Table of Contents
ISRN Applied Mathematics
VolumeΒ 2012, Article IDΒ 475463, 14 pages
http://dx.doi.org/10.5402/2012/475463
Research Article

Generalized 𝑀-Euler Numbers and Polynomials

Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea

Received 28 October 2011; Accepted 4 December 2011

Academic Editors: A. J.Β Kearsley and D.Β Kuhl

Copyright Β© 2012 H. Y. Lee et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We generalize the Euler numbers and polynomials by the generalized 𝑀-Euler numbers 𝐸𝑛,𝑀(π‘Ž) and polynomials 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž). For the complement theorem, 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) have interesting different properties from the Euler polynomials and we observe an interesting phenomenon of β€œscattering” of the zeros of the the generalized Euler polynomials 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) in complex plane.

1. Introduction

The Euler numbers and polynomials possess many interesting properties and arising in many areas of mathematics and physics. Recently, many mathematicians have studied in the area of the Euler numbers and polynomials (see [1–15]). In [14], we introduced that Euler equation 𝐸𝑛(π‘₯)=0 has symmetrical roots for π‘₯=1/2(see [14]). It is the aim of this paper to observe an interesting phenomenon of β€œscattering” of the zeros of the the generalized 𝑀-Euler polynomials 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) in complex plane. Throughout this paper we use the following notations. By ℀𝑝 we denote the ring of 𝑝-adic rational integers, β„šπ‘ denotes the field of 𝑝-adic rational numbers, ℂ𝑝 denotes the completion of algebraic closure of β„šπ‘,β„• denotes the set of natural numbers, β„€ denotes the ring of rational integers, β„š denotes the field of rational numbers, β„‚ denotes the set of complex numbers, and β„€+=β„•βˆͺ{0}. Let πœˆπ‘ be the normalized exponential valuation of ℂ𝑝 with |𝑝|𝑝=π‘βˆ’πœˆπ‘(𝑝)=π‘βˆ’1. When one talks of π‘ž-extension, π‘ž is considered in many ways such as an indeterminate, a complex number π‘žβˆˆβ„‚, or 𝑝-adic number π‘žβˆˆβ„‚π‘. If π‘žβˆˆβ„‚ one normally assumes that |π‘ž|<1. If π‘žβˆˆβ„‚π‘, we normally assume that |π‘žβˆ’1|𝑝<π‘βˆ’1/(π‘βˆ’1) so that π‘žπ‘₯=exp(π‘₯logπ‘ž) for |π‘₯|𝑝≀1.

For ξ€·β„€π‘”βˆˆπ‘ˆπ·π‘ξ€Έ=ξ€½π‘”π‘”βˆΆβ„€π‘βŸΆβ„‚π‘isuniformlydifferentiablefunctionξ€Ύ,(1.1) Kim defined the fermionic 𝑝-adic π‘ž-integral on ℀𝑝:πΌβˆ’1(ξ€œπ‘”)=℀𝑝𝑔(π‘₯)π‘‘πœ‡βˆ’1(π‘₯)=limπ‘π‘β†’βˆžπ‘βˆ’1π‘₯=0𝑔(π‘₯)(βˆ’1)π‘₯(1.2) (cf. [5–7]).

If we take 𝑔1(π‘₯)=𝑔(π‘₯+1) in (1.2), then we easily see thatπΌβˆ’1𝑔1ξ€Έ+πΌβˆ’1(𝑔)=2𝑔(0).(1.3) From (1.3), we obtainπΌβˆ’1𝑔𝑛+(βˆ’1)π‘›βˆ’1πΌβˆ’π‘ž(𝑔)=2π‘›βˆ’1𝑙=0(βˆ’1)π‘›βˆ’1βˆ’π‘™π‘”(𝑙),(1.4) where 𝑔𝑛(π‘₯)=𝑔(π‘₯+𝑛) (cf. [1–15]).

As a well-known definition, the Euler polynomials are defined by2𝐹(𝑑)=𝑒𝑑+1=𝑒𝐸𝑑=βˆžξ“π‘›=0𝐸𝑛𝑑𝑛,𝐹2𝑛!(𝑑,π‘₯)=𝑒𝑑𝑒+1π‘₯𝑑=𝑒𝐸(π‘₯)𝑑=βˆžξ“π‘›=0𝐸𝑛𝑑(π‘₯)𝑛,𝑛!(1.5) with the usual convention of replacing 𝐸𝑛(π‘₯) by 𝐸𝑛(π‘₯). In the special case, π‘₯=0,𝐸𝑛(0)=𝐸𝑛 are called the 𝑛th Euler numbers (cf. [1–15]).

Our aim in this paper is to define the generalized 𝑀-Euler numbers 𝐸𝑛,𝑀(π‘Ž) and polynomials 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž). We investigate some properties which are related to the generalized 𝑀-Euler numbers 𝐸𝑛,𝑀(π‘Ž) and polynomials 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž). In particular, distribution of roots for 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž)=0 is different from 𝐸𝑛(π‘₯)=0’s. We also derive the existence of a specific interpolation function which interpolate the generalized 𝑀-Euler numbers 𝐸𝑛,𝑀(π‘Ž) and polynomials𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž).

2. The Generalized 𝑀-Euler Numbers and Polynomials

Our primary goal of this section is to define the generalized 𝑀-Euler numbers 𝐸𝑛,𝑀(π‘Ž) and polynomials 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž). We also find generating functions of the generalized 𝑀-Euler numbers 𝐸𝑛,𝑀(π‘Ž) and polynomials 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž). Let π‘Ž be strictly positive real number.

The generalized 𝑀-Euler numbers and polynomials 𝐸𝑛,𝑀(π‘Ž),𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) are defined byβˆžξ“π‘›=0𝐸𝑛,𝑀(π‘‘π‘Ž)𝑛=ξ€œπ‘›!β„€π‘π‘€π‘Žπ‘₯π‘’π‘Žπ‘₯π‘‘π‘‘πœ‡βˆ’1(π‘₯),(2.1)βˆžξ“π‘›=0𝐸𝑛,𝑀(𝑑π‘₯βˆΆπ‘Ž)𝑛=ξ€œπ‘›!β„€π‘π‘€π‘Žπ‘¦π‘’(π‘Žπ‘¦+π‘₯)π‘‘π‘‘πœ‡βˆ’1(𝑦),forπ‘‘βˆˆβ„,π‘€βˆˆβ„‚,(2.2) respectively.

From above definition, we obtain𝐸𝑛,𝑀(ξ€œπ‘Ž)=β„€π‘π‘€π‘Žπ‘₯(π‘Žπ‘₯)π‘›π‘‘πœ‡βˆ’1(𝐸π‘₯),𝑛,π‘€ξ€œ(π‘₯βˆΆπ‘Ž)=β„€π‘π‘€π‘Žπ‘¦(π‘Žπ‘¦+π‘₯)π‘›π‘‘πœ‡βˆ’1(𝑦).(2.3)

Let 𝑔(π‘₯)=π‘€π‘Žπ‘₯π‘’π‘Žπ‘₯𝑑. By (1.3) and using 𝑝-adic integral on ℀𝑝, we haveπΌβˆ’1𝑔1ξ€Έ+πΌβˆ’1(ξ€œπ‘”)=β„€π‘π‘€π‘Ž(π‘₯+1)π‘’π‘Ž(π‘₯+1)π‘‘π‘‘πœ‡βˆ’1(ξ€œπ‘₯)+β„€π‘π‘€π‘Žπ‘₯π‘’π‘Žπ‘₯π‘‘π‘‘πœ‡βˆ’1(=𝑀π‘₯)π‘Žπ‘’π‘Žπ‘‘ξ€Έξ€œ+1β„€π‘π‘€π‘Žπ‘₯π‘’π‘Žπ‘₯π‘‘π‘‘πœ‡βˆ’1(π‘₯)=2.(2.4)

Hence, by (2.1), we obtainβˆžξ“π‘›=0𝐸𝑛,𝑀(π‘‘π‘Ž)𝑛=2𝑛!π‘€π‘Žπ‘’π‘Žπ‘‘.+1(2.5)

By (1.3), (2.2) and 𝑔(𝑦)=π‘€π‘Žπ‘¦π‘’(π‘Žπ‘¦+π‘₯)𝑑, we haveβˆžξ“π‘›=0𝐸𝑛,𝑀(𝑑π‘₯βˆΆπ‘Ž)𝑛=2𝑛!π‘€π‘Žπ‘’π‘Žπ‘‘π‘’+1π‘₯𝑑.(2.6)

After some elementary calculations, we obtainβˆžξ“π‘›=0𝐸𝑛,𝑀(𝑑π‘₯βˆΆπ‘Ž)𝑛𝑛!=2βˆžξ“π‘›=0(βˆ’1)π‘›π‘€π‘Žπ‘’π‘Žπ‘›π‘‘π‘’π‘₯𝑑.(2.7)

From (2.6), we have𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž)=𝑛𝑛=0βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ π‘₯π‘›βˆ’π‘˜πΈπ‘˜,𝑀=ξ€·(π‘Ž)π‘₯+𝐸𝑀(π‘Ž)𝑛,(2.8) with the usual convention of replacing (𝐸𝑀(π‘Ž))𝑛 by 𝐸𝑛,𝑀(π‘Ž).

3. Basic Properties for the Generalized 𝑀-Euler Numbers and Polynomials

By (2.5), we haveπœ•πœ•π‘₯βˆžξ“π‘›=0𝐸𝑛,𝑀(𝑑π‘₯βˆΆπ‘Ž)𝑛=πœ•π‘›!ξ‚€2πœ•π‘₯π‘€π‘Žπ‘’π‘Žπ‘‘π‘’+1π‘₯𝑑=π‘‘βˆžξ“π‘›=0𝐸𝑛,𝑀𝑑(π‘₯βˆΆπ‘Ž)𝑛=𝑛!βˆžξ“π‘›=0π‘›πΈπ‘›βˆ’1,𝑀𝑑(π‘₯βˆΆπ‘Ž)𝑛.𝑛!(3.1)

By (3.1), we have the following differential relation.

Theorem 3.1. For positive integers 𝑛, one has πœ•πΈπœ•π‘₯𝑛,𝑀(π‘₯βˆΆπ‘Ž)=π‘›πΈπ‘›βˆ’1,𝑀(π‘₯βˆΆπ‘Ž).(3.2)

By Theorem 3.1, we easily obtain the following corollary.

Corollary 3.2 (Integral formula). One has ξ€œπ‘žπ‘πΈπ‘›βˆ’1,𝑀1(π‘₯βˆΆπ‘Ž)𝑑π‘₯=𝑛𝐸𝑛,𝑀(π‘žβˆΆπ‘Ž)βˆ’πΈπ‘›,𝑀.(π‘βˆΆπ‘Ž)(3.3)

By (2.5), we obtainβˆžξ“π‘›=0𝐸𝑛,𝑀(𝑑π‘₯+π‘¦βˆΆπ‘Ž)𝑛=2𝑛!π‘€π‘Žπ‘’π‘Žπ‘‘π‘’+1(π‘₯+𝑦)𝑑=βˆžξ“π‘›=0𝐸𝑛,𝑀𝑑(π‘₯βˆΆπ‘Ž)𝑛𝑛!βˆžξ“π‘˜=0π‘¦π‘˜π‘‘π‘˜=π‘˜!βˆžξ“π‘›=0βŽ›βŽœβŽœβŽπ‘›ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ πΈπ‘˜,𝑀(π‘₯βˆΆπ‘Ž)π‘¦π‘›βˆ’π‘˜βŽžβŽŸβŽŸβŽ π‘‘π‘›.𝑛!(3.4)

By comparing coefficients of 𝑑𝑛/𝑛! in the above equation, we arrive at the following addition theorem.

Theorem 3.3 (Addition theorem). For π‘›βˆˆβ„€+, 𝐸𝑛,𝑀(π‘₯+π‘¦βˆΆπ‘Ž)=π‘›ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ πΈπ‘˜,𝑀(π‘₯βˆΆπ‘Ž)π‘¦π‘›βˆ’π‘˜.(3.5)

By (2.5), for π‘šβ‰‘1(mod2), we haveβˆžξ“π‘›=0ξƒ©π‘šπ‘›π‘šβˆ’1ξ“π‘˜=0(βˆ’1)π‘˜π‘€π‘Žπ‘˜πΈπ‘›,π‘€π‘šξ‚€π‘₯+π‘Žπ‘˜π‘šξ‚ξƒͺπ‘‘βˆΆπ‘Žπ‘›=𝑛!π‘šβˆ’1ξ“π‘˜=0(βˆ’1)π‘˜π‘€π‘Žπ‘˜ξƒ©βˆžξ“π‘›=0𝐸𝑛,π‘€π‘šξ‚€π‘₯+π‘Žπ‘˜π‘šξ‚ξƒͺβˆΆπ‘Ž(π‘šπ‘‘)𝑛=𝑛!π‘šβˆ’1ξ“π‘˜=0ξ‚€(βˆ’1)π‘˜π‘€π‘Žπ‘˜2π‘€π‘šπ‘Žπ‘’π‘šπ‘Žπ‘‘π‘’(π‘₯+π‘Žπ‘˜)𝑑=2π‘€π‘Žπ‘’π‘Žπ‘‘π‘’+1π‘₯𝑑=βˆžξ“π‘›=0𝐸𝑛,𝑀𝑑(π‘₯βˆΆπ‘Ž)𝑛.𝑛!(3.6) By comparing coefficients of 𝑑𝑛/𝑛! in the above equation, we arrive at the following multiplication theorem.

Theorem 3.4 (Multiplication theorem). For π‘š,π‘›βˆˆβ„•πΈπ‘›,𝑀(π‘₯βˆΆπ‘Ž)=π‘šπ‘›π‘šβˆ’1ξ“π‘˜=0(βˆ’1)π‘˜π‘€π‘Žπ‘˜πΈπ‘›,π‘€π‘šξ‚€π‘₯+π‘Žπ‘˜π‘šξ‚.βˆΆπ‘Ž(3.7)

From (1.3), we note thatξ€œ2=β„€π‘π‘€π‘Žπ‘₯+π‘Žπ‘’(π‘Žπ‘₯+π‘Ž)π‘‘π‘‘πœ‡βˆ’1(ξ€œπ‘₯)+β„€π‘π‘€π‘Žπ‘₯π‘’π‘Žπ‘₯π‘‘π‘‘πœ‡βˆ’1(=π‘₯)βˆžξ“π‘›=0ξƒ©π‘€π‘Žξ€œβ„€π‘π‘€π‘Žπ‘₯(π‘Žπ‘₯+π‘Ž)π‘›π‘‘πœ‡βˆ’1ξ€œ(π‘₯)+β„€π‘π‘€π‘Žπ‘₯(π‘Žπ‘₯)π‘›π‘‘πœ‡βˆ’1ξƒͺ𝑑(π‘₯)𝑛=𝑛!βˆžξ“π‘›=0ξ€·π‘€π‘ŽπΈπ‘›,𝑀(π‘ŽβˆΆπ‘Ž)+𝐸𝑛,𝑀(ξ€Έπ‘‘π‘Ž)𝑛.𝑛!(3.8) From the above, we obtain the following theorem.

Theorem 3.5. For π‘›βˆˆβ„€+, one has π‘€π‘ŽπΈπ‘›,𝑀(π‘ŽβˆΆπ‘Ž)+𝐸𝑛,𝑀(π‘Ž)=2,if𝑛=0,0,if𝑛>0.(3.9)

By (2.8) in the above, we arrive at the following corollary.

Corollary 3.6. For π‘›βˆˆβ„€+, one has π‘€π‘Žξ€·π‘Ž+𝐸𝑀(π‘Ž)𝑛+𝐸𝑛,𝑀(π‘Ž)=2,if𝑛=0,0,if𝑛>0,(3.10) with the usual convention of replacing (𝐸𝑀(π‘Ž))𝑛 by 𝐸𝑛,𝑀(π‘Ž).

From (1.4), we note thatβˆžξ“π‘š=02π‘›βˆ’1𝑙=0(βˆ’1)π‘›βˆ’1βˆ’π‘™π‘€π‘Žπ‘™(π‘Žπ‘™)π‘šξƒͺ𝑑𝑛=ξ€œπ‘š!β„€π‘π‘€π‘Žπ‘₯+π‘Žπ‘›π‘’(π‘Žπ‘₯+π‘Žπ‘›)π‘‘π‘‘πœ‡βˆ’1(π‘₯)+(βˆ’1)π‘›βˆ’1ξ€œβ„€π‘π‘€π‘Žπ‘₯π‘’π‘Žπ‘₯π‘‘π‘‘πœ‡βˆ’1=(π‘₯)βˆžξ“π‘š=0ξƒ©π‘€π‘Žπ‘›ξ€œβ„€π‘π‘€π‘Žπ‘₯(π‘Žπ‘₯+π‘Žπ‘›)π‘šπ‘‘πœ‡βˆ’1(π‘₯)+(βˆ’1)π‘›ξ€œβ„€π‘π‘€π‘Žπ‘₯(π‘Žπ‘₯)π‘šπ‘‘πœ‡βˆ’1ξƒͺ𝑑(π‘₯)π‘š=π‘š!βˆžξ“π‘š=0ξ€·π‘€π‘Žπ‘›πΈπ‘š,𝑀(π‘Žπ‘›βˆΆπ‘Ž)+(βˆ’1)π‘›βˆ’1πΈπ‘š,𝑀𝑑(π‘Ž)π‘š.π‘š!(3.11)

By comparing coefficients of 𝑑𝑛/𝑛! in the above equation, we arrive at the following theorem.

Theorem 3.7. For π‘›βˆˆβ„€+, one has π‘€π‘Žπ‘›πΈπ‘š,𝑀(π‘›π‘ŽβˆΆπ‘Ž)+(βˆ’1)π‘›βˆ’1πΈπ‘š,𝑀(π‘Ž)=2π‘›βˆ’1𝑙=0(βˆ’1)π‘›βˆ’1βˆ’π‘™π‘€π‘Žπ‘™(π‘Žπ‘™)π‘š.(3.12)

4. The Analogue of the Euler Zeta Function

By using the generalized 𝑀-Euler numbers and polynomials, the generalized 𝑀-Euler zeta function and the generalized Hurwitz 𝑀-Euler zeta functions are defined. These functions interpolate the generalized 𝑀-Euler numbers and 𝑀-Euler polynomials, respectively. Let𝐹𝑀(π‘₯βˆΆπ‘Ž)(𝑑)=2βˆžξ“π‘›=0(βˆ’1)π‘›π‘€π‘Žπ‘’π‘Žπ‘›π‘‘π‘’π‘₯𝑑=βˆžξ“π‘›=0𝐸𝑛,𝑀(𝑑π‘₯βˆΆπ‘Ž)𝑛.𝑛!(4.1)

By applying derivative operator, π‘‘π‘˜/π‘‘π‘‘π‘˜|𝑑=0 to the above equation, we haveπ‘‘π‘˜π‘‘π‘‘π‘˜πΉπ‘€||||(π‘₯βˆΆπ‘Ž)(𝑑)𝑑=0=2βˆžξ“π‘›=0(βˆ’1)π‘›π‘€π‘Žπ‘›(π‘Žπ‘›+π‘₯)π‘˜,(π‘˜βˆˆβ„•),(4.2)πΈπ‘˜,𝑀(π‘₯βˆΆπ‘Ž)=2βˆžξ“π‘›=0(βˆ’1)π‘›π‘€π‘Žπ‘›(π‘Žπ‘›+π‘₯)π‘˜.(4.3)

By using the above equation, we are now ready to define the generalized 𝑀-Euler zeta functions.

Definition 4.1. For π‘ βˆˆβ„‚, one defines πœπ‘€(π‘Ž)(π‘₯βˆΆπ‘ )=2βˆžξ“π‘›=1(βˆ’1)π‘›π‘€π‘Žπ‘›(π‘Žπ‘›+π‘₯)𝑠.(4.4)

Note that πœπ‘€(π‘Ž)(π‘₯,𝑠) is a meromorphic function on β„‚. Note that if 𝑀→1 and π‘Ž=1, then πœπ‘€(π‘Ž)(π‘₯βˆΆπ‘ )=𝜁(π‘₯βˆΆπ‘ ) which is the Hurwitz Euler zeta functions. Relation between πœπ‘€(π‘Ž)(π‘₯βˆΆπ‘ ) and πΈπ‘˜,𝑀(π‘₯βˆΆπ‘Ž) is given by the following theorem.

Theorem 4.2. For π‘˜βˆˆβ„•, one has πœπ‘€(π‘Ž)(π‘₯βˆΆβˆ’π‘ )=𝐸𝑠,𝑀(π‘₯βˆΆπ‘Ž).(4.5)

Observe that πœπ‘€(π‘Ž)(π‘₯βˆΆπ‘ ) function interpolates 𝐸𝑀(π‘₯βˆΆπ‘ ) numbers at nonnegative integers.

By using (4.2), we note thatπ‘‘π‘˜π‘‘π‘‘π‘˜πΉπ‘€||||(0βˆΆπ‘Ž)(𝑑)𝑑=0=2βˆžξ“π‘›=0(βˆ’1)π‘›π‘€π‘Žπ‘›(π‘Žπ‘›)π‘˜,(π‘˜βˆˆβ„•).(4.6) Hence, we obtainπΈπ‘˜,𝑀(π‘Ž)=2βˆžξ“π‘›=0(βˆ’1)π‘›π‘€π‘Žπ‘›(π‘Žπ‘›)π‘˜.(4.7)

By using the above equation, we are now ready to define the generalized Hurwitz 𝑀-Euler zeta functions.

Definition 4.3. Let π‘ βˆˆβ„‚. One defines πœπ‘€(π‘Ž)(𝑠)=2βˆžξ“π‘›=1(βˆ’1)π‘›π‘€π‘Žπ‘›(π‘Žπ‘›)𝑠.(4.8) Note that πœπ‘€(π‘Ž)(𝑠) is a meromorphic function on β„‚. Obverse that, if 𝑀→1 and π‘Ž=1, then πœπ‘€(π‘Ž)(𝑠)=𝜁(𝑠) which is the Euler zeta functions. Relation between πœπ‘€(π‘Ž)(𝑠) and πΈπ‘˜,𝑀(𝑠) is given by the following theorem.

Theorem 4.4. For π‘˜βˆˆβ„•, one has πœπ‘€(π‘Ž)(βˆ’π‘˜)=πΈπ‘˜,𝑀(π‘Ž).(4.9) Observe that πœπ‘€(π‘Ž)(βˆ’π‘˜) function interpolates πΈπ‘˜,𝑀(π‘Ž) numbers at nonnegative integers.

5. Zeros of the Generalized 𝑀-Euler Polynomials 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž)

In this section, we investigate the reflection symmetry of the zeros of the generalized 𝑀-Euler polynomials 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž).

In the special case, 𝑀=1,𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) are called generalized Euler polynomials 𝐸𝑛(π‘₯βˆΆπ‘Ž). Sinceβˆžξ“π‘›=0𝐸𝑛(π‘Žβˆ’π‘₯βˆΆπ‘Ž)(βˆ’π‘‘)𝑛=2𝑛!π‘’βˆ’π‘Žπ‘‘π‘’+1(π‘Žβˆ’π‘₯)(βˆ’π‘‘)=2π‘’π‘Žπ‘‘π‘’+1π‘₯𝑑=βˆžξ“π‘›=0𝐸𝑛𝑑(π‘₯βˆΆπ‘Ž)𝑛,𝑛!(5.1) we have𝐸𝑛(π‘₯βˆΆπ‘Ž)=(βˆ’1)𝑛𝐸𝑛(π‘Žβˆ’π‘₯βˆΆπ‘Ž),forπ‘›βˆˆβ„•.(5.2) We observe that 𝐸𝑛(π‘₯βˆΆπ‘Ž),π‘₯βˆˆβ„‚ has Re(π‘₯)=π‘Ž/2 reflection symmetry in addition to the usual Im(π‘₯)=0 reflection symmetry analytic complex functions.

Let𝐹𝑀,π‘Ž(2π‘₯βˆΆπ‘‘)=π‘€π‘Žπ‘’π‘Žπ‘‘π‘’+1π‘₯𝑑=βˆžξ“π‘›=0𝐸𝑛,𝑀(𝑑π‘₯βˆΆπ‘Ž)𝑛.𝑛!(5.3) Then we haveπΉπ‘€βˆ’1,π‘Ž2(π‘Žβˆ’π‘₯βˆΆβˆ’π‘‘)=π‘€βˆ’π‘Žπ‘’βˆ’π‘Žπ‘‘π‘’+1(π‘Žβˆ’π‘₯)(βˆ’π‘‘)=π‘€π‘Ž2π‘€π‘Žπ‘’π‘Žπ‘‘π‘’+1π‘₯𝑑=π‘€π‘Žβˆžξ“π‘›=0𝐸𝑛,𝑀(𝑑π‘₯βˆΆπ‘Ž)𝑛.𝑛!(5.4) Hence, we arrive at the following complement theorem.

Theorem 5.1 (Complement theorem). For π‘›βˆˆβ„•, 𝐸𝑛,π‘€βˆ’1(π‘Žβˆ’π‘₯βˆΆπ‘Ž)=(βˆ’1)π‘›π‘€π‘ŽπΈπ‘›,𝑀(π‘₯βˆΆπ‘Ž).(5.5)

Throughout the numerical experiments, we can finally conclude that 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž),π‘₯βˆˆβ„‚ has not Re(π‘₯)=π‘Ž/2 reflection symmetry analytic complex functions. However, we observe that 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž),π‘₯βˆˆβ„‚ has Im(π‘₯)=0 reflection symmetry (see Figures 1, 2, and 3). The obvious corollary is that the zeros of 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) will also inherit these symmetries:if𝐸𝑛,𝑀π‘₯0ξ€ΈβˆΆπ‘Ž=0,then𝐸𝑛,𝑀π‘₯βˆ—0ξ€ΈβˆΆπ‘Ž=0,(5.6) where βˆ— denotes complex conjugation (see Figures 1, 2 and 3).

fig1
Figure 1: Zeros of𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) for π‘Ž=1,2,3,4.
fig2
Figure 2: Real zeros of𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) for 1≀𝑛≀20.
fig3
Figure 3: Zeros of 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) for 𝑀=1,2,3,4.

We investigate the beautiful zeros of the generalized 𝑀-Euler polynomials 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) by using a computer. We plot the zeros of the generalized Euler polynomials 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) for 𝑛=30,π‘Ž=1,2,3,4, and π‘₯βˆˆβ„‚(Figure 1).

In Figure 1(a), we choose 𝑛=30,𝑀=1, and π‘Ž=1. In Figure 1(b), we choose 𝑛=30,𝑀=1, and π‘Ž=2. In Figure 1(c), we choose 𝑛=30,𝑀=3, and π‘Ž=3. In Figure 1(d), we choose 𝑛=30,𝑀=4, and π‘Ž=4.

Plots of real zeros of 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) for 1≀𝑛≀20 structure are presented (Figure 2).

In Figure 2(a), we choose 𝑀=1 and π‘Ž=1. In Figure 2(b), we choose 𝑀=1 and π‘Ž=2. In Figure 2(c), we choose 𝑀=3 and π‘Ž=3. In Figure 2(d), we choose 𝑀=4 and π‘Ž=4.

We investigate the beautiful zeros of the generalized 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) by using a computer. We plot the zeros of the generalized 𝑀-Euler polynomials 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) for 𝑛=30 and π‘₯βˆˆβ„‚(Figure 3).

In Figure 3(a), we choose π‘Ž=3 and 𝑀=1. In Figure 3(b), we choose π‘Ž=3 and 𝑀=2. In Figure 3(c), we choose π‘Ž=3 and 𝑀=3. In Figure 3(d), we choose π‘Ž=3 and 𝑀=4.

Stacks of zeros of 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) for 1≀𝑛≀30,𝑀=4,π‘Ž=3 from a 3D structure are presented (Figure 4).

475463.fig.004
Figure 4: Stacks of zeros of 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) for 1≀𝑛≀30.

Our numerical results for approximate solutions of real zeros of the generalized 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) are displayed (Tables 1 and 2).

tab1
Table 1: Numbers of real and complex zeros of 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž).
tab2
Table 2: Approximate solutions of 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž)=0,π‘₯βˆˆβ„.

We observe a remarkably regular structure of the complex roots of the generalized 𝑀-Euler polynomials 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž). We hope to verify a remarkably regular structure of the complex roots of the generalized 𝑀-Euler polynomials 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) (Table 1). Next, we calculated an approximate solution satisfying 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž),𝑀=2,π‘Ž=2,π‘₯βˆˆβ„. The results are given in Table 2.

The plot above shows the generalized 𝑀-Euler polynomials 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) for real 1β‰€π‘Žβ‰€7 and βˆ’5≀π‘₯≀5, with the zero contour indicated in black (Figure 5). In Figure 5(a), we choose 𝑛=2 and 𝑀=2. In Figure 5(b), we choose 𝑛=3 and 𝑀=3. In Figure 5(c), we choose 𝑛=4 and 𝑀=4. In Figure 5(d), we choose 𝑛=5 and 𝑀=5.

fig5
Figure 5: Zero contour of 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž).

Finally, we will consider the more general problems. How many roots does 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) have? This is an open problem. Prove or disprove: 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž)=0 has 𝑛 distinct solutions. Find the numbers of complex zeros 𝐢𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) of 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž),Im(π‘₯βˆΆπ‘Ž)β‰ 0. Since 𝑛 is the degree of the polynomial E𝑛,𝑀(π‘₯βˆΆπ‘Ž), the number of real zeros 𝑅𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) lying on the real plane Im(π‘₯βˆΆπ‘Ž)=0 is then 𝑅𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž)=π‘›βˆ’πΆπΈπ‘›,𝑀(π‘₯βˆΆπ‘Ž), where 𝐢𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) denotes complex zeros. See Table 1 for tabulated values of 𝑅𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) and 𝐢𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž). We plot the zeros of 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž), respectively (Figures 1–5). These figures give mathematicians an unbounded capacity to create visual mathematical investigations of the behavior of the roots of the 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž). Moreover, it is possible to create a new mathematical ideas and analyze them in ways that, generally, are not possible by hand. The authors have no doubt that investigation along this line will lead to a new approach employing numerical method in the field of research of 𝑀-Euler polynomials 𝐸𝑛,𝑀(π‘₯βˆΆπ‘Ž) to appear in mathematics and physics.

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