poly-Cauchy Numbers Research Articles

Poly-Cauchy Numbers with Higher Level

In this article, mainly from the analytical aspect, we introduce poly-Cauchy numbers with higher levels (level s) as a kind of extensions of poly-Cauchy numbers with level 2 and the original poly-Cauchy numbers and investigate their properties. Such poly-Cauchy numbers with higher levels are yielded from the inverse relationship with an s-step function of the exponential function. We show such a function with recurrence relations and give the expressions of poly-Cauchy numbers with higher levels. Poly-Cauchy numbers with higher levels can be also expressed in terms of iterated integrals and a combinatorial summation. Poly-Cauchy numbers with higher levels for negative indices have a double summation formula. In addition, Cauchy numbers with higher levels can be also expressed in terms of determinants.

On s-Stirling transform and poly-Cauchy numbers of the second kind with level 2

On s-Stirling transform and poly-Cauchy numbers of the second kind with level 2

Convolution identities of poly-Cauchy numbers with level 2

Poly-Cauchy numbers with level 2 are defined by inverse sine hyperbolic functions with the inverse relation from sine hyperbolic functions. In this paper, we introduce the Stirling numbers of the first kind with level 2 in order to establish some relations with poly- Cauchy numbers with level 2. Then, we show several convolution identities of poly-Cauchy numbers with level 2. In particular, that of three poly-Cauchy numbers with level 2 can be expressed as a simple form.

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.

A Combinatorial Approach to the Stirling Numbers of the First Kind with Higher Level

In this paper, we investigate a generalization of the classical Stirling numbers of the first kind by considering permutations over tuples with an extra condition on the minimal elements of the cycles. The main focus of this work is the analysis of combinatorial properties of these new objects. We give general combinatorial identities and some recurrence relations. We also show some connections with other sequences such as poly-Cauchy numbers with higher level and central factorial numbers. To obtain our results, we use pure combinatorial arguments and classical manipulations of formal power series.

Poly-Cauchy numbers of the second kind - the combinatorics behind

Poly-Cauchy numbers of the second kind - the combinatorics behind

Poly-Cauchy numbers with level 2

ABSTRACT 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 logarithms. On the contrary, poly-Bernoulli numbers can be from the inverse relation of logarithms and exponentials. In this similar stream, poly-Cauchy numbers with level 2 may be yielded from the inverse relation about the hyperbolic sine function, which is a 2-step function of the exponential function. In this article, we show several expressions, relations, and properties about poly-Cauchy numbers with level 2. Poly-Cauchy numbers with level 2 can be expressed in terms of multinomial coefficients with combinatorial summation, Stirling numbers of the first kind, or iterated integrals. We also give some recurrence relations for poly-Cauchy numbers with level 2. When the index is negative, the double summation may be formulated as a closed form. A simple case of Cauchy numbers with level 2 has some more relations with D numbers from higher-order Bernoulli numbers or complementary Euler numbers. We prove some more expressions in determinants, continued fractions or by Trudi's formula.

Some recurrence relations of poly-Cauchy numbers

Some recurrence relations of poly-Cauchy numbers

Restricted r-Stirling numbers and their combinatorial applications

We study set partitions with r distinguished elements and block sizes found in an arbitrary index set S. The enumeration of these (S, r)-partitions leads to the introduction of (S, r)-Stirling numbers, an extremely wide-ranging generalization of the classical Stirling numbers and the r-Stirling numbers. We also introduce the associated (S, r)-Bell and (S, r)-factorial numbers. We study fundamental aspects of these numbers, including recurrence relations and determinantal expressions. For S with some extra structure, we show that the inverse of the (S, r)-Stirling matrix encodes the Möbius functions of two families of posets. Through several examples, we demonstrate that for some S the matrices and their inverses involve the enumeration sequences of several combinatorial objects. Further, we highlight how the (S, r)-Stirling numbers naturally arise in the enumeration of cliques and acyclic orientations of special graphs, underlining their ubiquity and importance. Finally, we introduce related (S, r) generalizations of the poly-Bernoulli and poly-Cauchy numbers, uniting many past works on generalized combinatorial sequences.

Generalized Stirling numbers with poly-Bernoulli and poly-Cauchy numbers

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.

Generalized poly-Cauchy and poly-Bernoulli numbers by using incomplete $${\varvec{r}}$$ r -Stirling numbers

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.

Incomplete Poly-Bernoulli Numbers and Incomplete Poly-Cauchy Numbers Associated to the q-Hurwitz–Lerch Zeta Function

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.

The log-convexity of the poly-Cauchy numbers

In 2013, Komatsu introduced the poly-Cauchy numbers, which generalize Cauchy numbers. Several generalizations of poly-Cauchy numbers have been considered since then. One particular type of generalizations is that of multiparameter-poly-Cauchy numbers. In this paper, we study the log-convexity of the multiparameter-poly-Cauchy numbers of the first kind and of the second kind. In addition, we also discuss the log-behavior of multiparameter-poly-Cauchy numbers.

Generalized incomplete poly-Bernoulli and poly-Cauchy 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.

Generalizations of poly-Bernoulli and poly-Cauchy numbers

In this paper we consider some generalizations of poly-Bernoulli and poly-Cauchy numbers. The first is by means of the Hurwitz–Lerch zeta function. The second generalization is via weighted Stirling numbers. The third one is given with the help of degenerate Stirling numbers. All these generalizations lead to symmetries between various types of Stirling numbers, and enable us to investigate and expand algebraic properties of poly-Bernoulli and poly-Cauchy numbers. We also combine these generalizations and derive numerous combinatorial and arithmetical identities.

Incomplete poly-Cauchy numbers

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.

Formula omitted]-poly-Bernoulli numbers and [formula omitted]-poly-Cauchy numbers with a parameter by Jackson’s integrals

We define q-poly-Bernoulli polynomials Bn,ρ,q(k)(z) with a parameter ρ, q-poly-Cauchy polynomials of the first kind cn,ρ,q(k)(z) and of the second kind ĉn,ρ,q(k)(z) with a parameter ρ by Jackson’s integrals, which generalize the previously known numbers and polynomials, including poly-Bernoulli numbers Bn(k) and the poly-Cauchy numbers of the first kind cn(k) and of the second kind ĉn(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.

Sums of Products of Cauchy Numbers, Including Generalized Poly-Cauchy Numbers

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.

Poly-Bernoulli numbers and polynomials with a q parameter

We define poly-Bernoulli numbers and polynomials with a q parameter. These numbers and polynomials show a correspondence to poly-Cauchy numbers and polynomials with a q parameter recently defined by the second author. We study some arithmetical and number theoretical properties of these numbers and polynomials as well.

Multiparameter poly-Cauchy and poly-Bernoulli numbers and polynomials

Recently, Komastu introduced the concept of poly-Cauchy numbers and polynomials which generalize Cauchy numbers and polynomials. In this paper, we introduce new generaliza- tion of poly-Cauchy and poly-Bernoulli numbers and polynomials. Also, we introduce new generalizations of Cauchy numbers and polynomials. Moreover, we derive some identities involving the new numbers and polynomials and some types of Stirling numbers. These gives generalization of some relations poly-Cauchy and poly-Bernoulli numbers and poly- nomials. Furthermore, we obtain some relations between the multiparameter poly-Cauchy numbers and polynomials and new multiparameter poly-Bernoulli numbers and polyno- mials.