On New Sequences of $p$-Binomial and Catalan Transforms of the $k$-Mersenne Numbers and Associated Generating Functions
In this work, we investigate the binomial transforms and Catalan transform of the $k$-Mersenne and $k$-Mersenne-Lucas numbers and examine the new integer sequences. We apply the $p$-binomial, rising $p$-binomial, and falling $p$-binomial transforms to the $k$-Mersenne sequences and present the assoc...
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| Format: | Article |
| Language: | English |
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Emrah Evren KARA
2025-06-01
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| Series: | Universal Journal of Mathematics and Applications |
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| Online Access: | https://dergipark.org.tr/en/download/article-file/4614774 |
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| author | Kalika Prasad Hrishikesh Mahato Ritanjali Mohanty Munesh Kumari |
| author_facet | Kalika Prasad Hrishikesh Mahato Ritanjali Mohanty Munesh Kumari |
| author_sort | Kalika Prasad |
| collection | DOAJ |
| description | In this work, we investigate the binomial transforms and Catalan transform of the $k$-Mersenne and $k$-Mersenne-Lucas numbers and examine the new integer sequences. We apply the $p$-binomial, rising $p$-binomial, and falling $p$-binomial transforms to the $k$-Mersenne sequences and present the associated generating and exponential generating functions for these transforms. Lastly, we provide the corresponding Binet-type formulas and recurrence relations for binomial transforms of the $k$-Mersenne ($k$-Mersenne-Lucas) numbers. These results are supported by numerical illustrations. |
| format | Article |
| id | doaj-art-1d28fc4e5b9449b0936313b9aefca83e |
| institution | Kabale University |
| issn | 2619-9653 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | Emrah Evren KARA |
| record_format | Article |
| series | Universal Journal of Mathematics and Applications |
| spelling | doaj-art-1d28fc4e5b9449b0936313b9aefca83e2025-08-20T03:28:41ZengEmrah Evren KARAUniversal Journal of Mathematics and Applications2619-96532025-06-0182718010.32323/ujma.16410011225On New Sequences of $p$-Binomial and Catalan Transforms of the $k$-Mersenne Numbers and Associated Generating FunctionsKalika Prasad0https://orcid.org/0000-0002-3653-5854Hrishikesh Mahato1https://orcid.org/0000-0002-3769-0653Ritanjali Mohanty2https://orcid.org/0000-0002-5885-5613Munesh Kumari3https://orcid.org/0000-0002-6541-0284Government Engineering College BhojpurCentral University of JharkhandCentral University of JharkhandGovernment Engineering College BhojpurIn this work, we investigate the binomial transforms and Catalan transform of the $k$-Mersenne and $k$-Mersenne-Lucas numbers and examine the new integer sequences. We apply the $p$-binomial, rising $p$-binomial, and falling $p$-binomial transforms to the $k$-Mersenne sequences and present the associated generating and exponential generating functions for these transforms. Lastly, we provide the corresponding Binet-type formulas and recurrence relations for binomial transforms of the $k$-Mersenne ($k$-Mersenne-Lucas) numbers. These results are supported by numerical illustrations.https://dergipark.org.tr/en/download/article-file/4614774catalan transformgenerating functions$k$-mersenne and $k$-mersenne-lucas numbers$p$-binomial transforms |
| spellingShingle | Kalika Prasad Hrishikesh Mahato Ritanjali Mohanty Munesh Kumari On New Sequences of $p$-Binomial and Catalan Transforms of the $k$-Mersenne Numbers and Associated Generating Functions Universal Journal of Mathematics and Applications catalan transform generating functions $k$-mersenne and $k$-mersenne-lucas numbers $p$-binomial transforms |
| title | On New Sequences of $p$-Binomial and Catalan Transforms of the $k$-Mersenne Numbers and Associated Generating Functions |
| title_full | On New Sequences of $p$-Binomial and Catalan Transforms of the $k$-Mersenne Numbers and Associated Generating Functions |
| title_fullStr | On New Sequences of $p$-Binomial and Catalan Transforms of the $k$-Mersenne Numbers and Associated Generating Functions |
| title_full_unstemmed | On New Sequences of $p$-Binomial and Catalan Transforms of the $k$-Mersenne Numbers and Associated Generating Functions |
| title_short | On New Sequences of $p$-Binomial and Catalan Transforms of the $k$-Mersenne Numbers and Associated Generating Functions |
| title_sort | on new sequences of p binomial and catalan transforms of the k mersenne numbers and associated generating functions |
| topic | catalan transform generating functions $k$-mersenne and $k$-mersenne-lucas numbers $p$-binomial transforms |
| url | https://dergipark.org.tr/en/download/article-file/4614774 |
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