• 제목/요약/키워드: Euler polynomials and numbers

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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.

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.

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.

AN EXTENSION OF GENERALIZED EULER POLYNOMIALS OF THE SECOND KIND

  • Kim, Y.H.;Jung, H.Y.;Ryoo, C.S.
    • Journal of applied mathematics & informatics
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    • 제32권3_4호
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    • pp.465-474
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    • 2014
  • Many mathematicians have studied various relations beween Euler number $E_n$, Bernoulli number $B_n$ and Genocchi number $G_n$ (see [1-18]). They have found numerous important applications in number theory. Howard, T.Agoh, S.-H.Rim have studied Genocchi numbers, Bernoulli numbers, Euler numbers and polynomials of these numbers [1,5,9,15]. T.Kim, M.Cenkci, C.S.Ryoo, L. Jang have studied the q-extension of Euler and Genocchi numbers and polynomials [6,8,10,11,14,17]. In this paper, our aim is introducing and investigating an extension term of generalized Euler polynomials. We also obtain some identities and relations involving the Euler numbers and the Euler polynomials, the Genocchi numbers and Genocchi polynomials.

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.

DEGENERATE POLYEXPONENTIAL FUNCTIONS AND POLY-EULER POLYNOMIALS

  • Kurt, Burak
    • 대한수학회논문집
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    • 제36권1호
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    • pp.19-26
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    • 2021
  • Degenerate versions of the special polynomials and numbers since they have many applications in analytic number theory, combinatorial analysis and p-adic analysis. In this paper, we define the degenerate poly-Euler numbers and polynomials arising from the modified polyexponential functions. We derive explicit relations for these numbers and polynomials. Also, we obtain some identities involving these polynomials and some other special 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.