Pell and Pell–Lucas Sequences of Fractional Order
The purpose of this paper is to introduce the fractional Pell numbers, together with several properties, via a Grünwald–Letnikov fractional operator of orders <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow&g...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-06-01
|
| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/9/7/416 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849303552447479808 |
|---|---|
| author | Jagan Mohan Jonnalagadda Marius-F. Danca |
| author_facet | Jagan Mohan Jonnalagadda Marius-F. Danca |
| author_sort | Jagan Mohan Jonnalagadda |
| collection | DOAJ |
| description | The purpose of this paper is to introduce the fractional Pell numbers, together with several properties, via a Grünwald–Letnikov fractional operator of orders <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>. This paper also explores the fractional Pell–Lucas numbers and their properties.Due to the long-term memory property, fractional Pell sequences and fractional Pell–Lucas sequences present potential applications in modeling and computation. The closed form is deduced, and the numerical schemes are determined. The fractional characteristic equation is introduced, and it is shown that its solutions include a fractional silver ratio depending on the fractional order. In addition, the tiling problem and the concept of the fractional silver spiral are considered.A MATLAB program for applying the use of the fractional silver ratio is presented. |
| format | Article |
| id | doaj-art-1bf7d2a977fe404084e8a1ba37acaae1 |
| institution | Kabale University |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-1bf7d2a977fe404084e8a1ba37acaae12025-08-20T03:58:26ZengMDPI AGFractal and Fractional2504-31102025-06-019741610.3390/fractalfract9070416Pell and Pell–Lucas Sequences of Fractional OrderJagan Mohan Jonnalagadda0Marius-F. Danca1Department of Mathematics, Birla Institute of Technology & Science Pilani, Hyderabad 500078, Telangana, IndiaSTAR-UBB Institute, Babes-Bolyai University, 400084 Cluj-Napoca, RomaniaThe purpose of this paper is to introduce the fractional Pell numbers, together with several properties, via a Grünwald–Letnikov fractional operator of orders <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>. This paper also explores the fractional Pell–Lucas numbers and their properties.Due to the long-term memory property, fractional Pell sequences and fractional Pell–Lucas sequences present potential applications in modeling and computation. The closed form is deduced, and the numerical schemes are determined. The fractional characteristic equation is introduced, and it is shown that its solutions include a fractional silver ratio depending on the fractional order. In addition, the tiling problem and the concept of the fractional silver spiral are considered.A MATLAB program for applying the use of the fractional silver ratio is presented.https://www.mdpi.com/2504-3110/9/7/416Grünwald–Letnikov fractional operatorfractional Pell numbersfractional Pell–Lucas numbersfractional characteristic equationfractional silver ratio |
| spellingShingle | Jagan Mohan Jonnalagadda Marius-F. Danca Pell and Pell–Lucas Sequences of Fractional Order Fractal and Fractional Grünwald–Letnikov fractional operator fractional Pell numbers fractional Pell–Lucas numbers fractional characteristic equation fractional silver ratio |
| title | Pell and Pell–Lucas Sequences of Fractional Order |
| title_full | Pell and Pell–Lucas Sequences of Fractional Order |
| title_fullStr | Pell and Pell–Lucas Sequences of Fractional Order |
| title_full_unstemmed | Pell and Pell–Lucas Sequences of Fractional Order |
| title_short | Pell and Pell–Lucas Sequences of Fractional Order |
| title_sort | pell and pell lucas sequences of fractional order |
| topic | Grünwald–Letnikov fractional operator fractional Pell numbers fractional Pell–Lucas numbers fractional characteristic equation fractional silver ratio |
| url | https://www.mdpi.com/2504-3110/9/7/416 |
| work_keys_str_mv | AT jaganmohanjonnalagadda pellandpelllucassequencesoffractionalorder AT mariusfdanca pellandpelllucassequencesoffractionalorder |