• 제목/요약/키워드: q-Euler numbers and polynomials

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SOME PROPERTIES OF DEGENERATE CARLITZ-TYPE TWISTED q-EULER NUMBERS AND POLYNOMIALS

  • RYOO, CHEON SEOUNG
    • Journal of applied mathematics & informatics
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    • 제39권1_2호
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    • pp.1-11
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    • 2021
  • In this paper, we define degenerate Carlitz-type twisted q-Euler numbers and polynomials by generalizing the degenerate Euler numbers and polynomials, Carlitz's type degenerate q-Euler numbers and polynomials. We also give some interesting properties, explicit formulas, symmetric properties, a connection with degenerate Carlitz-type twisted q-Euler numbers and polynomials.

ON THE (p, q)-ANALOGUE OF EULER ZETA FUNCTION

  • RYOO, CHEON SEOUNG
    • Journal of applied mathematics & informatics
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    • 제35권3_4호
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    • pp.303-311
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    • 2017
  • 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.

SYMMETRIC IDENTITIES FOR DEGENERATE CARLITZ-TYPE q-EULER NUMBERS AND POLYNOMIALS

  • RYOO, CHEON SEOUNG
    • Journal of applied mathematics & informatics
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    • 제37권3_4호
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    • pp.259-270
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    • 2019
  • In this paper we define the degenerate Carlitz-type q-Euler polynomials by generalizing the degenerate Euler numbers and polynomials, degenerate Carlitz-type Euler numbers and polynomials. We also give some interesting properties, explicit formulas, a connection with degenerate Carlitz-type q-Euler numbers and polynomials.

A NOTE ON THE q-EULER NUMBERS AND POLYNOMIALS WITH WEAK WEIGHT α AND q-BERNSTEIN POLYNOMIALS

  • Lee, H.Y.;Jung, N.S.;Kang, J.Y.
    • Journal of applied mathematics & informatics
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    • 제31권3_4호
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    • pp.523-531
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    • 2013
  • In this paper we construct a new type of $q$-Bernstein polynomials related to $q$-Euler numbers and polynomials with weak weight ${\alpha}$ ; $E^{(\alpha)}_{n,q}$, $E^{(\alpha)}_{n,q}(x)$ respectively. Some interesting results and relationships are obtained.

SOME PROPERTIES OF GENERALIZED q-POLY-EULER NUMBERS AND POLYNOMIALS WITH VARIABLE a

  • KIM, A HYUN
    • Journal of applied mathematics & informatics
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    • 제38권1_2호
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    • pp.133-144
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    • 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.

IDENTITIES AND RELATIONS ON THE q-APOSTOL TYPE FROBENIUS-EULER NUMBERS AND POLYNOMIALS

  • Kucukoglu, Irem;Simsek, Yilmaz
    • 대한수학회지
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    • 제56권1호
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    • pp.265-284
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    • 2019
  • The main purpose of this paper is to investigate the q-Apostol type Frobenius-Euler numbers and polynomials. By using generating functions for these numbers and polynomials, we derive some alternative summation formulas including powers of consecutive q-integers. By using infinite series representation for q-Apostol type Frobenius-Euler numbers and polynomials including their interpolation functions, we not only give some identities and relations for these numbers and polynomials, but also define generating functions for new numbers and polynomials. Further we give remarks and observations on generating functions for these new numbers and polynomials. By using these generating functions, we derive recurrence relations and finite sums related to these numbers and polynomials. Moreover, by applying higher-order derivative to these generating functions, we derive some new formulas including the Hurwitz-Lerch zeta function, the Apostol-Bernoulli numbers and the Apostol-Euler numbers. Finally, for an application of the generating functions, we derive a multiplication formula, which is very important property in the theories of normalized polynomials and Dedekind type sums.

SYMMETRIC IDENTITIES INVOLVING THE MODIFIED (p, q)-HURWITZ EULER ZETA FUNCTION

  • KIM, A HYUN;AN, CHAE KYEONG;LEE, HUI YOUNG
    • Journal of applied mathematics & informatics
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    • 제36권5_6호
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    • pp.555-565
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    • 2018
  • The main subject of this paper is to introduce the (p, q)-Euler polynomials and obtain several interesting symmetric properties of the modified (p, q)-Hurwitz Euler Zeta function with regard to (p, q) Euler polynomials. In order to get symmetric properties, we introduce the new (p, q)-analogue of Euler polynomials $E_{n,p,q}(x)$ and numbers $E_{n,p,q}$.

A NOTE ON THE q-ANALOGUES OF EULER NUMBERS AND POLYNOMIALS

  • Choi, Jong-Sung;Kim, Tae-Kyun;Kim, Young-Hee
    • 호남수학학술지
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    • 제33권4호
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    • pp.529-534
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    • 2011
  • In this paper, we consider the q-analogues of Euler numbers and polynomials using the fermionic p-adic invariant integral on $\mathbb{Z}_p$. From these numbers and polynomials, we derive some interesting identities and properties on the q-analogues of Euler numbers and polynomials.

NUMERICAL INVESTIGATION OF ZEROS OF THE FULLY q-POLY-EULER NUMBERS AND POLYNOMIALS OF THE SECOND TYPE

  • JUNG YOOG KANG
    • Journal of Applied and Pure Mathematics
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    • 제5권5_6호
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    • pp.375-387
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    • 2023
  • In this paper, we construct a fully modified q-poly-Euler numbers and polynomials of the second type and give some properties. Finally, we investigate the zeros of the fully modified q-poly-Euler numbers and polynomials of the second type by using computer.