# 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)
- ${E}_{n}\left(x\right)={\sum}_{k=0}^{n}\left(\genfrac{}{}{0pt}{}{n}{k}\right){E}_{k}\left(0\right){x}^{n-k}$;

**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

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**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