• Journal of applied mathematics & informatics
• /
• 제35권5_6호
• /
• pp.599-609
• /
• 2017

We construct degenerated Euler polynomials of the second kind and find some basic properties of this polynomials. From this paper, we can see degenerated alternative power sum is defined and is related to degenerated Euler polynomials of the second kind. Using this power sum, we have a number of symmetric properties of degenerated Euler polynomials of the second kind.

• Journal of applied mathematics & informatics
• /
• 제38권1_2호
• /
• pp.37-56
• /
• 2020

In this paper we discuss some interesting properties of (p, q)-special polynomials and derive various relations. We gain some relations between (p, q)-zeta function and (p, q)-special polynomials by considering (p, q)-analogue Euler sum types. In addition, we derive the relationship between (p, q)-polylogarithm function and (p, q)-special polynomials.

• Journal of applied mathematics & informatics
• /
• 제38권1_2호
• /
• pp.133-144
• /
• 2020

In this paper, we discuss generalized q-poly-Euler numbers and polynomials. To do so, we define generalized q-poly-Euler polynomials with variable a and investigate its identities. We also represent generalized q-poly-Euler polynomials E^(k)_n,q(x; a) using Stirling numbers of the second kind. So we explore the relation between generalized q-poly-Euler polynomials and Stirling numbers of the second kind through it. At the end, we provide symmetric properties related to generalized q-poly-Euler polynomials using alternating power sum.

• 호남수학학술지
• /
• 제46권3호
• /
• pp.365-377
• /
• 2024

In this paper, we give some sums involving the generalized harmonic numbers H^r_n (σ) and the (q, r)-binomial coefficient $\left({L \atop k}\right)_{q,r}$ by using Euler's transform. For example, for (c, r) ∈ ℤ⁺ × ℝ⁺, $${\sum_{n=0}^{\infty}}{\sum_{k=0}^{n}}\,(-1)^k\,\left({n+r \atop n-k}\right)\frac{c^{n+1}H^{r-1}_k({\sigma})}{(n+1)(1+c)^{n+1}}=-(c+{\frac{1}{{\sigma}}})\,{\ln}\,(1+c{\sigma})+c,$$ and $${\sum_{k=0}^{n}}\left({n \atop k}\right)\left({L \atop k}\right)_{2,r}={\sum_{j=0}^{n}}{\sum_{k=0}^{j}}(-1)^k\left({j-k+2L+r \atop j-k}\right)\left({r \atop n-j}\right)\left({L \atop k}\right)_2,$$ where σ is appropriate parameter, H^r_n (σ) is the generalized hyperharmonic number of order r and $\left({L \atop k}\right)_q$ is the q-binomial coefficient.

• Journal of applied mathematics & informatics
• /
• 제34권1_2호
• /
• pp.107-114
• /
• 2016

In this paper we give some symmetric property of the generalized twisted q-Euler zeta functions and q-Euler polynomials.

• 호남수학학술지
• /
• 제36권3호
• /
• pp.647-657
• /
• 2014

In this paper, we study combinatoric convolution sums of divisor functions and get values of this sum when $n=2^mp$. We find that the value of this convolution sum is represented by a sum of powers of 2 and Bernoulli or Euler number.

• Journal of applied mathematics & informatics
• /
• 제38권5_6호
• /
• pp.591-600
• /
• 2020

This work is a continuation of our investigations for p-adic analogue of the alternating form Dirichlet L-functions $$L_E(s,{\chi})={\sum\limits_{n=1}^{\infty}}{\frac{(-1)^n{\chi}(n)}{n^s}},\;Re(s)>0$$. Let L_p,E(s, t; χ) be the p-adic Euler L-function of two variables. In this paper, for any α ∈ ℂ_p, |α|_p ≤ 1, we give a power series expansion of L_p,E(s, t; χ) in terms of the variable t. From this, we derive a power series expansion of the generalized Euler polynomials with negative index, that is, we prove that $$E_{-n,{\chi}}(t)={\sum\limits_{m=0}^{\infty}}\(\array{-n\\m}\)E_{-(m+n),{\chi}^{t^m}},\;n{\in}{\mathbb{N}}$$, where t ∈ ℂ_p with |t|_p < 1. Some further properties for L_p,E(s, t; χ) has also been shown.

• 대한수학회논문집
• /
• 제19권3호
• /
• pp.545-555
• /
• 2004

Ever since the time of Euler, the so-called Euler sums have been evaluated in many different ways. We give here a proof of the classical Euler sum by following Lewin's method. We also consider some related formulas involving Euler sums, which are evaluatable from some known definite integrals.

• 대한수학회보
• /
• 제51권5호
• /
• pp.1425-1432
• /
• 2014

The Euler zeta function is defined by ${\zeta}_E(s)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^8}$. The purpose of this paper is to find formulas of the Euler zeta function's values. In this paper, for $s{\in}\mathbb{N}$ we find the recurrence formula of ${\zeta}_E(2s)$ using the Fourier series. Also we find the recurrence formula of $\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{(2_{n-1})^{2s-1}}$, where $s{\geq}2({\in}\mathbb{N})$.

• 한국수학사학회지
• /
• 제20권4호
• /
• pp.71-84
• /
• 2007

베르누이가 처음으로 자연수 k에 대하여 합 $S_n(k)=\sum_{{\iota}=1}^n\;{\iota}^k$에 관한 공식들을 유도하는 방법을 발견하였다([4]). 그 이후, 리만 제타함수와 관련된 베르누이 수와 오일러 수에 관한 성질들이 연구되어왔다. 최근에 김태균은 $\mathbb{Z}_p$상에서 p-진 q-적분과 관련된 확장된 q-베르누이 수와 q-오일러 수, 연속된 q-정수의 멱수의 합에 관한 성질들을 밝혔다. 본 논문에서는 연속된 q-정수의 멱수의 합에 관한 역사적 배경과 발달과정을 고찰하고, 오일러 및 베르누이 수와 관련된 리만 제타함수가 해석적 함수로써 값을 가지는 문제를 q-확장된 부분의 이론으로 연구되어온 q-오일러 제타함수에 대해 체계적으로 논의한다.