Explicit Properties of <i>q</i>-Cosine and <i>q</i>-Sine Euler Polynomials Containing Symmetric Structures
In this paper, we introduce <i>q</i>-cosine and <i>q</i>-sine Euler polynomials and determine identities for these polynomials. From these polynomials, we obtain some special properties using a power series of <i>q</i>-trigonometric functions, properties of <i&...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-07-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/12/8/1247 |
| _version_ | 1797561076980645888 |
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| author | Cheon Seoung Ryoo Jung Yoog Kang |
| author_facet | Cheon Seoung Ryoo Jung Yoog Kang |
| author_sort | Cheon Seoung Ryoo |
| collection | DOAJ |
| description | In this paper, we introduce <i>q</i>-cosine and <i>q</i>-sine Euler polynomials and determine identities for these polynomials. From these polynomials, we obtain some special properties using a power series of <i>q</i>-trigonometric functions, properties of <i>q</i>-exponential functions, and <i>q</i>-analogues of the binomial theorem. We investigate the approximate roots of <i>q</i>-cosine Euler polynomials that help us understand these polynomials. Moreover, we display the approximate roots movements of <i>q</i>-cosine Euler polynomials in a complex plane using the Newton method. |
| first_indexed | 2024-03-10T18:09:10Z |
| format | Article |
| id | doaj.art-7cf1d800c15a4bdfa95ba3f7b6ca2f50 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T18:09:10Z |
| publishDate | 2020-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-7cf1d800c15a4bdfa95ba3f7b6ca2f502023-11-20T08:16:17ZengMDPI AGSymmetry2073-89942020-07-01128124710.3390/sym12081247Explicit Properties of <i>q</i>-Cosine and <i>q</i>-Sine Euler Polynomials Containing Symmetric StructuresCheon Seoung Ryoo0Jung Yoog Kang1Department of Mathematics, Hanman University, Daejeon 10216, KoreaDepartment of Mathematics Education, Silla University, Busan 469470, KoreaIn this paper, we introduce <i>q</i>-cosine and <i>q</i>-sine Euler polynomials and determine identities for these polynomials. From these polynomials, we obtain some special properties using a power series of <i>q</i>-trigonometric functions, properties of <i>q</i>-exponential functions, and <i>q</i>-analogues of the binomial theorem. We investigate the approximate roots of <i>q</i>-cosine Euler polynomials that help us understand these polynomials. Moreover, we display the approximate roots movements of <i>q</i>-cosine Euler polynomials in a complex plane using the Newton method.https://www.mdpi.com/2073-8994/12/8/1247q-cosine Euler polynomialsq-sine Euler polynomialsq-trigonometric functionq-exponential function |
| spellingShingle | Cheon Seoung Ryoo Jung Yoog Kang Explicit Properties of <i>q</i>-Cosine and <i>q</i>-Sine Euler Polynomials Containing Symmetric Structures Symmetry q-cosine Euler polynomials q-sine Euler polynomials q-trigonometric function q-exponential function |
| title | Explicit Properties of <i>q</i>-Cosine and <i>q</i>-Sine Euler Polynomials Containing Symmetric Structures |
| title_full | Explicit Properties of <i>q</i>-Cosine and <i>q</i>-Sine Euler Polynomials Containing Symmetric Structures |
| title_fullStr | Explicit Properties of <i>q</i>-Cosine and <i>q</i>-Sine Euler Polynomials Containing Symmetric Structures |
| title_full_unstemmed | Explicit Properties of <i>q</i>-Cosine and <i>q</i>-Sine Euler Polynomials Containing Symmetric Structures |
| title_short | Explicit Properties of <i>q</i>-Cosine and <i>q</i>-Sine Euler Polynomials Containing Symmetric Structures |
| title_sort | explicit properties of i q i cosine and i q i sine euler polynomials containing symmetric structures |
| topic | q-cosine Euler polynomials q-sine Euler polynomials q-trigonometric function q-exponential function |
| url | https://www.mdpi.com/2073-8994/12/8/1247 |
| work_keys_str_mv | AT cheonseoungryoo explicitpropertiesofiqicosineandiqisineeulerpolynomialscontainingsymmetricstructures AT jungyoogkang explicitpropertiesofiqicosineandiqisineeulerpolynomialscontainingsymmetricstructures |