A New Type of Euler Polynomials and Numbers

Kim-Kim [12] introduced the new type of degenerate Bernoulli numbers and polynomials arising from the degenerate logarithm function. In this paper, we introduce a new type of degenerate poly-Euler polynomials and numbers, are called degenerate poly-Euler polynomials and numbers, by using the degenerate polylogarithm function and derive several properties on the degenerate poly-Euler polynomials and numbers. In the last section, we also consider the degenerate unipoly-Euler polynomials attached to an arithmetic function, by using the degenerate polylogarithm function and investigate some identities of those polynomials. In particular, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

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Type 2 Degenerate Poly-Euler Polynomials

Symmetry ◽

10.3390/sym12061011 ◽

2020 ◽

Vol 12 (6) ◽

pp. 1011 ◽

Cited By ~ 3

Author(s):

Dae Sik Lee ◽

Hye Kyung Kim ◽

Lee-Chae Jang

Keyword(s):

Final Section ◽

Bernoulli Polynomials ◽

Stirling Numbers ◽

Euler Polynomials ◽

Special Polynomials ◽

Special Numbers And Polynomials ◽

Euler Polynomials And Numbers ◽

New Type ◽

Explicit Expressions

In recent years, many mathematicians have studied the degenerate versions of many special polynomials and numbers. The polyexponential functions were introduced by Hardy and rediscovered by Kim, as inverses to the polylogarithms functions. The paper is divided two parts. First, we introduce a new type of the type 2 poly-Euler polynomials and numbers constructed from the modified polyexponential function, the so-called type 2 poly-Euler polynomials and numbers. We show various expressions and identities for these polynomials and numbers. Some of them involving the (poly) Euler polynomials and another special numbers and polynomials such as (poly) Bernoulli polynomials, the Stirling numbers of the first kind, the Stirling numbers of the second kind, etc. In final section, we introduce a new type of the type 2 degenerate poly-Euler polynomials and the numbers defined in the previous section. We give explicit expressions and identities involving those polynomials in a similar direction to the previous section.

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New Type of Degenerate Poly-Frobenius-Euler Polynomials and Numbers

10.20944/preprints202007.0319.v1 ◽

2020 ◽

Author(s):

Waseem Khan

Keyword(s):

Bernoulli Polynomials ◽

Euler Polynomials ◽

Polylogarithm Function ◽

Euler Polynomials And Numbers ◽

New Type ◽

Explicit Expressions

Motivated by Kim-Kim [19] introduced the new type of degenerate poly- Bernoulli polynomials by means of the degenerate polylogarithm function. In this paper, we define the degenerate poly-Frobenius-Euler polynomials, called the new type of degenerate poly-Frobenius-Euler polynomials, by means of the degenerate polylogarithm function. Then, we derive explicit expressions and some identities of those numbers and polynomials.

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Remarks on Sum of Products of (h,q)-Twisted Euler Polynomials and Numbers

Journal of Inequalities and Applications ◽

10.1155/2008/816129 ◽

2008 ◽

Vol 2008 ◽

pp. 1-8 ◽

Cited By ~ 13

Author(s):

Hacer Ozden ◽

Ismail Naci Cangul ◽

Yilmaz Simsek

Keyword(s):

Euler Polynomials ◽

Euler Polynomials And Numbers ◽

Sum Of Products

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Explicit formulas for the Bernoulli and Euler polynomials and numbers

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg ◽

10.1007/bf02950761 ◽

1991 ◽

Vol 61 (1) ◽

pp. 175-180 ◽

Cited By ~ 3

Author(s):

P. G. Todorov

Keyword(s):

Euler Polynomials ◽

Explicit Formulas ◽

Euler Polynomials And Numbers

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Some Identities of Degenerate Bell Polynomials

Mathematics ◽

10.3390/math8010040 ◽

2020 ◽

Vol 8 (1) ◽

pp. 40 ◽

Cited By ~ 6

Author(s):

Taekyun Kim ◽

Dae San Kim ◽

Han Young Kim ◽

Jongkyum Kwon

Keyword(s):

Stirling Numbers ◽

Bernoulli Numbers ◽

Bell Polynomials ◽

Euler Polynomials ◽

New Type ◽

Degenerate Version

The new type degenerate of Bell polynomials and numbers were recently introduced, which are a degenerate version of Bell polynomials and numbers and are different from the previously introduced partially degenerate Bell polynomials and numbers. Several expressions and identities on those polynomials and numbers were obtained. In this paper, as a further investigation of the new type degenerate Bell polynomials, we derive several identities involving those degenerate Bell polynomials, Stirling numbers of the second kind and Carlitz’s degenerate Bernoulli or degenerate Euler polynomials. In addition, we obtain an identity connecting the degenerate Bell polynomials, Cauchy polynomials, Bernoulli numbers, Stirling numbers of the second kind and degenerate Stirling numbers of the second kind.

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