• Title/Summary/Keyword: Euler numbers and polynomials

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A NEW FAMILY OF FUBINI TYPE NUMBERS AND POLYNOMIALS ASSOCIATED WITH APOSTOL-BERNOULLI NUMBERS AND POLYNOMIALS

  • Kilar, Neslihan;Simsek, Yilmaz
    • Journal of the Korean Mathematical Society
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    • v.54 no.5
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    • pp.1605-1621
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    • 2017
  • The purpose of this paper is to construct a new family of the special numbers which are related to the Fubini type numbers and the other well-known special numbers such as the Apostol-Bernoulli numbers, the Frobenius-Euler numbers and the Stirling numbers. We investigate some fundamental properties of these numbers and polynomials. By using generating functions and their functional equations, we derive various formulas and relations related to these numbers and polynomials. In order to compute the values of these numbers and polynomials, we give their recurrence relations. We give combinatorial sums including the Fubini type numbers and the others. Moreover, we give remarks and observation on these numbers and polynomials.

ON THE (p, q)-POLY-KOROBOV POLYNOMIALS AND RELATED POLYNOMIALS

  • KURT, BURAK;KURT, VELI
    • Journal of applied mathematics & informatics
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    • v.39 no.1_2
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    • pp.45-56
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    • 2021
  • D.S. Kim et al. [9] considered some identities and relations for Korobov type numbers and polynomials. In this work, we investigate the degenerate Korobov type Changhee polynomials and the (p,q)-poly-Korobov polynomials. We give a generalization of the Korobov type Changhee polynomials and the (p,q) poly-Korobov polynomials. We prove some properties and identities and explicit relations for these polynomials.

q-DEDEKIND-TYPE DAEHEE-CHANGHEE SUMS WITH WEIGHT α ASSOCIATED WITH MODIFIED q-EULER POLYNOMIALS WITH WEIGHT α

  • Seo, Jong Jin;Araci, Serkan;Acikgoz, Mehmet
    • Journal of the Chungcheong Mathematical Society
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    • v.27 no.1
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    • pp.1-8
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    • 2014
  • 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.

Bernoulli and Euler Polynomials in Two Variables

  • Claudio Pita-Ruiz
    • Kyungpook Mathematical Journal
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    • v.64 no.1
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    • pp.133-159
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    • 2024
  • In a previous work we studied generalized Stirling numbers of the second kind S(a2,b2,p2)a1,b1 (p1, k), where a1, a2, b1, b2 are given complex numbers, a1, a2 ≠ 0, and p1, p2 are non-negative integers given. In this work we use these generalized Stirling numbers to define Bernoulli polynomials in two variables Bp1,p2 (x1, x2), and Euler polynomials in two variables Ep1p2 (x1, x2). By using results for S(1,x2,p2)1,x1 (p1, k), we obtain generalizations, to the bivariate case, of some well-known properties from the standard case, as addition formulas, difference equations and sums of powers. We obtain some identities for bivariate Bernoulli and Euler polynomials, and some generalizations, to the bivariate case, of several known identities for Bernoulli and Euler numbers and polynomials of the standard case.

A NOTE ON THE q-EULER NUMBERS AND POLYNOMIALS WITH WEIGHT (α,ω)

  • Rim, Seog-Hoon;Jeong, Joo-Hee
    • Honam Mathematical Journal
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    • v.34 no.2
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    • pp.183-190
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    • 2012
  • The main purpose of this paper is to introduce a new type of $q$-Euler numbers and polynomials with weak weight (${\alpha}$,${\omega}$): $\tilde{E}^{({\alpha},{\omega})}_{n,q}$ and $\tilde{E}^{({\alpha},{\omega})}_{n,q}(x)$, respectively. By using the fermionic $p$-adic $q$-integral on $\mathbb{Z}_p$, we can obtain some results and derive some recurrence identities for the $q$-Euler numbers and polynomials with weak weight (${\alpha}$,${\omega}$).

ON HIGHER ORDER (p, q)-FROBENIUS-GENOCCHI NUMBERS AND POLYNOMIALS

  • KHAN, WASEEM A.;KHAN, IDREES A.;KANG, J.Y.
    • Journal of applied mathematics & informatics
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    • v.37 no.3_4
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    • pp.295-305
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    • 2019
  • In the present paper, we introduce (p, q)-Frobenius-Genocchi numbers and polynomials and investigate some basic identities and properties for these polynomials and numbers including addition theorems, difference equations, derivative properties, recurrence relations and so on. Then, we provide integral representations, implicit and explicit formulas and relations for these polynomials and numbers. We consider some relationships for (p, q)-Frobenius-Genocchi polynomials of order ${\alpha}$ associated with (p, q)-Bernoulli polynomials, (p, q)-Euler polynomials and (p, q)-Genocchi polynomials.

A NOTE ON (p, q)-ANALOGUE TYPE OF FROBENIUS-GENOCCHI NUMBERS AND POLYNOMIALS

  • Khan, Waseem A.;Khan, Idrees A.
    • East Asian mathematical journal
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    • v.36 no.1
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    • pp.13-24
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    • 2020
  • The main purpose of this paper is to introduce Apostol type (p, q)-Frobenius-Genocchi numbers and polynomials of order α and investigate some basic identities and properties for these polynomials and numbers including addition theorems, difference equations, derivative properties, recurrence relations. We also obtain integral representations, implicit and explicit formulas and relations for these polynomials and numbers. Furthermore, we consider some relationships for Apostol type (p, q)-Frobenius-Genocchi polynomials of order α associated with (p, q)-Apostol Bernoulli polynomials, (p, q)-Apostol Euler polynomials and (p, q)-Apostol Genocchi polynomials.