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

Munesh Kumari; Kalika Prasad; Ritanjali Mohanty; Hrishikesh Mahato

doi:10.32323/ujma.1641001

EN

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

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

References

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Details

Primary Language

English

Subjects

Numerical and Computational Mathematics (Other), Algebra and Number Theory

Journal Section

Research Article

Authors

Munesh Kumari
0000-0002-6541-0284
India

Kalika Prasad ^*
0000-0002-3653-5854
India

Ritanjali Mohanty
0000-0002-5885-5613
India

Hrishikesh Mahato
0000-0002-3769-0653
India

Early Pub Date

May 29, 2025

Publication Date

June 27, 2025

Submission Date

February 16, 2025

Acceptance Date

May 6, 2025

Published in Issue

Year 2025 Volume: 8 Number: 2

DOI

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

IZ

https://izlik.org/JA59TJ88PU

APA

Kumari, M., Prasad, K., Mohanty, R., & Mahato, H. (2025). On New Sequences of $p$-Binomial and Catalan Transforms of the $k$-Mersenne Numbers and Associated Generating Functions. Universal Journal of Mathematics and Applications, 8(2), 71-80. https://doi.org/10.32323/ujma.1641001

AMA

1.Kumari M, Prasad K, Mohanty R, Mahato H. On New Sequences of $p$-Binomial and Catalan Transforms of the $k$-Mersenne Numbers and Associated Generating Functions. Univ. J. Math. Appl. 2025;8(2):71-80. doi:10.32323/ujma.1641001

Chicago

Kumari, Munesh, Kalika Prasad, Ritanjali Mohanty, and Hrishikesh Mahato. 2025. “On New Sequences of $p$-Binomial and Catalan Transforms of the $k$-Mersenne Numbers and Associated Generating Functions”. Universal Journal of Mathematics and Applications 8 (2): 71-80. https://doi.org/10.32323/ujma.1641001.

EndNote

Kumari M, Prasad K, Mohanty R, Mahato H (June 1, 2025) On New Sequences of $p$-Binomial and Catalan Transforms of the $k$-Mersenne Numbers and Associated Generating Functions. Universal Journal of Mathematics and Applications 8 2 71–80.

IEEE

[1]M. Kumari, K. Prasad, R. Mohanty, and H. Mahato, “On New Sequences of $p$-Binomial and Catalan Transforms of the $k$-Mersenne Numbers and Associated Generating Functions”, Univ. J. Math. Appl., vol. 8, no. 2, pp. 71–80, June 2025, doi: 10.32323/ujma.1641001.

ISNAD

Kumari, Munesh - Prasad, Kalika - Mohanty, Ritanjali - Mahato, Hrishikesh. “On New Sequences of $p$-Binomial and Catalan Transforms of the $k$-Mersenne Numbers and Associated Generating Functions”. Universal Journal of Mathematics and Applications 8/2 (June 1, 2025): 71-80. https://doi.org/10.32323/ujma.1641001.

JAMA

1.Kumari M, Prasad K, Mohanty R, Mahato H. On New Sequences of $p$-Binomial and Catalan Transforms of the $k$-Mersenne Numbers and Associated Generating Functions. Univ. J. Math. Appl. 2025;8:71–80.

MLA

Kumari, Munesh, et al. “On New Sequences of $p$-Binomial and Catalan Transforms of the $k$-Mersenne Numbers and Associated Generating Functions”. Universal Journal of Mathematics and Applications, vol. 8, no. 2, June 2025, pp. 71-80, doi:10.32323/ujma.1641001.

Vancouver

1.Munesh Kumari, Kalika Prasad, Ritanjali Mohanty, Hrishikesh Mahato. On New Sequences of $p$-Binomial and Catalan Transforms of the $k$-Mersenne Numbers and Associated Generating Functions. Univ. J. Math. Appl. 2025 Jun. 1;8(2):71-80. doi:10.32323/ujma.1641001