On (<i>p</i>,<i>q</i>)–Fibonacci and (<i>p</i>,<i>q</i>)–Lucas Polynomials Associated with Changhee Numbers and Their Properties
Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties have been studied in the literature with the help of generating functions and their functional equations. In this paper, using the <inline-formula><math xmlns="http://www.w3.org...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-04-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/4/851 |
| _version_ | 1797603427696508928 |
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| author | Chuanjun Zhang Waseem Ahmad Khan Can Kızılateş |
| author_facet | Chuanjun Zhang Waseem Ahmad Khan Can Kızılateş |
| author_sort | Chuanjun Zhang |
| collection | DOAJ |
| description | Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties have been studied in the literature with the help of generating functions and their functional equations. In this paper, using the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>–Fibonacci polynomials, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>–Lucas polynomials, and Changhee numbers, we define the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>–Fibonacci–Changhee polynomials and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>–Lucas–Changhee polynomials, respectively. We obtain some important identities and relations of these newly established polynomials by using their generating functions and functional equations. Then, we generalize the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>–Fibonacci–Changhee polynomials and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>–Lucas–Changhee polynomials called generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>–Fibonacci–Lucas–Changhee polynomials. We derive a determinantal representation for the generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>–Fibonacci–Lucas–Changhee polynomials in terms of the special Hessenberg determinant. Finally, we give a new recurrent relation of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>–Fibonacci–Lucas–Changhee polynomials. |
| first_indexed | 2024-03-11T04:29:04Z |
| format | Article |
| id | doaj.art-65ad592f6664418aa31a7352b3d4bb2a |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-11T04:29:04Z |
| publishDate | 2023-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-65ad592f6664418aa31a7352b3d4bb2a2023-11-17T21:33:38ZengMDPI AGSymmetry2073-89942023-04-0115485110.3390/sym15040851On (<i>p</i>,<i>q</i>)–Fibonacci and (<i>p</i>,<i>q</i>)–Lucas Polynomials Associated with Changhee Numbers and Their PropertiesChuanjun Zhang0Waseem Ahmad Khan1Can Kızılateş2School of Mathematics and Big Data, Guizhou Normal College, Guiyang 550018, ChinaDepartment of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi ArabiaDepartment of Mathematics, Zonguldak Bülent Ecevit University, Zonguldak 67100, TurkeyMany properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties have been studied in the literature with the help of generating functions and their functional equations. In this paper, using the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>–Fibonacci polynomials, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>–Lucas polynomials, and Changhee numbers, we define the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>–Fibonacci–Changhee polynomials and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>–Lucas–Changhee polynomials, respectively. We obtain some important identities and relations of these newly established polynomials by using their generating functions and functional equations. Then, we generalize the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>–Fibonacci–Changhee polynomials and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>–Lucas–Changhee polynomials called generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>–Fibonacci–Lucas–Changhee polynomials. We derive a determinantal representation for the generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>–Fibonacci–Lucas–Changhee polynomials in terms of the special Hessenberg determinant. Finally, we give a new recurrent relation of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>–Fibonacci–Lucas–Changhee polynomials.https://www.mdpi.com/2073-8994/15/4/851(<i>p</i>,<i>q</i>)–Fibonacci polynomials(<i>p</i>,<i>q</i>)–Lucas polynomialsChanghee numbersgenerating functionHessenberg determinant |
| spellingShingle | Chuanjun Zhang Waseem Ahmad Khan Can Kızılateş On (<i>p</i>,<i>q</i>)–Fibonacci and (<i>p</i>,<i>q</i>)–Lucas Polynomials Associated with Changhee Numbers and Their Properties Symmetry (<i>p</i>,<i>q</i>)–Fibonacci polynomials (<i>p</i>,<i>q</i>)–Lucas polynomials Changhee numbers generating function Hessenberg determinant |
| title | On (<i>p</i>,<i>q</i>)–Fibonacci and (<i>p</i>,<i>q</i>)–Lucas Polynomials Associated with Changhee Numbers and Their Properties |
| title_full | On (<i>p</i>,<i>q</i>)–Fibonacci and (<i>p</i>,<i>q</i>)–Lucas Polynomials Associated with Changhee Numbers and Their Properties |
| title_fullStr | On (<i>p</i>,<i>q</i>)–Fibonacci and (<i>p</i>,<i>q</i>)–Lucas Polynomials Associated with Changhee Numbers and Their Properties |
| title_full_unstemmed | On (<i>p</i>,<i>q</i>)–Fibonacci and (<i>p</i>,<i>q</i>)–Lucas Polynomials Associated with Changhee Numbers and Their Properties |
| title_short | On (<i>p</i>,<i>q</i>)–Fibonacci and (<i>p</i>,<i>q</i>)–Lucas Polynomials Associated with Changhee Numbers and Their Properties |
| title_sort | on i p i i q i fibonacci and i p i i q i lucas polynomials associated with changhee numbers and their properties |
| topic | (<i>p</i>,<i>q</i>)–Fibonacci polynomials (<i>p</i>,<i>q</i>)–Lucas polynomials Changhee numbers generating function Hessenberg determinant |
| url | https://www.mdpi.com/2073-8994/15/4/851 |
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