scholarly journals Notes on Several Families of Differential Equations Related to the Generating Function for the Bernoulli Numbers of the Second Kind

2018 ◽  
Vol 6 (2) ◽  
pp. 40-42 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Differential Equations ◽  
Generating Function ◽  
Bernoulli Numbers

scholarly journals New type of degenerate Daehee polynomials of the second kind

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Sunil Kumar Sharma ◽  
Waseem A. Khan ◽  
Serkan Araci ◽  
Sameh S. Ahmed
Keyword(s):  
Generating Function ◽  
Special Class ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Bernoulli Numbers And Polynomials ◽  
New Type ◽  
Order Degenerate ◽  
Math Phys

Abstract Recently, Kim and Kim (Russ. J. Math. Phys. 27(2):227–235, 2020) have studied new type degenerate Bernoulli numbers and polynomials by making use of degenerate logarithm. Motivated by (Kim and Kim in Russ. J. Math. Phys. 27(2):227–235, 2020), we consider a special class of polynomials, which we call a new type of degenerate Daehee numbers and polynomials of the second kind. By using their generating function, we derive some new relations including the degenerate Stirling numbers of the first and second kinds. Moreover, we introduce a new type of higher-order degenerate Daehee polynomials of the second kind. We also derive some new identities and properties of this type of polynomials.

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. 

scholarly journals Labels distance in bucket recursive trees with variable capacities of buckets

2021 ◽  
Vol 13 (2) ◽  
pp. 413-426
Author(s):  
S. Naderi ◽  
R. Kazemi ◽  
M. H. Behzadi
Keyword(s):  
Partial Differential Equations ◽  
Probability Distribution ◽  
Differential Equations ◽  
Generating Function ◽  
Generating Functions ◽  
Function Approach ◽  
Closed Formula ◽  
Tree Model ◽  
Partial Differential ◽  
Recursive Trees

Abstract The bucket recursive tree is a natural multivariate structure. In this paper, we apply a trivariate generating function approach for studying of the depth and distance quantities in this tree model with variable bucket capacities and give a closed formula for the probability distribution, the expectation and the variance. We show as j → ∞, lim-iting distributions are Gaussian. The results are obtained by presenting partial differential equations for moment generating functions and solving them.

scholarly journals Differential Equations Arising from the Generating Function of the (r, β)-Bell Polynomials and Distribution of Zeros of Equations

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 736 ◽  
Author(s):  
Kyung-Won Hwang ◽  
Cheon Seoung Ryoo ◽  
Nam Soon Jung
Keyword(s):  
Differential Equations ◽  
Generating Function ◽  
Numerical Experiments ◽  
Bell Polynomials ◽  
Distribution Of Zeros

In this paper, we study differential equations arising from the generating function of the ( r , β ) -Bell polynomials. We give explicit identities for the ( r , β ) -Bell polynomials. Finally, we find the zeros of the ( r , β ) -Bell equations with numerical experiments.

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 A note on p-adic Carlitz's q-Bernoulli numbers

2000 ◽  
Vol 62 (2) ◽  
pp. 227-234 ◽  
Author(s):  
Taekyun Kim ◽  
Seog-Hoon Rim
Keyword(s):  
Invariant Measure ◽  
Generating Function ◽  
Bernoulli Number ◽  
Bernoulli Numbers ◽  
Adic Case

In a recent paper I have shown that Carlitz's q-Bernoulli number can be represented as an integral by the q-analogue μq of the ordinary p-adic invariant measure. In the p-adic case, J. Satoh could not determine the generating function of q-Bernoulli numbers. In this paper, we give the generating function of q-Bernoulli numbers in the p-adic case.

scholarly journals On a Dual Model with Barrier Strategy

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Yuzhen Wen ◽  
Chuancun Yin
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Generating Function ◽  
Risk Model ◽  
Moment Generating Function ◽  
Dual Model ◽  
Barrier Strategy ◽  
Dividend Payments ◽  
The Moment

We consider the dual of the generalized Erlang(n)risk model with a barrier dividend strategy. We derive integro-differential equations with boundary conditions satisfied by the expectation of the sum of discounted dividends until ruin and the moment-generating function of the discounted dividend payments until ruin, respectively. The results are illustrated by several examples.

scholarly journals Simplifying Coefficients in Differential Equations Associated with Higher Order Bernoulli Numbers of the Second Kind

2017 ◽  
Author(s):  
Feng Qi ◽  
Da-Wei Niu ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Inversion Formula ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Bell Polynomials

In the paper, by virtue of the Faà di Bruno formula, some properties of the Bell polynomials of the second kind, and an inversion formula for the Stirling numbers of the first and second kinds, the authors establish meaningfully and significantly two identities which simplify coefficients in a family of ordinary differential equations associated with higher order Bernoulli numbers of the second kind.

Sign in / Sign up

Export Citation Format

Share Document