Search

We consider two types of continued fraction expansions of the generating functions of Cauchy numbers. One type has been often studied by many authors, but another has not. In this paper, we give several continued fraction expansions of hypergeometric Cauchy numbers, shifted Cauchy numbers and leaping Cauchy numbers. In special cases, continued fraction expansions of the classical Cauchy numbers of both kinds are deduced.

Carlitz defined the degenerate Bernoulli polynomials β n ( λ , x ) by means of the generating function t ((1 + λ t ) 1/ λ − 1) −1 (1 + λ t ) x / λ . In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy and Euler numbers. In this paper, we show some expressions and properties of hypergeometric degenerate Bernoulli polynomials β N , n ( λ , x ) and numbers, in particular, in terms of determinants. The coefficients of the polynomial β n ( λ , 0) were completely determined by Howard in 1996. We determine the coefficients of the polynomial β N , n ( λ , 0) . Hypergeometric Bernoulli numbers and hypergeometric Cauchy numbers appear in the coefficients.

We introduce a (p, q)-analog of the poly-Euler polynomials and numbers by using the (p, q)-polylogarithm function. These new sequences are generalizations of the poly-Euler numbers and polynomials. We present several combinatorial identities and properties of these new polynomials and also show some relations with (p, q)-poly-Bernoulli polynomials and (p, q)-poly-Cauchy polynomials. The (p, q)-analogs generalize the well-known concept of q-analog.

We give continued fraction expansions of the generating functions of Bernoulli numbers, Cauchy numbers, Euler numbers, harmonic numbers, and their generalized or related numbers. In particular, we focus on explicit forms of the convergents of these continued fraction expansions. Linear fractional transformations of such continued fractions are also discussed. We show more continued fraction expansions for different numbers and types, in particular, on Cauchy numbers.

In 1985, Howard introduced degenerate Cauchy polynomials together with degenerate Bernoulli polynomials. His degenerate Bernoulli polynomials have been studied by many authors, but his degenerate Cauchy polynomials have been forgotten. In this paper, we introduce some kinds of hypergeometric degenerate Cauchy numbers and polynomials from the different viewpoints. By studying the properties of the first one, we give their expressions and determine the coefficients. Concerning the second one, called H-degenerate Cauchy polynomials, we show several identities and study zeta functions interpolating these polynomials.

There are many identities including Fibonacci numbers. However, few determinants of $$n\times n$$ matrix which is equal to the Fibonacci number have been known. In 1974, Proskuryakov showed the first such an example in his Linear Algebra book, though it is believed that the first person is Lucas. Nevertheless, in 1875, Glaisher gave several determinants of matrices which are equal to the Bernoulli, Euler, Cauchy and more numbers. By studying Cameron’s operator in terms of determinants, we introduce the technique to produce many examples of $$n\times n$$ matrix which is equal to the Fibonacci-like numbers.

The objective of this article is to study the periodicity of Lucas-balancing numbers modulo any positive integer. Some relations between the periodicity of balancing and Lucas-balancing numbers are also discussed. Further, in this study the p-adic order of balancing numbers is completely characterized .

We introduce poly-Cauchy numbers with level 2. Poly-Cauchy numbers may be interpreted as a kind of generalizations of the classical Cauchy numbers by using the inverse relation of exponentials and ...

We introduce and study shifted Cauchy numbers as different natural extension of the classical Cauchy numbers, in particular, in terms of determinantal expressions. We give several arithmetical or c...

We study some properties of restricted and associated Fubini numbers. In particular, they have the natural extensions of the original Fubini numbers in the sense of determinants. We also introduce modified Bernoulli and Cauchy numbers and study characteristic properties., Comment: An. \c{S}tiin\c{t}. Univ. "Ovidius" Constan\c{t}a Ser. Mat

We introduce generalized hypergeometric Bernoulli numbers for Dirichlet characters. We study their properties, including relations, expressions and determinants. At the end in Appendix we derive first few expressions of these numbers., Comment: Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas

In this paper, we introduce a parametric type of Bernoulli polynomials with level 3 and study their characteristic and combinatorial properties. In particular, we show some determinant expressions of these polynomials. We also give determinant expressions of a parametric type of Bernoulli polynomials with level 2.

Several articles deal with tilings with squares and dominoes of the well-known regular square mosaic in Euclidean plane, but not any with the hyperbolic regular square mosaics. In this article, we examine the tiling problem with colored squares and dominoes of one type of the possible hyperbolic generalization of $(2\times n)$-board., Comment: 10 pages, 8 figures

By using the generalized Stirling numbers studied by Hsu and Shiue, we define a new kind of generalized poly-Bernoulli and poly-Cauchy numbers. By using the formulae of the generalized Stirling numbers, we give their characteristic and combinatorial properties.

We define a truncated Euler polynomial E m,n (x) as a generalization of the classical Euler polynomial En (x). In this paper we give its some properties and relations with the hypergeometric Bernoulli polynomial.

We give elementary proofs of three formulas involving Bell numbers, including a generalization of the Gould-Quaintance formula and a generalization of Spivey?s formula. We find variants for two of our formulas which involve some well-known sequences, among them the Fibonacci, Bernoulli and Euler numbers.

We introduce leaping Cauchy numbers, that are generalizations of the analogous numbers to Euler numbers, as Cauchy numbers corresponds to Bernoulli numbers, in particular, in terms of determinant expressions. We also give several properties including sums of products.

In this paper we introduce restricted r-Stirling numbers of the first kind. Together with restricted r-Stirling numbers of the second kind and the associated r-Stirling numbers of both kinds, by giving more arithmetical and combinatorial properties, we introduce a new generalization of incomplete poly-Cauchy numbers of both kinds and incomplete poly-Bernoulli numbers.

For positive integers N and M, the general hypergeometric Cauchy polynomials c M,N,n (z) (M, N ≥ 1; n ≥ 0) are defined by $$\frac{1}{(1+t)^z} \frac{1}{{}_2F_1(M,N;N+1;-t)}=\sum_{n=0}^\infty c_{M,N,n}(z)\, \frac{t^n}{n!}\,, $$ where $${{}_2 F_1(a,b;c;z)}$$ is the Gauss hypergeometric function. When M = N = 1, c n = c 1,1,n are the classical Cauchy numbers. In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy and Euler numbers. In the aspect of determinant expressions, hypergeometric Cauchy numbers are the natural extension of the classical Cauchy numbers, though many kinds of generalizations of the Cauchy numbers have been considered by many authors. In this paper, we show some interesting expressions of generalized hypergeometric Cauchy numbers. We also give a convolution identity for generalized hypergeometric Cauchy polynomials.

For any positive integer k and prime p, we define an explicit set A k , p of positive integers n, having positive upper and lower density, on which the p-adic valuation of the Stirling number of the first kind s ( n + 1 , k + 1 ) is a nondecreasing function of n and is given exactly by a simple formula. This is motivated by, and extends, some recent results of Lengyel.

For an integer k, define poly-Euler numbers of the second kind $$\widehat{E}_n^{(k)}$$ ( $$n=0,1,\ldots $$ ) by $$\begin{aligned} \frac{{\mathrm{Li}}_k(1-e^{-4 t})}{4\sinh t}=\sum _{n=0}^\infty \widehat{E}_n^{(k)}\frac{t^n}{n!}. \end{aligned}$$ When $$k=1$$ , $$\widehat{E}_n=\widehat{E}_n^{(1)}$$ are Euler numbers of the second kind or complimentary Euler numbers defined by $$\begin{aligned} \frac{t}{\sinh t}=\sum _{n=0}^\infty \widehat{E}_n\frac{t^n}{n!}. \end{aligned}$$ Euler numbers of the second kind were introduced as special cases of hypergeometric Euler numbers of the second kind in Komatsu and Zhu (Hypergeometric Euler numbers, 2016, arXiv:1612.06210 ), so that they would supplement hypergeometric Euler numbers. In this paper, we study generalized Euler numbers of the second kind and give several properties and applications.

By using the restricted and associated Stirling numbers of the first kind, we define the generalized restricted and associated poly-Cauchy polynomials. By using the restricted and associated Stirling numbers of the second kind, we define the generalized restricted and associated poly-Bernoulli polynomials. These polynomials are generalizations of original poly-Cauchy polynomials and original poly-Bernoulli polynomials, respectively. We also study their characteristic and combinatorial properties.

In 2013, Komatsu introduced the poly-Cauchy numbers, which generalizeCauchy numbers. Several generalizations of poly-Cauchy numbers have been con-sidered since then. One particular type of generalizations is that of multiparameter-poly-Cauchy numbers. In this paper, we study the log-convexity of the multiparame-ter-poly-Cauchy numbers of the rst kind and of the second kind. In addition, wealso discuss the log-behavior of multiparameter-poly-Cauchy numbers.Mathematics Subject Classication (2010): 05A19, 05A20, 11B83.Key words: Cauchy numbers, poly-Cauchy numbers, multiparameter-poly-Cauchy num-bers, log-convexity, log-concavity.

Using the restricted and associated Stirling numbers of the second kind, we define the incomplete multi-poly-Bernoulli numbers which generalize poly-Bernoulli numbers. We study their analytic and combinatorial properties. As an application, we present a new infinite series representation of the multiple zeta values via the Lambert W-function. We also give similar results for incomplete Bernoulli numbers of Hurwitz type and incomplete multi-poly-Bernoulli-star numbers.

We prove characterizations of Appell polynomials by means of symmetric property. For these polynomials, we establish a simple linear expression in terms of Bernoulli and Euler polynomials. As applications, we give interesting examples. In addition, from our study, we obtain Fourier expansions of Appell polynomials. This result recovers Fourier expansions known for Bernoulli and Euler polynomials and obtains the Fourier expansions for higher order Bernoulli-Euler's one.

In this paper, the n-th convergent of the generating function of Cauchy numbers is explicitly given. As an application, we give some new identities of Cauchy numbers in terms of binomial coefficients and harmonic numbers.

In this paper, we define the truncated Bernoulli-Carlitz numbers and the truncated Cauchy-Carlitz numbers as analogues of hypergeometric Bernoulli numbers and hypergeometric Cauchy numbers, and as extensions of Bernoulli-Carlitz numbers and the Cauchy-Carlitz numbers. These numbers can be expressed explicitly in terms of incomplete Stirling-Carlitz numbers.

By using the restricted and associated Stirling numbers of the first kind and by generalizing the (unsigned) Stirling numbers of the first kind, we define the generalized incomplete poly-Cauchy numbers by combining the generalized and the incomplete poly-Cauchy numbers, and study their arithmetical and combinatorial properties. We also study the corresponding generalized incomplete poly-Bernoulli numbers.

By using the restricted Stirling numbers and associated Stirling numbers, we introduce two kinds of incomplete Cauchy numbers, which are generalizations that of the classical Cauchy numbers. We also study several arithmetical and combinatorial properties.

By using the associated and restricted Stirling numbers of the second kind, we give some generalizations of the poly-Bernoulli numbers. We also study their arithmetical and combinatorial properties. As an application, at the end of the paper we present a new infinite series representation of the Riemann zeta function via the Lambert $W$.

The integer values of Cauchy polynomials are expressed in terms of $${r}$$ -Stirling numbers of the first kind. Several relations between the integral values of Bernoulli polynomials and those of Cauchy polynomials are obtained in terms of $${r}$$ -Stirling numbers of both kinds. Also, we find a relation between the Cauchy polynomials and hyperharmonic numbers.

We define q -poly-Bernoulli polynomials B n , ρ , q ( k ) ( z ) with a parameter ρ , q -poly-Cauchy polynomials of the first kind c n , ρ , q ( k ) ( z ) and of the second kind c n , ρ , q ( k ) ( z ) with a parameter ρ by Jackson’s integrals, which generalize the previously known numbers and polynomials, including poly-Bernoulli numbers B n ( k ) and the poly-Cauchy numbers of the first kind c n ( k ) and of the second kind c n ( k ) . We investigate their properties connected with usual Stirling numbers and weighted Stirling numbers. We also give the relations between generalized poly-Bernoulli polynomials and two kinds of generalized poly-Cauchy polynomials.

The $n$-th Cauchy number $c_n$ ($n\ge 0$) are defined by the generating function $x/\ln(1+x)=\sum_{n=0}^\infty c_n x^n/n!$. In this paper, we deal with formulae of the type $$ \sum_{l_1+\cdots+l_m=\mu\atop l_1,\dots,l_m\ge 0}\frac{\mu!}{l_1!\cdots l_m!}(c_{l_1}+\cdots+c_{l_m})^n=a_0 c_{n+\mu}+\cdots+a_{m-1}c_{n+\mu-m+1}\,, $$ where the $a_i$ are suitable rational numbers, the $c_i$ are Cauchy numbers and $$ (c_{l_1}+\cdots+c_{l_m})^n:=\sum_{k_1+\cdots+k_m=n\atop k_1,\dots,k_m\ge 0}\frac{n!}{k_1!\cdots k_m!}c_{k_1+l_1}\cdots c_{k_m+l_m}\,. $$ In special, we give explicit formulae for $m=3$ and $m=4$.

In this note, by using the Hasse-Teichm\"uller derivatives, we obtain two explicit expressions for the related numbers of higher order Appell polynomials. One of them presents a determinant expression for the related numbers of higher order Appell polynomials, which involves several determinant expressions of special numbers, such as the higher order generalized hypergeometric Bernoulli and Cauchy numbers, thus recovers the classical determinant expressions of Bernoulli and Cauchy numbers stated in an article by Glaisher in 1875., Comment: 14 pages

In this paper, we give several characteristics of hypergeometric Bernoulli numbers, including several identities for hypergeometric Bernoulli numbers which the convergents of the continued fraction expansion of the generating function of the hypergeometric Bernoulli numbers entail. We show an analog of Kummer’s congruences in the classical Bernoulli numbers. We also give some determinant expressions of hypergeometric Bernoulli numbers and some relations between the hypergeometric and the classical Bernoulli numbers. By applying Trudi’s formula, we have some different expressions and inversion relations.

By using the restricted and associated Stirling numbers of the first kind by generalizing the (unsigned) Stirling numbers of the first kind, we define the incomplete poly-Cauchy numbers by generalizing the classical Cauchy numbers, and study their arithmetical and combinatorial properties.

In this paper we introduce a q-analogue of the incomplete poly-Bernoulli numbers and incomplete poly-Cauchy numbers by using the q-Hurwitz–Lerch zeta Function. Then we study several combinatorial properties of these new sequences. Moreover, we give some relations between the q-Hurwitz type incomplete poly-Bernoulli numbers, the q-Hurwitz type incomplete poly-Cauchy numbers and the incomplete Stirling numbers of both kinds.

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 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.

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

In a recent paper, de Mathan and Teulié asked whether lim infq→+∞q⋅‖qα‖⋅|q|p = 0 holds for every badly approximable real number α and every prime number p. After a survey of the known results on this open problem, we study the analogous question in fields of power series

Given a set of positive integers A = {a_1,...,a_n}, we study the number p_A (t) of nonnegative integer solutions (m_1,...,m_n) to m_1 a_1 + ... m_n a_n = t. We derive an explicit formula for the polynomial part of p_A., 5 pages