# Corrections: Kim, T.; et al. Some Identities for Euler and Bernoulli Polynomials and Their Zeros. Axioms 2018, 7, 56

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## 1. Corrigendum

- Theorem 1 in [1] and Results (13) and (14) in [2] are identical: for $n\ge 0$,$${E}_{n}^{\left(C\right)}(x,y)=\sum _{l=0}^{n}\left(\genfrac{}{}{0pt}{}{n}{l}\right){E}_{l}{C}_{n-l}(x,y)\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{E}_{n}^{\left(S\right)}(x,y)=\sum _{l=0}^{n}\left(\genfrac{}{}{0pt}{}{n}{l}\right){E}_{l}{S}_{n-l}(x,y).$$
- Theorem 3 in [1] and Proposition 2.1 in [2] state the same outcome: for $n\ge 0$,$${E}_{n}^{\left(C\right)}(1-x,y)={(-1)}^{n}{E}_{n}^{\left(C\right)}(x,y)\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{E}_{n}^{\left(S\right)}(1-x,y)={(-1)}^{n+1}{E}_{n}^{\left(S\right)}(x,y).$$
- Theorem 4 in [1] and Proposition 2.2 in [2] present identical results: for $n\ge 0$,$${E}_{n}^{\left(C\right)}(x+1,y)+{E}_{n}^{\left(C\right)}(x,y)=2{C}_{n}(x,y)\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{E}_{n}^{\left(S\right)}(x+1,y)+{E}_{n}^{\left(S\right)}(x,y)=2{S}_{n}(x,y).$$
- Corollary 1 in [1] and Corollary 2.2 in [2] show matching expressions: for $n\ge 0$,$${E}_{2n}^{\left(C\right)}(1,y)+{E}_{2n}^{\left(C\right)}(0,y)=2{(-1)}^{n}{y}^{2n}\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{E}_{2n+1}^{\left(S\right)}(1,y)+{E}_{2n+1}^{\left(S\right)}(0,y)=2{(-1)}^{n}{y}^{2n+1}.$$
- Theorem 5 in [1] and Proposition 2.3 in [2] have matching results: for $n\ge 0\phantom{\rule{3.33333pt}{0ex}}r\in \mathbb{N}$,$${E}_{n}^{\left(C\right)}(x+r,y)=\sum _{k=0}^{n}\left(\genfrac{}{}{0pt}{}{n}{k}\right){E}_{k}^{\left(C\right)}(x,y){r}^{n-k}\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{E}_{n}^{\left(S\right)}(x+r,y)=\sum _{k=0}^{n}\left(\genfrac{}{}{0pt}{}{n}{k}\right){E}_{k}^{\left(S\right)}(x,y){r}^{n-k}.$$

## 2. Corrections

## Acknowledgments

## References

- Kim, T.; Ryoo, C.S. Some Identities for Euler and Bernoulli Polynomials and Their Zeros. Axioms
**2018**, 7, 56. [Google Scholar] [CrossRef] - Masjed-Jamei, M.; Beyki, M.R.; Koepf, W. A New Type of Euler Polynomials and Numbers. Mediterr. J. Math.
**2018**, 15, 138. [Google Scholar] [CrossRef] - Masjed-Jamei, M.; Beyki, M.R.; Koepf, W. An extension of the Euler-Maclaurin quadrature formula using a parametric type of Bernoulli polynomials. Bull. Sci. Math.
**2019**. to appear. [Google Scholar] [CrossRef]

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## Share and Cite

**MDPI and ACS Style**

Kim, T.; Ryoo, C.S.
Corrections: Kim, T.; et al. Some Identities for Euler and Bernoulli Polynomials and Their Zeros. *Axioms* 2018, *7*, 56. *Axioms* **2019**, *8*, 107.
https://doi.org/10.3390/axioms8040107

**AMA Style**

Kim T, Ryoo CS.
Corrections: Kim, T.; et al. Some Identities for Euler and Bernoulli Polynomials and Their Zeros. *Axioms* 2018, *7*, 56. *Axioms*. 2019; 8(4):107.
https://doi.org/10.3390/axioms8040107

**Chicago/Turabian Style**

Kim, Taekyun, and Cheon Seoung Ryoo.
2019. "Corrections: Kim, T.; et al. Some Identities for Euler and Bernoulli Polynomials and Their Zeros. *Axioms* 2018, *7*, 56" *Axioms* 8, no. 4: 107.
https://doi.org/10.3390/axioms8040107