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Takao Komatsu

Publications and source records attributed to Takao Komatsu.

At least 19 recordsLinked to original sources
arXiv2026

Finite $q$-multiple harmonic sums at roots of unity: Symmetrized identities for $1$-$2$ index multisets

Closed-form results for finite $q$-multiple harmonic sums are abundant for uniformly repeated indices, whereas mixed-weight tuples pose substantial combinatorial challenges. Building on recent progress for patterns containing a single weight-$2$ entry embedded among weight-$1$ indices, we treat multisets composed of arbitrary numbers of $1$s and $2$s at roots of unity. Since individual ordered sums resist simple evaluation, we analyze their symmetrized sum over all permutations. Using newly derived evaluations for $q$-multiple sums with negative powers together with multiplicative identities, we obtain compact binomial-based closed forms. These enlarge the repertoire of explicitly computable finite $q$-multiple zeta-type objects and lay groundwork for further mixed-index investigations.

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arXiv2026

$p$-numerical semigroup of the sequence of consecutive odd integers

We prove the $p$-Frobenius problems proposed as Conjectures 7.1 and 7.5 developed by T. Komatsu and R. Pandey (Bull. Korean Math. Soc. 2025;62:1397--1409.) for two families of consecutive odd integers. For integers $r,L,n\ge0$, the bounded restricted partition function $p_{\le r}^{(\le L)}(\le n)$ counts partitions of $n$ into at most $r$ parts, each at most $L$. Thus the bounded restricted partition functions $p_{\le 3}^{(\le a)}(\le s)$and $p_{\le 3}^{(\le a+1)}(\le s)$ play central roles in the proofs. Their generating functions are Gaussian polynomials, whose symmetry and unimodality provide a common tool for treating both families.

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arXiv2026

Tribonacci properties of identities, matrices, and determinants

This paper considers the properties of Tribonacci numbers on identities, matrices, and determinants. In the first front part, we obtain several symmetric identities of Tribonacci numbers by a matrix-based approach and binomial inversion technique. In the core section of the latter half, we present a determinant representation of Tribonacci numbers in a slightly modified Toeplitz--Hessenberg form derived from Bell polynomials.

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arXiv2026

Fibonomial determinants

In this paper, we find several determinants expressing the Fibonomial coefficients. We also give the generating functions, Vandermonde identity, and continued fractions about Fibonomial coefficients.

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arXiv2026

Error terms for continued fractions of $e^{1/s}$ and $\sqrt{\frac{v}{u}}\tanh\!\Bigl(\frac{1}{\sqrt{uv}}\Bigr)$

Many classical identities arise from nothing more mysterious than looking at the same object in two different ways. A number, a function, or a combinatorial object may admit several natural decompositions, and by disassembling it in one way and reassembling it in another, we often obtain unexpected corollaries. Telescoping sums provide a particularly vivid incarnation of this principle: by arranging terms so that successive contributions cancel, one performs a conceptual ``cut-and-paste'' that often admits a clean geometric interpretation. Generating functions offer a complementary perspective. Encoding a problem into a formal power series and then evaluating that series at a prescribed point naturally expresses the same quantity as an infinite (or finite) expansion, and equating these representations yields a wealth of identities. For example, for a real number \(α\) given by its continued fraction expansion $α= [a_0, a_1,a_2,\dots]$, with convergents \(p_n/q_n\) and error terms $E_n := p_n - αq_n$, one can obtain ``additive'' decompositions of the form $\sum_{n\ge-1} a_{n+1}\,\lvert E_n\rvert \;=\; α+ 1$, $\sum_{n\ge-1} a_{n+1}\,E_n^{2} \;=\; α$. Thus $α$ and $α+1$ themselves appear as weighted sums of the local approximation errors of their convergents. In this note we explore what such decompositions yield in two explicit cases: the continued fraction \[ e^{1/s} = [1;\,{\overline{(2k-1)s-1,1,1}}]_{k=1}^{\infty} \] and the continued fraction \[ \frac{s}{u}\tanh\!\Bigl(\frac{1}{s}\Bigr) = [\,0;\,\overline{(4k-3)u,\,(4k-1)\tfrac{s^{2}}{u}}\,]_{k=1}^{\infty} \]

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arXiv2026

Continued fractions, determinant expressions, and identities

In this paper, we clarified the relationship between continued fractions, determinants, and identities, making it easier to apply these methods systematically in other settings. In particular, we studied finite continued fractions from the perspective of incomplete numbers (restricted or associated numbers) and also explored their relationships with determinant representations and identities. Most of the new results in this paper concern $q$-analogues of special numbers, whereas the classical cases mainly serve to illustrate and unify the general framework. The framework developed here is flexible and allows one to derive continued fractions, determinant formulas, and coefficient identities in a uniform way for several new $q$-families, and it is expected to be applicable to other families of special numbers, as well.

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arXiv2026

Finite $q$-multiple harmonic sums on $1-\cdots-1,A,1-\cdots-1$ indices

In this paper, we give explicit expressions about $q$-harmonic sums on $1-\cdots-1,A,1-\cdots-1$ indices. When $A=1$, many previous authors have studied and showed the identities, expressions, and properties. There are many results for explicit expressions about $q$-multiple zeta values or $q$-harmonic sums on $A-\cdots-A$ indices. Though there is the way to treat $q$-multiple zeta values unless the indices are the same, it has been successful to get the explicit expression of $q$-harmonic sums on $1-\cdots-1,A,1-\cdots-1$ indices when $A=2$. In this paper, we shall consider more general results when $A\ge 3$.

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arXiv2025

Some explicit values of a $q$-multiple zeta function whose denominator power is not uniform

One of the generalizations of multiple zeta values is the $q$-version, and in the case of finite sums, they may be expressed explicitly in polynomial form. Several results have been found when the powers of the factors in the denominator are equal and when they are small. In this paper, we give explicit formulas for the case when the powers are unequal and are small.

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arXiv2025

Primes of the form $ax+by$ in certain intervals with small solutions

Let $1 0.005\cdot \frac{1}{\ell+1}\frac{g_{\ell,a,b}}{\log g_{\ell,a,b}}. \end{equation*} Let $π_{\ell,a,b}^{*}$ be the number of primes $p\leq g_{\ell,a,b}$ having at most $\ell$ solutions for (1). For an integer $a\ge 3$ and a large sufficiently integer $b$ with $\gcd(a,b)=1$, we also prove that $$ π^{*}_{\ell,a,b}>\frac{(2\ell+1)a}{2(\ell a+a-1)}\frac{g_{\ell,a,b}}{\log g_{\ell,a,b}}. $$ Moreover, if $\ell \frac{\ell+0.02}{\ell+1}\frac{g_{\ell,a,b}}{\log g_{\ell,a,b}}. \end{equation*} These results generalize the previous ones of Chen and Zhu (2025), who established the results for the case $\ell=0$.

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arXiv2025

Some explicit values of a $q$-multiple zeta-star function at roots of unity

In this paper, we show some expressions of certain $q$-multiple zeta-star values at roots of unity. These explicit formulas are expressed by using the determinants or Bell polynomials. Explicit formulas for other types of values can be found from recurrence relations obtained using generating functions.

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arXiv2025

Some explicit values of a $q$-multiple zeta function at roots of unity

In this paper, we give the values of a certain kind of $q$-multiple zeta functions at roots of unity. Various multiple zeta values have been proposed and studied by many researchers, but these multiple zeta values naturally arise from generalizations of Stirling numbers. It is interesting, but by no means easy, to show the values explicitly in certain cases. We give explicit formulas by using Bell polynomials, determinants, $r$-Stirling numbers, etc.

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arXiv2025

Congruence properties of Lehmer-Euler numbers

Certain generalization of Euler numbers was defined in 1935 by Lehmer using cubic roots of unity, as a natural generalization of Bernoulli and Euler numbers. In this paper, Lehmer's generalized Euler numbers are studied to give certain congruence properties together with recurrence and explicit formulas of the numbers. We also show a new polynomial sequence and its properties. Some identities including Euler and central factorial numbers are obtained.

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arXiv2024

The Frobenius number for the triple of the 2-step star numbers

In this paper, we give closed form expressions of the Frobenius number for the triple of the $2$-step star numbers $an(n-2) + 1$ for an integer $a \geq 4$. These numbers have been studied from different aspects for some $a$'s. These numbers can also be considered as variations of the well known star numbers of the form $6n(n-1) + 1$. We also give closed form expressions of the Sylvester number (genus) for the triple of the $2$-step star numbers.

math.CO↗
arXiv2024

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

Recently, the concept of the $p$-numerical semigroup with $p$-symmetric properties has been introduced. When $p=0$, the classical numerical semigroup with symmetric properties is recovered. In this paper, we further study the $p$-numerical semigroup with $p$-almost symmetric properties. We also give $p$-generalized formulas of Watanabe and Johnson, and introduce $p$-Arf numerical semigroup and study its properties.

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arXiv2024

Analytic aspects of $q,r$-analogue of poly-Stirling numbers of both kinds

The Stirling numbers of type $B$ of the second kind count signed set partitions. In this paper we provide new combinatorial and analytical identities regarding these numbers as well as Broder's $r$-version of these numbers. Among these identities one can find recursions, explicit formulas based on the inclusion-exclusion principle, and also exponential generating functions. These Stirling numbers can be considered as members of a wider family of triangles of numbers that are characterized using results of Comtet and Lancaster. We generalize these theorems, which present equivalent conditions for a triangle of numbers to be a triangle of generalized Stirling numbers, to the case of the $q,r$-poly Stirling numbers, which are $q$-analogues of the restricted Stirling numbers defined by Broder and having a polynomial value appearing in their defining recursion. There are two ways to do this and these ways are related by a nice identity.

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arXiv2024

The Frobenius number for shifted geometric sequences associated with the number of solutions

For a non-negative integer $p$, one of the generalized Frobenius numbers, that is called the $p$-Frobenius number, is the largest integer that is represented at most in $p$ ways as a linear combination with nonnegative integer coefficients of a given set of positive integers whose greatest common divisor is one. The famous so-called Frobenius number proposed by Frobenius is reduced to the $0$-Frobenius number when $p=0$. The explicit formula for the Frobenius number with two variables was found in the 19th century, but a formula with more than two variables is very difficult to find, and closed formulas of Frobenius numbers have been found only in special cases such as geometric, Thabit, Mersenne, and so on. The case of $p>0$ was even more difficult, and not a single formula was known. However, most recently, we have finally succeeded in giving the $p$-Frobenius numbers as closed-form expressions of the triangular number triplet , repunits, Fibonacci triplet and Jacobsthal triplet. In this paper, we give closed-form expressions of the $p$-Frobenius number for the finite sequence $\{a b^n-c\}_n$, where $a$, $b$ and $c$ are integers with $a\ge 1$, $b\ge 2$ and $c\ne 0$. This sequence includes the cases for geometric, Thabit and Mersenne as well as their variations.

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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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