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Euler's famous formula written in symbolic notation as $(B_0+B_0)^n=-n B_{n-1}-(n-1)B_n$ was extended to $(B_{l_1}+\cdots+B_{l_m})^n$ for $m\ge 2$ and arbitrary fixed integers $l_1,\dots,l_m\ge 0$. In this paper, we consider the higher-order recurrences for Cauchy numbers $(c_{l_1}+\cdots+c_{l_m})^n$, where the $n$-th Cauchy number $c_n$ ($n\ge 0$) is defined by the generating function $x/\ln(1+x)=\sum_{n=0}^\infty c_n x^n/n!$. In special, we give an explicit expression in the case $l_1=\cdots=l_m=0$ for any integers $n\ge 1$ and $m\ge 2$. We also discuss the case for Cauchy numbers of the second kind $\widehat c_n$ in similar ways.

We prove several explicit formulae for the $n$-th Bernoulli polynomial $B_{n}(x)$, in which $B_{n}(x)$ is equal to an affine combination of the polynomials $(x-1)^{n}$, $(x-2)^{n}$, $ldots$, $(x-k-1)^{n}$, where $k$ is any fixed positive integer greater or equal than $n$.

In this paper, we introduce the hypergeometric Euler number as an analogue of the hypergeometric Bernoulli number and the hypergeometric Cauchy number. We study several expressions and sums of products of hypergeometric Euler numbers. We also introduce complementary hypergeometric Euler numbers and give some characteristic properties. There are strong reasons why these hypergeometric numbers are important. The hypergeometric numbers have one of the advantages that yield the natural extensions of determinant expressions of the numbers, though many kinds of generalizations of the Euler numbers have been considered by many authors.

We prove nonlinear relation on multiple Hurwitz-Riemann zeta functions. Using analytic continuation of these multiple Hurwitz-Riemann zeta function, we quote at negative integers Euler's nonlinear relation for generalized Bernoulli polynomials and numbers.

Denote the nth convergent of the continued fraction α=[a0;a1,a2,…] by pn/qn=[a0;a1,…,an]. In this paper we give exact formulae for the quantities Dn:=qnα−pn in several typical types of Tasoev continued fractions. A simple example of the type of Tasoev continued fraction considered is α=[0;ua,ua2,ua3,…].

Consider linear three-term recurrence relation Z n = a 1 Z n−1 + a 2 Z n−2 (n ≥ 2). Then, from our former theorem, the three-term leaping recurrence relation holds for any positive integer k. In this article we shall consider the linear four-term recurrence relation Z n = a 1 Z n−1 + a 2 Z n−2 + a 3 Z n−3 (n ≥ 3). Then we can have the four-term leaping recurrence relation Z n = b 1 Z n−k + b 2 Z n−2k + b 3 Z n−3k for a given positive integer k with 1 < k < n/3.

Extending the work of Burger et al., here we show that every quasi-periodic simple continued fraction α can be transformed into a quasi-periodic non-simple continued fraction having period length one. Moreover, a certain kind of quasi-periodic non-simple continued fraction is equivalent to a quasi-periodic N-continued fraction. The results of this paper follow from arguments of Burger et al. but we apply our version to offer new continued fractions for certain classes of real numbers.

Many new types of Hurwitz continued fractions have been studied by the author. In this paper we show that all of these closed forms can be expressed by using confluent hypergeometric functions 0 F1(;c;z). In the application we study some new Hurwitz continued fractions whose closed form can be expressed by using confluent hypergeometric functions.

Nous calculons des bornes superieures et inferieures pour l'approximation de fonctions hyperboliques aux points 1/s (s = 1,2,...) par des rationnels x/y, tels que x,y satisfassent une equation quadratique. Par exemple, tous les entiers positifs x,y avec y = 0 (mod 2), solutions de l'equation de Pythagore x 2 + y 2 = z 2 , satisfont |ysinh(1/s)-x|>>loglogy logy Reciproquement, pour chaque s = 1,2,..., il existe une infinite d'entiers x, y, premiers entre eux, tels que |ysinh(1/s)-x|<

Many researchers have investigated sums of products of Bernoulli numbers and related types of sums of products. Recently, the concept of poly-Cauchy numbers has been introduced by the author as a generalization of the classical Cauchy number. In this paper, we investigate sums of products of Cauchy numbers including generalized poly-Cauchy numbers. We give a relation among these sums and explicit expressions of sums of any arbitrary products. In addition, we also show the similar results in the case of poly-Cauchy numbers of the second kind and those of mixed ones of both kinds.

In 1935 Carlitz introduced Bernoulli–Carlitz numbers as analogues of Bernoulli numbers for the rational function field F r ( T ) . In this paper, we introduce Cauchy–Carlitz numbers as analogues of Cauchy numbers. By using Stirling–Carlitz numbers, we give their arithmetical and combinatorial properties and relations with Bernoulli–Carlitz numbers for F r ( T ) . Several new identities are also obtained by using Hasse–Teichmuller derivatives.

In this note, we study the divisibility relation $U_m\mid U_{n+k}^s-U_n^s$, where ${\bf U}:=\{U_n\}_{n\ge 0}$ is the Lucas sequence of characteristic polynomial $x^2-ax\pm 1$ and $k,m,n,s$ are positive integers., Comment: 19 pags. Submitted

The tanh-type, tan-type, and e-type Hurwitz continued fractions have been generalized by the author. In this paper, we study a generalized form of e2-type Hurwitz continued fractions by using confluent hypergeometric functions. We also obtain a similar type of Tasoev continued fractions.

To clear up both algebraic and geometric structures for integrable systems derived from self-consistent field theory, in particular, a geometrical aspect of the random phase approximation (RPA) equation is presented from the viewpoint of symmetry of the evolution equation. The RPA equation for an infinite-dimensional Grassmannian is constructed. It gives us a simple geometrical interpretation that the collective submanifold is a rotator on a curved surface.

Rational approximations to Tasoev continued fractions, the exponential quasi-periodic continued fractions, are given. These values are the best possible for all integers.

The continued fractions studied by Tasoev are not widely known although their characteristics are very similar to those of Hurwitz continued fractions. Recently, the author found several general forms of Tasoev continued fractions, and by applying this method he also obtained some more general forms of Hurwitz continued fractions belonging to so called tanh-type and tan-type. In this paper, we constitute a new class of general forms of Hurwitz continued fractions of e-type. The known continued fraction expansions of e1/a (a 1), ae1/a and (1/a)e1/a are included as special cases. The corresponding Tasoev continued fractions are also derived.

We show some new variations on Tasoev's continued fractions [ 0 ; a k , … , a k ︸ m ¯ ] k = 1 ∞ , where the periodic parts include the exponentials in k instead of the polynomials in k . We also mention some relations with other kinds of continued fractions, in particular, with Rogers–Ramanujan continued fractions.

The hypergeometric series reduce to many elementary functions. Many of them are known to have the continued fraction expansions. Few of them for certain values, however, are known to have the simple continued fraction expansions which keep reg- ularities. In this paper we show more simple continued fraction expansions holding regularities.

(Une approximation des irrationnels quadratiques). - Soit a > 1 un irrationnel. Plusieurs auteurs ont etudie les nombres m (α) = inf{|y|: y ∈ Λ m , y ¬= 0}, ou m est un entier positif et Λ m est l'ensemble de tous les reels de la forme y = ∈ 0 α n + ∈ 1 α n-1 + ... + ∈ n-1 α + ∈ n avec des |∈ i | ≤ m entiers. La valeur de 1 (α) a ete precisee pour beaucoup de nombres de Pisot et m (α) pour le nombre d'or. Dans cet article, on determine m (α) lorsque a est un irrationnel qui satisfait α 2 = aα ± 1 avec a entier positif.

In this paper, by considering Barnes-type Narumi polynomials of the second kind and Barnes-type Peters polynomials of the second kind, we define and investigate the hybrid polynomials of these polynomials. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities.

In this paper, by considering Barnes-type Daehee polynomials of the first kind as well as poly-Cauchy polynomials of the first kind, we define and investigate the mixed-type polynomials of these polynomials. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities. MSC:05A15, 05A40, 11B68, 11B75, 65Q05.

We show that \begin{equation*} \gcd(2!S(2n+1,2),…,(2n+1)!S(2n+1,2n+1))=\text{denominator of $B_{2n}$}, \end{equation*} where $S(n,k)$ is the Stirling number of the second kind and $B_{n}$ is the Bernoulli number.

In this paper, we consider Barnes' multiple Frobenius-Euler and poly-Bernoulli mixed-type polynomials. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities.

In this paper, we establish more properties of generalized poly-Euler polynomials with three parameters and we investigate a kind of symmetrized generalization of poly- Euler polynomials. Moreover, we introduce a more general form of multi poly-Euler polynomials and obtain some identities parallel to those of the generalized poly-Euler polynomials., 17 pages

Toward a unified algebraic understanding of physical concepts of the so-called particle and collective motions, a new theory for unified description of both the motions is proposed with the use of time-dependent Hartree-Fock theory on a circle S1. The theory simply and clearly elucidates a collective motion induced by a TD mean-field potential. It also describes symmetry breaking of fermion systems and successive occurrence of the collective motion due to recovery of the symmetry. The theoretical frame asserts that the Fock space of finite-dimensional fermions has an algebraic structure to be embedded into that of infinite-dimensional fermions.

A unified aspect of the self-consistent field (SCF) method and the τ-functional method is presented. SCF theory in the τ-functional space F∞ manifestly results in a gauge theory of fermions and then a collective motion appears as a motion of fermion gauges with a common factor. This provides a new algebraic tool for the microscopic understanding of fermion many-body systems.

We consider the values |q|→∞|q|‖qθ − φ‖, where θ is a Hurwitzian number, in other words, its continued fraction expansion includes a quasi-periodic form. Especially, we set θ = e1/s, where s is a positive integer.

and M−(θ, φ) = lim inf q→+∞ q‖qθ + φ‖ = lim inf q→−∞ |q|‖qθ − φ‖. ThenM(θ, φ) = min(M+(θ, φ),M−(θ, φ)). These notations are introduced by Cusick, Rockett and Szusz [2].M(θ, φ) orM+(θ, φ) has been treated by Cassels [1], Descombes [3], Sos [9], Cusick et al. [2] and the author [5] by using several algorithms for inhomogeneous Diophantine approximation in which φ is expressed by the continued fraction expansion of θ. However, it is not easy to evaluate M(θ, φ) if it exists for any given pair of θ and φ. In this paper we establish the relationship between M(θ, φ) and the algorithm of Nishioka, Shiokawa and Tamura. Indeed, this was hinted at in [5] but has not been proved yet. If we use this result, we can find the

We consider a new algorithm yielding the values M (θ, φ)=lim inf|q|→∞ |q|‖qθ−φ‖. Specifically, we compute M (θ, 1/a), M (θ, 1/2a), M (θ, 1/ a 2 +4 ) , and M (θ, 1/2) when θ=( a 2 +4 −a)/2=[0, a, a, ...] .

Let $\xi=\sqrt{v/u}\tanh(uv)^{-1/2}$, where $u$ and $v$ are positive integers, and let $\eta=|h(\xi)|$, where $h(t)$ is a non-constant rational function with algebraic coefficients. We compute upper and lower bounds for the approximation of certain values $\eta$ of hyperbolic functions by rationals $x/y$ such that $x$ and $y$ satisfy Diophantine equations. We show that there are infinitely many coprime integers $x$ and $y$ such that $|y\eta-x|\ll\log\log y/\log y$ and a Diophantine equation holds simultaneously relating $x$ and $y$ and some integer $z$. Conversely, all positive integers $x$ and $y$ with $y\ge c_0$ solving the Diophantine equation satisfy $|y\eta-x|\gg\log\log y/\log y$.

In Komatsu's work (2013), the concept of poly-Cauchy numbers is introduced as an analogue of that of poly-Bernoulli numbers. Both numbers are extensions of classical Cauchy numbers and Bernoulli numbers, respectively. There are several generalizations of poly-Cauchy numbers, including poly-Cauchy numbers with a q parameter and shifted poly-Cauchy numbers. In this paper, we give a further generalization of poly-Cauchy numbers and investigate several arithmetical properties. We also give the corresponding generalized poly-Bernoulli numbers so that both numbers have some relations.

A recent paper (J. Number Theory42(1992), 61–87) announced various arithmetical properties of the Mahler functionf(θ, φ;x, y)=∑∞k=1∑1≤m≤kθ+φxkym. Unfortunately the arguments of that paper are marred by an error whereby the arguments hold only forφ=0 (or whenbn=1 for all positive integersn). We show how to correct the arguments so that they do hold for generalφ. Moreover, another paper (J. Number Theory43(1993), 293–318) soon afterwards happened again to discuss the subject using different expressions for the power series forf. We show that the corrected results here do coincide with those later results notwithstanding our alternative presentation

In this article, we give upper bounds for cyclotomic numbers of order e over a finite field with q elements, where e is a divisor of q-1. In particular, we show that under certain assumptions, cyclotomic numbers are at most $\lceil\frac{k}{2}\rceil$, and the cyclotomic number (0,0) is at most $\lceil\frac{k}{2}\rceil-1$, where k=(q-1)/e. These results are obtained by using a known formula for the determinant of a matrix whose entries are binomial coefficients., 11 pages, minor revision