scholarly journals A Parametric Type of Cauchy Polynomials with Higher Level

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 207
Author(s):  
Takao Komatsu
Keyword(s):  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Combinatorial Properties ◽  
Cauchy Numbers ◽  
Level 3

There are many kinds of generalizations of Cauchy numbers and polynomials. Recently, a parametric type of the Bernoulli numbers with level 3 was introduced and studied as a kind of generalization of Bernoulli polynomials. A parametric type of Cauchy numbers with level 3 is its analogue. In this paper, as an analogue of a parametric type of Bernoulli polynomials with level 3 and its extension, we introduce a parametric type of Cauchy polynomials with a higher level. We present their characteristic and combinatorial properties. By using recursions, we show some determinant expressions.

scholarly journals Some Identities of Symmetry for the Generalized Bernoulli Numbers and Polynomials

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Taekyun Kim ◽  
Seog-Hoon Rim ◽  
Byungje Lee
Keyword(s):  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Power Sums ◽  
Invariant Integral ◽  
Generalized Bernoulli Polynomials ◽  
Symmetric Properties

By the properties ofp-adic invariant integral onℤp, we establish various identities concerning the generalized Bernoulli numbers and polynomials. From the symmetric properties ofp-adic invariant integral onℤp, we give some interesting relationship between the power sums and the generalized Bernoulli polynomials.

scholarly journals Two types of hypergeometric degenerate Cauchy numbers

2020 ◽  
Vol 18 (1) ◽  
pp. 417-433
Author(s):  
Takao Komatsu
Keyword(s):  
Zeta Functions ◽  
Bernoulli Polynomials ◽  
Cauchy Numbers

Abstract 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.

scholarly journals Symmetric Identities of Hermite-Bernoulli Polynomials and Hermite-Bernoulli Numbers Attached to a Dirichlet Character χ

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 675 ◽  
Author(s):  
Serkan Araci ◽  
Waseem Khan ◽  
Kottakkaran Nisar
Keyword(s):  
Bernoulli Polynomials ◽  
Dirichlet Character ◽  
Bernoulli Numbers ◽  
Complex Order ◽  
Arbitrary Complex ◽  
Symmetric Identities

We aim to introduce arbitrary complex order Hermite-Bernoulli polynomials and Hermite-Bernoulli numbers attached to a Dirichlet character χ and investigate certain symmetric identities involving the polynomials, by mainly using the theory of p-adic integral on Z p . The results presented here, being very general, are shown to reduce to yield symmetric identities for many relatively simple polynomials and numbers and some corresponding known symmetric identities.

scholarly journals Obtaining Easily Sums of Powers on Arithmetic Progressions and Properties of Bernoulli Polynomials by Operator Calculus

2017 ◽  
Vol 9 (5) ◽  
pp. 73
Author(s):  
Do Tan Si
Keyword(s):  
Differential Operator ◽  
Generating Function ◽  
Arithmetic Progression ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Arithmetic Progressions ◽  
Operator Calculus ◽  
Sums Of Powers ◽  
The Way

We show that a sum of powers on an arithmetic progression is the transform of a monomial by a differential operator and that its generating function is simply related to that of the Bernoulli polynomials from which consequently it may be calculated. Besides, we show that it is obtainable also from the sums of powers of integers, i.e. from the Bernoulli numbers which in turn may be calculated by a simple algorithm.By the way, for didactic purpose, operator calculus is utilized for proving in a concise manner the main properties of the Bernoulli polynomials. 

A generalized class of restricted Stirling and Lah numbers

2018 ◽  
Vol 68 (4) ◽  
pp. 727-740 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Generating Function ◽  
Stirling Numbers ◽  
Combinatorial Properties ◽  
Exponential Generating Function ◽  
Cauchy Numbers ◽  
Restricted Stirling Numbers

Abstract In this paper, we consider a polynomial generalization, denoted by $\begin{array}{} u_m^{a,b} \end{array}$ (n, k), of the restricted Stirling numbers of the first and second kind, which reduces to these numbers when a = 1 and b = 0 or when a = 0 and b = 1, respectively. If a = b = 1, then $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) gives the cardinality of the set of Lah distributions on n distinct objects in which no block has cardinality exceeding m with k blocks altogether. We derive several combinatorial properties satisfied by $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) and some additional properties in the case when a = b = 1. Our results not only generalize previous formulas found for the restricted Stirling numbers of both kinds but also yield apparently new formulas for these numbers in several cases. Finally, an exponential generating function formula is derived for $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) as well as for the associated Cauchy numbers.

scholarly journals Voros coefficients for the hypergeometric differential equations and Eynard–Orantin’s topological recursion: Part II: For confluent family of hypergeometric equations

2019 ◽  
Vol 4 (1) ◽  
Author(s):  
Kohei Iwaki ◽  
Tatsuya Koike ◽  
Yumiko Takei
Keyword(s):  
Free Energy ◽  
Differential Equations ◽  
Classical Limit ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Spectral Curves ◽  
Topological Recursion ◽  
Hypergeometric Equations ◽  
Explicit Expressions

Abstract We show that each member of the confluent family of the Gauss hypergeometric equations is realized as quantum curves for appropriate spectral curves. As an application, relations between the Voros coefficients of those equations and the free energy of their classical limit computed by the topological recursion are established. We will also find explicit expressions of the free energy and the Voros coefficients in terms of the Bernoulli numbers and Bernoulli polynomials. Communicated by: Youjin Zhang

scholarly journals Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 847 ◽  
Author(s):  
Dmitry V. Dolgy ◽  
Dae San Kim ◽  
Jongkyum Kwon ◽  
Taekyun Kim
Keyword(s):  
Random Variables ◽  
Infinite Family ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Integer Power ◽  
Bernoulli Numbers And Polynomials ◽  
Power Sums ◽  
Power Sum ◽  
Invariant Integrals

In this paper, we investigate some identities on Bernoulli numbers and polynomials and those on degenerate Bernoulli numbers and polynomials arising from certain p-adic invariant integrals on Z p . In particular, we derive various expressions for the polynomials associated with integer power sums, called integer power sum polynomials and also for their degenerate versions. Further, we compute the expectations of an infinite family of random variables which involve the degenerate Stirling polynomials of the second and some value of higher-order Bernoulli polynomials.

scholarly journals On p-adic Integral Representation of q-Bernoulli Numbers Arising from Two Variable q-Bernstein Polynomials

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 451 ◽  
Author(s):  
Dae Kim ◽  
Taekyun Kim ◽  
Cheon Ryoo ◽  
Yonghong Yao
Keyword(s):  
Integral Representation ◽  
Bernstein Polynomials ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Finite Product ◽  
Bernoulli Numbers And Polynomials ◽  
Special Values ◽  
Invariant Integrals ◽  
Evaluation Problem ◽  
Double Integrals

The q-Bernoulli numbers and polynomials can be given by Witt’s type formulas as p-adic invariant integrals on Z p . We investigate some properties for them. In addition, we consider two variable q-Bernstein polynomials and operators and derive several properties for these polynomials and operators. Next, we study the evaluation problem for the double integrals on Z p of two variable q-Bernstein polynomials and show that they can be expressed in terms of the q-Bernoulli numbers and some special values of q-Bernoulli polynomials. This is generalized to the problem of evaluating any finite product of two variable q-Bernstein polynomials. Furthermore, some identities for q-Bernoulli numbers are found.

scholarly journals Old and New Identities for Bernoulli Polynomials via Fourier Series

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Luis M. Navas ◽  
Francisco J. Ruiz ◽  
Juan L. Varona
Keyword(s):  
Fourier Series ◽  
Linear Combination ◽  
Legendre Polynomials ◽  
Fourier Coefficients ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Gegenbauer Polynomials ◽  
Linear Combinations ◽  
The Given

The Bernoulli polynomialsBkrestricted to[0,1)and extended by periodicity haventh sine and cosine Fourier coefficients of the formCk/nk. In general, the Fourier coefficients of any polynomial restricted to[0,1)are linear combinations of terms of the form1/nk. If we can make this linear combination explicit for a specific family of polynomials, then by uniqueness of Fourier series, we get a relation between the given family and the Bernoulli polynomials. Using this idea, we give new and simpler proofs of some known identities involving Bernoulli, Euler, and Legendre polynomials. The method can also be applied to certain families of Gegenbauer polynomials. As a result, we obtain new identities for Bernoulli polynomials and Bernoulli numbers.

scholarly journals On the Symmetries of theq-Bernoulli Polynomials

2008 ◽  
Vol 2008 ◽  
pp. 1-7 ◽  
Author(s):  
Taekyun Kim
Keyword(s):  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Reflection Symmetry ◽  
Bernoulli Numbers And Polynomials ◽  
Power Sum ◽  
Congruence Properties

Kupershmidt and Tuenter have introduced reflection symmetries for theq-Bernoulli numbers and the Bernoulli polynomials in (2005), (2001), respectively. However, they have not dealt with congruence properties for these numbers entirely. Kupershmidt gave a quantization of the reflection symmetry for the classical Bernoulli polynomials. Tuenter derived a symmetry of power sum polynomials and the classical Bernoulli numbers. In this paper, we study the new symmetries of theq-Bernoulli numbers and polynomials, which are different from Kupershmidt's and Tuenter's results. By using our symmetries for theq-Bernoulli polynomials, we can obtain some interesting relationships betweenq-Bernoulli numbers and polynomials.

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