem of Frobenius. $0$-Frobenius number is the classical Frobenius number. Our formula is not only a natural extension of the existing classical formulas, but also has the great advantage that the explicit expressions of values such as the $p$-Frobenius and related numbers can be obtained systematically. The concept and formula of the weighted sum has been given recently. We also give a $p$-generalized formula for such weighted sums. The central role is the $p$-Apéry set, which is a generalization of the classical Apéry set.

On the determination of $p$-Frobenius and related numbers using the $p$-Apéry set

TL;DR

This work generalizes the Frobenius problem by introducing the p-Apéry set and defining the p-Frobenius, p-Sylvester, and related numerical semigroup invariants through counting representations up to multiplicity . It provides a unified, explicit framework: a central formula expresses sums of representable numbers in terms of the -Apéry set and Bernoulli numbers, from which , , and follow. The authors extend the theory to weighted sums with weight , deriving closed forms involving Eulerian numbers and Bernoulli numbers, and discuss degenerate and root-of-unity cases. The results enable systematic, direct computation of p-Frobenius and related quantities and are demonstrated with concrete examples, including new explicit formulas for several special sequences.

Abstract

In this paper, we give convenient formulas in order to obtain explicit expressions of a generalized Frobenius number called the -Frobenius number as well as its related values. Here, for a non-negative integer , the -Frobenius number is the largest integer whose number of solutions of the linear diophantine equation in terms of positive integers with is at most . When , the problem is reduced to the famous and classical linear Diophantine problem of Frobenius. -Frobenius number is the classical Frobenius number. Our formula is not only a natural extension of the existing classical formulas, but also has the great advantage that the explicit expressions of values such as the -Frobenius and related numbers can be obtained systematically. The concept and formula of the weighted sum has been given recently. We also give a -generalized formula for such weighted sums. The central role is the -Apéry set, which is a generalization of the classical Apéry set.
Paper Structure (7 sections, 8 theorems, 63 equations, 1 table)

This paper contains 7 sections, 8 theorems, 63 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of Theorem \ref{['th-mp']}
  4. Examples
  5. Weighted sums
  6. Proof of Theorem \ref{['th-hh']}
  7. Examples

Key Result

Theorem 1

Let $k$, $p$ and $\mu$ be integers with $k\ge 2$, $p\ge 0$ and $\mu\ge 0$. Assume that $\gcd(a_1,a_2,\dots,a_k)=1$. We have where $B_n$ are Bernoulli numbers, defined in (eq:ber), and $\{m_0^{(p)},m_1^{(p)},\dots,m_{a_1-1}^{(p)}\}$ is the $p$-apéry set of $A=\{a_1,a_2,\dots,a_k\}$ and $a_1=\min_{1\le i\le k}a_i$.

Theorems & Definitions (10)

  • Theorem 1
  • Corollary 1
  • proof : Proof of Corollary \ref{['cor-mp']}.
  • Corollary 2: bb20
  • Theorem 2
  • Theorem 3
  • Corollary 3
  • Theorem 4
  • proof
  • Corollary 4