# On the Decomposability of the Linear Combinations of Euler Polynomials with Odd Degrees

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

## 2. Auxiliary Results

**Lemma**

**1.**

- (a)
- ${E}_{n}\left(x\right)={(-1)}^{n}{E}_{n}(1-x)$;
- (b)
- ${E}_{n}(x+1)+{E}_{n}\left(x\right)=2{x}^{n}$;
- (c)
- ${E}_{n}^{\prime}\left(x\right)=n{E}_{n-1}\left(x\right)$;
- (d)
- ${E}_{2n-1}(1/2)={E}_{2n}\left(0\right)={E}_{2n}\left(1\right)=0$ for $n\in \mathbb{N}$;
- (e)
- ${E}_{n}\left(x\right)={\sum}_{k=0}^{n}\left(\right)open="("\; close=")">\genfrac{}{}{0pt}{}{n}{k}$;

**Proof.**

**Lemma**

**2**

**(Kreso and Rakaczki [14]).**

- ${\tilde{F}}_{1}\circ {\tilde{F}}_{2}$ and ${F}_{1}\circ {F}_{2}$ are equivalent over $\mathbb{L}$,
- ${\tilde{F}}_{1}\left(x\right)$ and ${\tilde{F}}_{2}\left(x\right)$ are monic polynomials with coefficients in $\mathbb{K},$
- $\mathrm{coeff}\phantom{\rule{4pt}{0ex}}({x}^{\mathrm{deg}{\tilde{F}}_{1}-1},\phantom{\rule{4pt}{0ex}}{\tilde{F}}_{1}\left(x\right))=0.$

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

## 3. Proof of the Theorem

## 4. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Abramowitz, M.; Stegun, I. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables; Dover: New York, NY, USA, 1972. [Google Scholar]
- Brillhart, J. On the Euler and Bernoulli polynomials. J. Reine Angew. Math.
**1969**, 234, 45–64. [Google Scholar] - Koblitz, N. p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd ed.; Graduate Texts in Mathematics; Springer: New York, NY, USA, 1984. [Google Scholar]
- Bagarello, F.; Trapani, C.; Triolo, S. Representable states on quasilocal quasi*-algebras. J. Math. Phys.
**2011**, 52, 013510. [Google Scholar] [CrossRef] - Ballentine, L.E.; McRae, S.M. Moment equations for probability distributions in classical and quantum mechanics. Phys. Rev. A
**1998**, 58, 1799–1809. [Google Scholar] [CrossRef] - Trapani, C.; Triolo, S. Representations of modules over a*-algebra and related seminorms. Stud. Math.
**2008**, 184, 133–148. [Google Scholar] [CrossRef][Green Version] - Triolo, S. WQ* algebras of measurable operators. Indian J. Pure Appl. Math.
**2012**, 43, 601–617. [Google Scholar] [CrossRef] - Dujella, A.; Gusíc, I. Indecomposability of polynomials and related Diophantine equations. Q. J. Math.
**2006**, 57, 193–201. [Google Scholar] [CrossRef] - Dujella, A.; Gusíc, I. Decomposition of a recursive family of polynomials. Monatshefte für Mathematik
**2007**, 152, 97–104. [Google Scholar] [CrossRef][Green Version] - Dujella, A.; Gusic, I.; Tichy, R.F. On the indecomposability of polynomials. Osterreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II
**2005**, 214, 81–88. [Google Scholar] [CrossRef] - Dujella, A.; Tichy, R.F. Diophantine equations for second-order recursive sequences of polynomials. Quart. J. Math.
**2001**, 52, 161–169. [Google Scholar] [CrossRef] - Kreso, D.; Tichy, R.F. Functional composition of polynomials: Indecomposability, Diophantine equations and lacunary polynomials. Graz. Math. Ber.
**2015**, 363, 143–170. [Google Scholar] - Bilu, Y.; Brindza, B.; Kirschenhofer, P.; Pintér, Á.; Tichy, R. Diophantine Equations and Bernoulli Polynomials. Compositio Math.
**2002**, 131, 173–188. [Google Scholar] [CrossRef][Green Version] - Kreso, D.; Rakaczki, C. Diophantine equations with Euler polynomials. Acta Arith.
**2013**, 161, 267–281. [Google Scholar] [CrossRef][Green Version] - Pintér, Á.; Rakaczki, C. On the decomposability of linear combinations of Bernoulli. Monatshefte Math.
**2016**, 180, 631–648. [Google Scholar] [CrossRef] - Pintér, Á.; Rakaczki, C. On the decomposability of the linear combinations of Euler polynomials. Miskolc Math. Notes
**2017**, 18, 407–415. [Google Scholar] [CrossRef]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pintér, Á.; Rakaczki, C.
On the Decomposability of the Linear Combinations of Euler Polynomials with Odd Degrees. *Symmetry* **2019**, *11*, 739.
https://doi.org/10.3390/sym11060739

**AMA Style**

Pintér Á, Rakaczki C.
On the Decomposability of the Linear Combinations of Euler Polynomials with Odd Degrees. *Symmetry*. 2019; 11(6):739.
https://doi.org/10.3390/sym11060739

**Chicago/Turabian Style**

Pintér, Ákos, and Csaba Rakaczki.
2019. "On the Decomposability of the Linear Combinations of Euler Polynomials with Odd Degrees" *Symmetry* 11, no. 6: 739.
https://doi.org/10.3390/sym11060739