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ON THE (p, q)-ANALOGUE OF EULER ZETA FUNCTION
RYOO, CHEON SEOUNG The Korean Society for Computational and Applied M 2017 Journal of applied mathematics & informatics Vol.35 No.3
In this paper we define (p, q)-analogue of Euler zeta function. In order to define (p, q)-analogue of Euler zeta function, we introduce the (p, q)-analogue of Euler numbers and polynomials by generalizing the Euler numbers and polynomials, Carlitz's type q-Euler numbers and polynomials. We also give some interesting properties, explicit formulas, a connection with (p, q)-analogue of Euler numbers and polynomials. Finally, we investigate the zeros of the (p, q)-analogue of Euler polynomials by using computer.
ON THE (p, q)-ANALOGUE OF EULER ZETA FUNCTION
유천성 한국전산응용수학회 2017 Journal of applied mathematics & informatics Vol.35 No.3
In this paper we define (p, q)-analogue of Euler zeta function. In order to define (p, q)-analogue of Euler zeta function, we introduce the (p, q)-analogue of Euler numbers and polynomials by generalizing the Euler numbers and polynomials, Carlitz’s type q-Euler numbers and polynomials. We also give some interesting properties, explicit formulas, a connection with (p, q)-analogue of Euler numbers and polynomials. Finally, we investigate the zeros of the (p, q)-analogue of Euler polynomials by using computer.
THE q-ANALOGUE OF TWISTED LERCH TYPE EULER ZETA FUNCTIONS
Jang, Lee-Chae Korean Mathematical Society 2010 대한수학회보 Vol.47 No.6
q-Volkenborn integrals ([8]) and fermionic invariant q-integrals ([12]) are introduced by T. Kim. By using these integrals, Euler q-zeta functions are introduced by T. Kim ([18]). Then, by using the Euler q-zeta functions, S.-H. Rim, S. J. Lee, E. J. Moon, and J. H. Jin ([25]) studied q-Genocchi zeta functions. And also Y. H. Kim, W. Kim, and C. S. Ryoo ([7]) investigated twisted q-zeta functions and their applications. In this paper, we consider the q-analogue of twisted Lerch type Euler zeta functions defined by $${\varsigma}E,q,\varepsilon(s)=[2]q \sum\limits_{n=0}^\infty\frac{(-1)^n\epsilon^nq^{sn}}{[n]_q}$$ where 0 < q < 1, $\mathfrak{R}$(s) > 1, $\varepsilon{\in}T_p$, which are compared with Euler q-zeta functions in the reference ([18]). Furthermore, we give the q-extensions of the above twisted Lerch type Euler zeta functions at negative integers which interpolate twisted q-Euler polynomials.
김태균,김대산,김혜경 장전수학회 2022 Advanced Studies in Contemporary Mathematics Vol.32 No.3
Recently, we introduced the λ -q-umbral calculus which centers around the λ -q-Sheffer sequences and the degenerate q-Sheffer sequences by introducing λ -q-linear functionals and λ - q-differential operators, respectively, instead of λ -linear functionals and λ -differential operators. In this paper, we introduce the degenerate poly-q-Bernoulli polynomials and degenerate poly-q- Euler polynomials and numbers by using the q-polylogarithm function. These new sequences are q-analogues of the degenerate poly-Bernoulli polynomials and degenerate poly-Euler polynomials, respectively. We give interesting combinatorial identities and properties of these new polynomials by using λ -q-umbral calculus.
Patrick Njionou Sadjang 대한수학회 2018 대한수학회지 Vol.55 No.5
Several addition formulas for a general class of $q$-Appell sequences are proved. The $q$-addition formulas, which are derived, involved not only the generalized $q$-Bernoulli, the generalized $q$-Euler and the generalized $q$-Genocchi polynomials, but also the $q$-Stirling numbers of the second kind and several general families of hypergeometric polynomials. Some $q$-umbral calculus generalizations of the addition formulas are also investigated.
Calculating zeros of the q-Euler polynomials
유천성 장전수학회 2009 Proceedings of the Jangjeon mathematical society Vol.12 No.2
In [1-6], we treated q-analogues of ordinary Euler polynomials which is called q-Euler polynomials En,q(x). One purpose of this paper is to consider the reflection symmetries of the q-Euler polynomials En,q(x). Using numerical investigation, some interesting results are obtained.
THE q-ANALOGUE OF TWISTED LERCH TYPE EULER ZETA FUNCTIONS
장이채 대한수학회 2010 대한수학회보 Vol.47 No.6
q-Volkenborn integrals ([8]) and fermionic invariant q-integ-rals ([12]) are introduced by T. Kim. By using these integrals, Euler q-zeta functions are introduced by T. Kim ([18]). Then, by using the Euler q-zeta functions, S.-H. Rim, S. J. Lee, E. J. Moon, and J. H. Jin ([25])studied q-Genocchi zeta functions. And also Y. H. Kim, W. Kim, and C. S. Ryoo ([7]) investigated twisted q-zeta functions and their applications. In this paper, we consider the q-analogue of twisted Lerch type Euler zeta functions defined by [수식]E;q;ε(s) = [2]q ∞Σn=0(-)nεnqsn [n]q where 0 < q < 1; R(s) > 1, ε∈ Tp, which are compared with Euler q-zeta functions in the reference ([18]). Furthermore, we give the q-extensions of the above twisted Lerch type Euler zeta functions at negative integers which interpolate twisted q-Euler polynomials.
Sadjang, Patrick Njionou Korean Mathematical Society 2018 대한수학회지 Vol.55 No.5
Several addition formulas for a general class of q-Appell sequences are proved. The q-addition formulas, which are derived, involved not only the generalized q-Bernoulli, the generalized q-Euler and the generalized q-Genocchi polynomials, but also the q-Stirling numbers of the second kind and several general families of hypergeometric polynomials. Some q-umbral calculus generalizations of the addition formulas are also investigated.
Seo, Jong Jin,Araci, Serkan,Acikgoz, Mehmet Chungcheong Mathematical Society 2014 충청수학회지 Vol.27 No.1
Recently, q-Dedekind-type sums related to q-Euler polynomials was studied by Kim in [T. Kim, Note on q-Dedekind-type sums related to q-Euler polynomials, Glasgow Math. J. 54 (2012), 121-125]. It is aim of this paper to consider a p-adic continuous function for an odd prime to inside a p-adic q-analogue of the higher order Dedekind-type sums with weight related to modified q-Euler polynomials with weight by using Kim's p-adic q-integral.
q-analogues of some results for the Apostol-Euler polynomials
Qiu-Ming Luo 장전수학회 2010 Advanced Studies in Contemporary Mathematics Vol.20 No.1
We investigate some bacic properties and the generating functions of q-Apostol-Euler polynomials of higher order. Several interesting relationships between the q-Apostol-Euler polynomials and q-Hurwitz-Lerch Zeta function are obtained. We also try to give some conjectures concerning the multiplication formulas of q-Apostol-Euler polynomials of higher order.