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Axioms
, Volume 8 (4) – Oct 1, 2019

/lp/multidisciplinary-digital-publishing-institute/corrections-kim-t-et-al-some-identities-for-euler-and-bernoulli-ZmsI1PDo2J

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- Multidisciplinary Digital Publishing Institute
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- © 1996-2019 MDPI (Basel, Switzerland) unless otherwise stated Terms and Conditions Privacy Policy
- ISSN
- 2075-1680
- DOI
- 10.3390/axioms8040107
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axioms Correction Corrections: Kim, T.; et al. Some Identities for Euler and Bernoulli Polynomials and Their Zeros. Axioms 2018, 7, 56. 1 2, Taekyun Kim and Cheon Seoung Ryoo * Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea; tkkim@kw.ac.kr Department of Mathematics, Hannam University, Daejeon 306-791, Korea * Correspondence: ryoocs@hnu.kr Received: 17 September 2019; Accepted: 29 September 2019; Published: 1 October 2019 1. Corrigendum The authors, Kim and Ryoo in [1], studied Euler polynomials and Bernoulli polynomials with an extended variable to a complex variable, replacing real variable x by complex variable x + iy, and achieved several useful identities and properties. The authors would like to note that these results can also be derived from a different approach by considering Euler polynomials and Bernoulli polynomials with a pair of two variables, as shown in [2], instead of a complex variable. For example, Masjed-Jamei, Beyki and Koepf in [2] introduced the new type Euler polynomials given by pt ¥ n 2e t (c) cos(qt) = E (p, q) , å n e + 1 n! n=0 (1) pt n 2e t (s) sin(qt) = E (p, q) , å n e + 1 n! n=0 which are considered without a complex variable. On the other hand, the authors in [1] considered the Euler polynomials and Bernoulli polynomials with a complex variable instead of x variable as follows: 2 t (x+iy)t e = E (x + iy) , e + 1 n! n=0 and ¥ n t t (x+iy)t e = B (x + iy) , e 1 n! n=0 which imply the equivalence deﬁnitions to Equation (1) as ¥ n ¥ n 2 E (x + iy) + E (x iy) t t n n (c) xt e cos(yt) = = E (x, y) , å å e + 1 2 n! n! n=0 n=0 (2) ¥ n ¥ n 2 E (x + iy) E (x iy) t t n n (s) xt e sin(yt) = = E (x, y) å å n e + 1 2i n! n! n=0 n=0 and ¥ n ¥ n t B (x + iy) + B (x iy) t t n n (c) xt e cos(yt) = = B (x, y) , å å n e 1 2 n! n! n=0 n=0 (3) ¥ n ¥ n t B (x + iy) B (x iy) t t n n (s) xt e sin(yt) = = B (x, y) . å å n e 1 2i n! n! n=0 n=0 Axioms 2019, 8, 107; doi:10.3390/axioms8040107 www.mdpi.com/journal/axioms Axioms 2019, 8, 107 2 of 3 Here, the authors considered the Euler polynomials and Bernoulli polynomials of a complex variable, by treating the real and imaginary parts separately, which are able to introduce the cosine Euler polynomials, the sine Euler polynomials, the cosine Bernoulli polynomials, and the sine Bernoulli polynomials such as Equations (2) and (3). After the paper “Some Identities for Euler and Bernoulli Polynomials and Their Zeros in Axioms 2018, 7, 56.” by T. Kim and C.S. Ryoo was published, we realized that some results of the paper “A New Type of Euler Polynomials and Numbers in Mediterr. J. Math. (2018) 15: 138.” by M. Masjed-Jamei, M.R. Beyki, and W. Koepf were published ahead with some identical results, which are consistent with the ones in the paper [1]. The authors in [1], after the publication, were aware of that Hacéne Belbachir, the reviewer of the paper [2], left the question related to the extension of a variable in Mathematical Reviews (MR3808565) of the American Mathematical Society: “Is it possible to obtain their results by considering the classical Euler polynomials of complex variable x + iy, and treating the real part and the imaginary part separately?” The approach in Equation (2) can be an afﬁrmative answer to the question. Thus, we want to inform our readers that some results of Reference [2] have been published before the paper [1]. In addition, their related works are presented in [3], in which some similar results are shown as their consistent works in [2]. The authors conclusively note that some of the results in both [1,2] are derived from these two different approaches mentioned above. In addition, the identical results in both [1,2] are listed as follows. 1. Theorem 1 in [1] and Results (13) and (14) in [2] are identical: for n 0, n n n n (C) (S) E (x, y) = E C (x, y) and E (x, y) = E S (x, y). n å l nl n å l nl l l l=0 l=0 2. Theorem 3 in [1] and Proposition 2.1 in [2] state the same outcome: for n 0, (C) (C) (S) (S) n n+1 E (1 x, y) = (1) E (x, y) and E (1 x, y) = (1) E (x, y). n n n n 3. Theorem 4 in [1] and Proposition 2.2 in [2] present identical results: for n 0, (C) (C) (S) (S) E (x + 1, y) + E (x, y) = 2C (x, y) and E (x + 1, y) + E (x, y) = 2S (x, y). n n n n n n 4. Corollary 1 in [1] and Corollary 2.2 in [2] show matching expressions: for n 0, (C) (C) (S) (S) n 2n n 2n+1 E (1, y) + E (0, y) = 2(1) y and E (1, y) + E (0, y) = 2(1) y . 2n 2n 2n+1 2n+1 5. Theorem 5 in [1] and Proposition 2.3 in [2] have matching results: for n 0 r 2 N, n n n n (C) (C) (S) (S) nk nk E (x + r, y) = E (x, y)r and E (x + r, y) = E (x, y)r . n å n å k k k k k=0 k=0 2. Corrections In addition, while reviewing our paper, we found some typing errors: Equation (11) should be revised by E (x + iy) E (x iy) n n 2i and Equation (31) should be also replaced by B (x + iy) B (x iy) n n 2i Axioms 2019, 8, 107 3 of 3 Acknowledgments: The authors would like to thank the authors of [2,3] and Luna Shen for noticing that some results in the paper [2] are released ahead in [1]. References 1. Kim, T.; Ryoo, C.S. Some Identities for Euler and Bernoulli Polynomials and Their Zeros. Axioms 2018, 7, 56. 2. Masjed-Jamei, M.; Beyki, M.R.; Koepf, W. A New Type of Euler Polynomials and Numbers. Mediterr. J. Math. 2018, 15, 138, doi:10.1007/s00009-018-1181-1. 3. 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. c 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/).

Axioms – Multidisciplinary Digital Publishing Institute

**Published: ** Oct 1, 2019

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