scholarly journals SYMMETRIC PROPERTIES FOR THE GENERALIZED TWISTED $(h, q)$-EULER POLYNOMIALS ASSOCIATED WITH $p$-ADIC INVARIANT $q$-INTEGRAL ON $\Bbb Z_p$

2015 ◽  
Vol 102 (2) ◽  
Author(s):  
C.S. Ryoo
Keyword(s):  
Euler Polynomials ◽  
Symmetric Properties

scholarly journals Some New Identities on the Bernoulli and Euler Numbers

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Dae San Kim ◽  
Taekyun Kim ◽  
Sang-Hun Lee ◽  
D. V. Dolgy ◽  
Seog-Hoon Rim
Keyword(s):  
Euler Polynomials ◽  
Euler Numbers ◽  
Symmetric Properties

We give some new identities on the Bernoulli and Euler numbers by using the bosonicp-adic integral onZpand reflection symmetric properties of Bernoulli and Euler polynomials.

scholarly journals Some Symmetry Identities for Carlitz’s Type Degenerate Twisted (p,q)-Euler Polynomials Related to Alternating Twisted (p,q)-Sums

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1371
Author(s):  
Cheon-Seoung Ryoo
Keyword(s):  
Euler Polynomials ◽  
Euler Numbers ◽  
Explicit Formulas ◽  
Euler Numbers And Polynomials ◽  
New Form ◽  
Symmetric Properties

In this paper, we define a new form of Carlitz’s type degenerate twisted (p,q)-Euler numbers and polynomials by generalizing the degenerate Euler numbers and polynomials, Carlitz’s type degenerate q-Euler numbers and polynomials. Some interesting identities, explicit formulas, symmetric properties, a connection with Carlitz’s type degenerate twisted (p,q)-Euler numbers and polynomials are obtained. Finally, we investigate the zeros of the Carlitz’s type degenerate twisted (p,q)-Euler polynomials by using computer.

scholarly journals Symmetric Properties for Dirichlet-Type Multiple (p,q)-L-Function

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 95
Author(s):  
Kyung-Won Hwang ◽  
Ravi P. Agarwal ◽  
Cheon Seoung Ryoo
Keyword(s):  
Higher Order ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Dirichlet Type ◽  
Euler Numbers And Polynomials ◽  
Symmetric Identities ◽  
Symmetric Properties

The main aim of this article is to investigate some interesting symmetric identities for the Dirichlet-type multiple (p,q)-L function. We use this function to examine the symmetry of the generalized higher-order (p,q)-Euler polynomials related to χ. First, the generalized higher-order (p,q)-Euler numbers and polynomials related to χ are defined. We also give a few new symmetric properties for the Dirichlet-type multiple (p,q)-L-function and generalized higher-order (p,q)-Euler polynomials related to χ.

scholarly journals Some Symmetric Properties and Location Conjecture of Approximate Roots for (p,q)-Cosine Euler Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1520
Author(s):  
Cheon Seoung Ryoo ◽  
Jung Yoog Kang
Keyword(s):  
Euler Polynomials ◽  
Symmetric Properties

In this paper, we introduce (p,q)-cosine Euler polynomials. From these polynomials, we find several properties and identities. Moreover, we find the circle equations of approximate roots for (p,q)-cosine Euler polynomials by using a computer.

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