scholarly journals On $q$-Hypergeometric Bernoulli polynomials and numbers

2021 ◽  
pp. 1-17
Author(s):  
Salifou MBOUTNGAM ◽  
Patrick NJİONOU SADJANG
Keyword(s):  
Bernoulli Polynomials ◽  
Bernoulli Polynomials And Numbers

scholarly journals A Note on the Multiple Twisted Carlitz's Type -Bernoulli Polynomials

2008 ◽  
Vol 2008 ◽  
pp. 1-7 ◽  
Author(s):  
Lee-Chae Jang ◽  
Cheon-Seoung Ryoo
Keyword(s):  
Bernoulli Polynomials ◽  
Bernoulli Polynomials And Numbers

We give the twisted Carlitz's type -Bernoulli polynomials and numbers associated with -adic -inetgrals and discuss their properties. Furthermore, we define the multiple twisted Carlitz's type -Bernoulli polynomials and numbers and obtain the distribution relation for them.

scholarly journals New degenerate Bernoulli, Euler, and Genocchi polynomials

2020 ◽  
Vol 29 (1) ◽  
pp. 1-16
Author(s):  
Orli Herscovici ◽  
Toufik Mansour
Keyword(s):  
Exponential Function ◽  
Bernoulli Polynomials ◽  
Genocchi Polynomials ◽  
Bernoulli Polynomials And Numbers ◽  
Euler Identity

AbstractWe introduce new generalizations of the Bernoulli, Euler, and Genocchi polynomials and numbers based on the Carlitz-Tsallis degenerate exponential function. Also, we present generalizations of some familiar identities and connection between these types of Bernoulli, Euler, and Genocchi polynomials. Moreover, we establish new analogues of the Euler identity for degenerate Bernoulli polynomials and numbers.

scholarly journals Formulas involving sums of powers, special numbers and polynomials arising from p-adic integrals, trigonometric and generating functions

2020 ◽  
Vol 108 (122) ◽  
pp. 103-120
Author(s):  
Neslihan Kilar ◽  
Yilmaz Simsek
Keyword(s):  
Generating Functions ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Sums Of Powers ◽  
Alternating Sums ◽  
Special Numbers And Polynomials ◽  
Bernoulli Polynomials And Numbers ◽  
Euler Polynomials And Numbers ◽  
Historical Remarks ◽  
Positive Integers

The formula for the sums of powers of positive integers, given by Faulhaber in 1631, is proven by using trigonometric identities and some properties of the Bernoulli polynomials. Using trigonometric functions identities and generating functions for some well-known special numbers and polynomials, many novel formulas and relations including alternating sums of powers of positive integers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Fubini numbers, the Stirling numbers, the tangent numbers are also given. Moreover, by applying the Riemann integral and p-adic integrals involving the fermionic p-adic integral and the Volkenborn integral, some new identities and combinatorial sums related to the aforementioned numbers and polynomials are derived. Furthermore, we serve up some revealing and historical remarks and observations on the results of this paper.

scholarly journals Degenerate Catalan-Daehee numbers and polynomials of order $ r $ arising from degenerate umbral calculus

2021 ◽  
Vol 7 (3) ◽  
pp. 3845-3865
Author(s):  
Hye Kyung Kim ◽  
◽  
Dmitry V. Dolgy ◽  
Keyword(s):  
Bernoulli Polynomials ◽  
Umbral Calculus ◽  
Bell Polynomials ◽  
Psychological Burden ◽  
Special Polynomials ◽  
Sheffer Sequences ◽  
Special Numbers And Polynomials ◽  
Bernoulli Polynomials And Numbers ◽  
Euler Polynomials And Numbers ◽  
Degenerate Version

<abstract><p>Many mathematicians have studied degenerate versions of some special polynomials and numbers that can take into account the surrounding environment or a person's psychological burden in recent years, and they've discovered some interesting results. Furthermore, one of the most important approaches for finding the combinatorial identities for the degenerate version of special numbers and polynomials is the umbral calculus. The Catalan numbers and the Daehee numbers play important role in connecting relationship between special numbers.</p> <p>In this paper, we first define the degenerate Catalan-Daehee numbers and polynomials and aim to study the relation between well-known special polynomials and degenerate Catalan-Daehee polynomials of order $ r $ as one of the generalizations of the degenerate Catalan-Daehee polynomials by using the degenerate Sheffer sequences. Some of them include the degenerate and other special polynomials and numbers such as the degenerate falling factorials, the degenerate Bernoulli polynomials and numbers of order $ r $, the degenerate Euler polynomials and numbers of order $ r $, the degenerate Daehee polynomials of order $ r $, the degenerate Bell polynomials, and so on.</p></abstract>

Upper bounds for analytic summand functions and related inequalities

2021 ◽  
Vol 71 (5) ◽  
pp. 1103-1112
Author(s):  
Soodeh Mehboodi ◽  
M. H. Hooshmand
Keyword(s):  
Special Functions ◽  
Bernoulli Polynomials ◽  
Upper Bounds ◽  
Bernoulli Numbers ◽  
Natural Numbers ◽  
Bernoulli Polynomials And Numbers

Abstract The topic of analytic summability of functions was introduced and studied in 2016 by Hooshmand. He presented some inequalities and upper bounds for analytic summand functions by applying Bernoulli polynomials and numbers. In this work we apply upper bounds, represented by Hua-feng, for Bernoulli numbers to improve the inequalities and related results. Then, we observe that the inequalities are sharp and leave a conjecture about them. Also, as some applications, we use them for some special functions and obtain many particular inequalities. Moreover, we arrived at the inequality 1 p + 2 p + 3 p + ⋯ + r p ≤ 1 2 r p + 1 3 r p + 1 ( p + 1 ) + 2 3 p ! π p + 1 sinh ⁡ ( π r ) $1^p + 2^p + 3^p + \dots + r^p \leq \frac{1}{2}r^p + \frac{1}{3}\frac{r^{p+1}}{(p+1)} + \frac{2}{3}\frac{p!}{\pi^{p+1}}\sinh(\pi r)$ , for r sums of power of natural numbers, if p ∈ ℕ e and analogously for the odd case.

scholarly journals Modified Degenerate Carlitz’s q-Bernoulli Polynomials and Numbers with Weight (α, β)

2017 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz
Keyword(s):  
Recurrence Relations ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Bernoulli Polynomials And Numbers

The main goal of the present paper is to construct some families of the Carlitz&rsquo;s q-Bernoulli polynomials and numbers. We firstly introduce the modified Carlitz&rsquo;s q-Bernoulli polynomials and numbers with weight&nbsp;(&alpha;, &beta;) and investigate their some explicit properties and identities arising from the bosonic q-Volkenborn integral on&nbsp;ℤp. We then define the modified degenerate Carlitz&rsquo;s q-Bernoulli polynomials and numbers with weight&nbsp;(&alpha;, &beta;) and obtain some recurrence relations and other identities. Moreover, we derive some correlations with the modified Carlitz&rsquo;s q-Bernoulli polynomials with weight (&alpha;, &beta;), the modified degenerate Carlitz&rsquo;s q-Bernoulli polynomials with weight (&alpha;, &beta;), the Stirling numbers of the first kind and second kind.

p-Bernoulli and geometric polynomials

2018 ◽  
Vol 14 (02) ◽  
pp. 595-613 ◽  
Author(s):  
Levent Kargın
Keyword(s):  
Integral Representation ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Bernoulli Polynomials And Numbers

We relate geometric polynomials and [Formula: see text]-Bernoulli polynomials with an integral representation, then obtain several properties of [Formula: see text]-Bernoulli polynomials. These results yield new identities for Bernoulli numbers. Moreover, we evaluate a Faulhaber-type summation in terms of [Formula: see text]-Bernoulli polynomials. Finally, we introduce poly-[Formula: see text]-Bernoulli polynomials and numbers, then study some arithmetical and number theoretical properties of them.

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