On New Sequences of $p$-Binomial and Catalan Transforms of the $k$-Mersenne Numbers and Associated Generating Functions

Year 2025, Volume: 8 Issue: 2, 71 - 80, 27.06.2025

Munesh Kumari , Kalika Prasad , Ritanjali Mohanty , Hrishikesh Mahato

https://doi.org/10.32323/ujma.1641001

Abstract

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.

Keywords

Catalan transform , Generating functions , $k$-Mersenne and $k$-Mersenne-Lucas numbers , $p$-Binomial transforms

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