arXiv subjects

Takao Komatsu

Publications and source records attributed to Takao Komatsu.

At least 37 records · Page 2Linked to original sources

arXiv2024

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

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.

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arXiv2024

$p$-numerical semigroups of Pell triples

For a nonnegative integer $p$, the $p$-numerical semigroup $S_p$ is defined as the set of integers whose nonnegative integral linear combinations of given positive integers $a_1,a_2,\dots,a_κ$ with $\gcd(a_1,a_2,\dots,a_κ)=1$ are expressed in more than $p$ ways. When $p=0$, $S=S_0$ is the original numerical semigroup. The largest element and the cardinality of $\mathbb N_0\backslash S_p$ are called the $p$-Frobenius number and the $p$-genus, respectively. Their explicit formulas are known for $κ=2$, but those for $κ\ge 3$ have been found only in some special cases. For some known cases, such as the Fibonacci and the Jacobsthal triplets, similar techniques could be applied and explicit formulas such as the $p$-Frobenius number could be found. In this paper, we give explicit formulas for the $p$-Frobenius number and the $p$-genus of Pell numerical semigroups $\bigl(P_i(u),P_{i+2}(u),P_{i+k}(u)\bigr)$. Here, for a given positive integer $u$, Pell-type numbers $P_n(u)$ satisfy the recurrence relation $P_n(u)=u P_{n-1}(u)+P_{n-2}(u)$ ($n\ge 2$) with $P_0(u)=0$ and $P_1(u)=1$. The $p$-Apéry set is used to find the formulas, but it shows a different pattern from those in the known results, and some case by case discussions are necessary.

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arXiv2023

On a conjecture of Ram\'ırez Alfons\'ın and Skałba III

Let $1<c<d$ be two relatively prime integers. For a non-negative integer $\ell$, let $g_\ell(c,d)$ be the largest integer $n$ such that $n=c x+d y$ has at most $\ell$ non-negative solutions $(x,y)$. In this paper we prove that $$ π_{\ell,c,d}\sim\frac{π\bigl(g_\ell(c,d)\bigr)}{2 \ell+2}\quad(\text{as}~ c\to\infty)\,, $$ where $π_{\ell,c,d}$ is the number of primes $n$ having more than $\ell$ distinct non-negative solutions to $n=c x+d y$ with $n\le g_\ell(c,d)$, and $π(x)$ denotes the number of all primes up to $x$. The case where $\ell=0$ has been proved by Ding, Zhai and Zhao recently, which was conjectured formerly by Ram\'ırez Alfons\'ın and Skałba.

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arXiv2023

Polynomial identities and Fermat quotients

We prove some polynomial identities from which we deduce congruences modulo $p^2$ for the Fermat quotient $\frac{2^p-2}{p}$ for any odd prime $p$ (Proposition 1 and Theorem 1). These congruences are simpler than the one obtained by Jothilingam in 1985 which involves listing quadratic residues in some order. On the way, we also observe some more congruences for the Fermat quotient that generalize Eisenstein's classical congruence. Using such polynomial identities, we obtain some sums involving harmonic numbers. We also prove formulae for binomial sums of harmonic numbers of higher order.

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arXiv2023

The $p$-numerical semigroup of the triple of arithmetic progressions

For given positive integers $a_1,a_2,\dots,a_k$ with $\gcd(a_1,a_2,\dots,a_k)=1$, the denumerant $d(n)=d(n;a_1,a_2,\dots,a_k)$ is the number of nonnegative solutions $(x_1,x_2,\dots,x_k)$ of the linear equation $a_1 x_1+a_2 x_2+\dots+a_k x_k=n$ for a positive integer $n$. For a given nonnegative integer $p$, let $S_p=S_p(a_1,a_2,\dots,a_k)$ be the set of all nonnegative integers $n$'s such that $d(n)>p$. In this paper, we are interested in the $p$-Frobenius number, which is the maximum of the set of gaps $\mathbb N_0\backslash S_p$. Here $\mathbb N_0$ denotes the set of nonnegative integers. When $p=0$, $S=S_0$ is the original numerical semigroup, and the $0$-Frobenius number is the original Frobenius number. The explicit formula for two variables is known not only for $p=0$ but also for $p>0$, but when there are three or more variables, it is difficult even in the special case of $p=0$. For $p>0$, it is not only more difficult, but no explicit formula had been found. In this paper, explicit formulas of the $p$-Frobenius number and related values are given for the triple of arithmetic progressions. The main tool is to determine the elements of the $p$-Apéry set.

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arXiv2023

$p$-numerical semigroups with $p$-symmetric properties

The so-called Frobenius number in the famous linear Diophantine problem of Frobenius is the largest integer such that the linear equation $a_1 x_1+\cdots+a_k x_k=n$ ($a_1,\dots,a_k$ are given positive integers with $\gcd(a_1,\dots,a_k)=1$) does not have a non-negative integer solution $(x_1,\dots,x_k)$. The generalized Frobenius number (called the $p$-Frobenius number) is the largest integer such that this linear equation has at most $p$ solutions. That is, when $p=0$, the $0$-Frobenius number is the original Frobenius number. In this paper, we introduce and discuss $p$-numerical semigroups by developing a generalization of the theory of numerical semigroups based on this flow of the number of representations. That is, for a certain non-negative integer $p$, $p$-gaps, $p$-symmetric semigroups, $p$-pseudo-symmetric semigroups, and the like are defined, and their properties are obtained. When $p=0$, they correspond to the original gaps, symmetric semigroups, and pseudo-symmetric semigroups, respectively.

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arXiv2023

$p$-numerical semigroup of generalized Fibonacci triples

For a nonnegative integer $p$, we give explicit formulas for the $p$-Frobenius number and the $p$-genus of generalized Fibonacci numerical semigroups. Here, the $p$-numerical semigroup $S_p$ is defined as the set of integers whose nonnegative integral linear combinations of given positive integers $a_1,a_2,\dots,a_k$ are expressed more than $p$ ways. When $p=0$, $S_0$ with the $0$-Frobenius number and the $0$-genus is the original numerical semigroup with the Frobenius number and the genus. In this paper, we consider the $p$-numerical semigroup involving Jacobsthal polynomials, which include Fibonacci numbers as special cases. We can also treat with the Jacobsthal-Lucas polynomials, including Lucas numbers accordingly. One of the applications on the $p$-Hilbert series is mentioned.

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arXiv2022

The $p$-Frobenius and $p$-Sylvester numbers for Fibonacci and Lucas triplets

In this paper we study a certain kind of generalized linear Diophantine problem of Frobenius. Let $a_1,a_2,\dots,a_l$ be positive integers such that their greatest common divisor is one. For a nonnegative integer $p$, denote the $p$-Frobenius number by $g_p(a_1,a_2,\dots,a_l)$, which is the largest integer that can be represented at most $p$ ways by a linear combination with nonnegative integer coefficients of $a_1,a_2,\dots,a_l$. When $p=0$, $0$-Frobenius number is the classical Frobenius number. When $l=2$, $p$-Frobenius number is explicitly given. However, when $l=3$ and even larger, even in special cases, it is not easy to give the Frobenius number explicitly, and it is even more difficult when $p>0$, and no specific example has been known. However, very recently, we have succeeded in giving explicit formulas for the case where the sequence is of triangular numbers or of repunits for the case where $l=3$. In this paper, we show the explicit formula for the Fibonacci triple when $p>0$. In addition, we give an explicit formula for the $p$-Sylvester number, that is, the total number of nonnegative integers that can be represented in at most $p$ ways. Furthermore, explicit formulas are shown concerning the Lucas triple.

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arXiv2022

Sylvester sums on the Frobenius set in arithmetic progression with initial gaps

Let $a_1,a_2,\dots,a_k$ be positive integers with $\gcd(a_1,a_2,\dots,a_k)=1$. Frobenius number is the largest positive integer that is NOT representable in terms of $a_1,a_2,\dots,a_k$. When $k\ge 3$, there is no explicit formula in general, but some formulae may exist for special sequences $a_1,a_2,\dots,a_k$, including, those forming arithmetic progressions and their modifications. In this paper we give explicit formulae for the sum of nonrepresentable positive integers (Sylvester sum) as well as Frobenius numbers and the number of nonrepresentable positive integers (Sylverster number) for $a_1,a_2,\dots,a_k$ forming arithmetic progressions with initial gaps.

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arXiv2022

The Frobenius number for sequences of triangular numbers associated with number of solutions

The famous linear diophantine problem of Frobenius is the problem to determine the largest integer (Frobenius number) whose number of representations in terms of $a_1,\dots,a_k$ is at most zero, that is not representable. In other words, all the integers greater than this number can be represented for at least one way. One of the natural generalizations of this problem is to find the largest integer (generalized Frobenius number) whose number of representations is at most a given nonnegative integer $p$. It is easy to find the explicit form of this number in the case of two variables. However, no explicit form has been known even in any special case of three variables. In this paper we are successful to show explicit forms of the generalized Frobenius numbers of the triples of triangular numbers. When $p=0$, their Frobenius number is given by Robles-Pérez and Rosales in 2018.

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arXiv2022

Sylvester power and weighted sums on the Frobenius set in arithmetic progression

Let $a_1,a_2,\dots,a_k$ be positive integers with $\gcd(a_1,a_2,\dots,a_k)=1$. Frobenius number is the largest positive integer that is NOT representable in terms of $a_1,a_2,\dots,a_k$. When $k\ge 3$, there is no explicit formula in general, but some formulae may exist for special sequences $a_1,a_2,\dots,a_k$, including, those forming arithmetic progressions and their modifications. In this paper, we give formulae for the power and weighted sum of nonrepresentable positive integers. As applications, we show explicit expressions of these sums for $a_1,a_2,\dots,a_k$ forming arithmetic progressions.

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arXiv2022

Sylvester sums on the Frobenius set in arithmetic progression

Let $a_1,a_2,\dots,a_k$ be positive integers with $\gcd(a_1,a_2,\dots,a_k)=1$. The concept of the weighted sum $\sum_{n\in{\rm NR}}λ^{n}$ is introduced in \cite{KZ0,KZ}, where ${\rm NR}={\rm NR}(a_1,a_2,\dots,a_k)$ denotes the set of positive integers nonrepresentable in terms of $a_1,a_2,\dots,a_k$. When $λ=1$, such a sum is often called Sylvester sum. The main purpose of this paper is to give explicit expressions of the Sylvester sum ($λ=1$) and the weighed sum ($λ\ne 1$), where $a_1,a_2,\dots,a_k$ forms arithmetic progressions. As applications, various other cases are also considered, including weighted sums, almost arithmetic sequences, arithmetic sequences with an additional term, and geometric-like sequences. Several examples illustrate and confirm our results.

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arXiv2021

Cameron's operator in terms of determinants, and hypergeometric numbers

By studying Cameron's operator in terms of determinants, two kinds of "integer" sequences of incomplete numbers were introduced. One was the sequence of restricted numbers, including $s$-step Fibonacci sequences. Another was the sequence of associated numbers, including Lamé sequences of higher order. By the classical Trudi's formula and the inverse relation, more expressions were able to be obtained. These relations and identities can be extended to those of sequence of negative integers or rational numbers. As applications, we consider hypergeometric Bernoulli, Cauchy and Euler numbers with some modifications.

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arXiv2021

Asymmetric Circular Graph with Hosoya Index and Negative Continued Fractions

It has been known that the Hosoya index of caterpillar graph can be calculated as the numerator of the simple continued fraction. Recently, the author \cite{Komatsu2020} introduces a more general graph called caterpillar-bond graph and shows that its Hosoya index can be calculated as the numerator of the general continued fraction. In this paper, we show how the Hosoya index of the graph with non-uniform ring structure can be calculated from the negative continued fraction. We also give the relation between some radial graphs and multidimensional continued fractions in the sense of the Hosoya index.

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arXiv2021

Weighted Sylvester sums on the Frobenius set in more variables

Let $a_1,a_2,\dots,a_k$ be positive integers with $\gcd(a_1,a_2,\dots,a_k)=1$. Let ${\rm NR}={\rm NR}(a_1,a_2,\dots,a_k)$ denote the set of positive integers nonrepresentable in terms of $a_1,a_2,\dots,a_k$. The largest nonrepresentable integer $\max{\rm NR}$, the number of nonrepresentable positive integers $\sum_{n\in{\rm NR}}1$ and the sum of nonrepresentable positive integers $\sum_{n\in{\rm NR}}n$ have been widely studied for a long time as related to the famous Frobenius problem. In this paper by using Eulerian numbers, we give formulas for the weighted sum $\sum_{n\in{\rm NR}}λ^{n}n^μ$, where $μ$ is a nonnegative integer and $λ$ is a complex number. We also examine power sums of nonrepresentable numbers and some formulae for three variables. Several examples illustrate and support our results.

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arXiv2021

On several kinds of sums of balancing numbers

The balancing numbers $B_n$ ($n=0,1,\cdots$) are solutions of the binary recurrence $B_n=6B_{n-1}-B_{n-2}$ ($n\ge 2$) with $B_0=0$ and $B_1=1$. In this paper we show several relations about the sums of product of two balancing numbers of the type $\sum_{m=0}^n B_{k m+r}B_{k(n-m)+r}$ ($k>r\ge 0$) and the alternating sum of reciprocal of balancing numbers $\left\lfloor\left(\sum_{k=n}^\infty\frac{1}{B_{l k}}\right)^{-1}\right\rfloor$. Similar results are also obtained for Lucas-balancing numbers $C_n$ ($n=0,1,\cdots$), satisfying the binary recurrence $C_n=6C_{n-1}-C_{n-2}$ ($n\ge 2$) with $C_0=1$ and $C_1=3$. Some binomial sums involving these numbers are also explored.

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arXiv2021

Recurrence relations of poly-Cauchy numbers by the $r$-Stirling transform

We give some formulas of poly-Cauchy numbers by the $r$-Stirling transform. In the case of the classical or poly-Bernoulli numbers, the formulas are with Stirling numbers of the first kind. In our case of the classical or poly-Cauchy numbers, the formulas are with Stirling numbers of the second kind. We also discuss annihilation formulas for poly-Cauchy number with negative indices.

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arXiv2021

Stirling numbers with level $2$ and poly-Bernoulli numbers with level $2$

In this paper, we introduce poly-Bernoulli numbers with level $2$, related to the Stirling numbers of the second kind with level $2$, and study several properties of poly-Bernoulli numbers with level $2$ from their expressions, relations, and congruences. Poly-Bernoulli numbers with level $2$ have strong connections with poly-Cauchy numbers with level $2$. In a special case, we can determine the denominators of Bernoulli numbers with level $2$ by showing a von Staudt-Clausen like theorem.

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