Open Access Research Article

Taylor Expansions of q−Pochhammer Symbols via the Leverrier–Takeno Method

Authors

Juan José Diaz Bulnes * Department of Exact and Technological Sciences, Federal University of Amapá, Rod. J. Kubitschek, 68903-419, Macapá, AP, Brazil.
Irfan Nurhidayat Department, International Engineering and Technology Institute, Hong Kong, China.
Mahmood Ahmad Pathan Department of Mathematics, Aligarh Muslim University, Aligarh, India-202 002.
Hemant Kumar Department of Mathematics, D. A-V. Postgraduate College, Kanpur, Uttar Pradesh, India-208 001.
José Luis López-Bonilla ESIME-Zacatenco, Instituto Politécnico Nacional, Edif. 4, 1er. Piso, Col. Lindavista CP 07738, CDMX, México.

https://doi.org/10.22105/kmisj.v2i1.88

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Abstract

This article presents a new approach to the study of generalized q−Pochhammer symbols through the Leverrier–Takeno method. By linking the derivatives of these products to the coefficients of characteristic polynomials, the work provides a systematic framework for constructing Taylor expansions in terms of combinatorial objects such as Bell polynomials and q−binomial coefficients. The method naturally leads to several well-known results in q−series, including the q−binomial theorem, Euler’s identity, and the Jacobi triple product identity. In addition, the approach reveals close connections with partition functions and recurrence relations, offering new perspectives on classical number theoretic results. The findings highlight the effectiveness of combining linear algebraic methods with combinatorial techniques to deepen the understanding of q−series and their applications in mathematics.

Keywords

Characteristic polynomial q−binomial theorem Leverrier-takeno’s procedure Taylor expansion Complete bell polynomials Partition function Newton’s recurrence expression q−series Jacobi triple product identity

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Published 2025-03-11
Issue Vol. 2 No. 1 (2025): kmisj
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Diaz Bulnes, J. J., Nurhidayat, I. ., Pathan, M. A., Kumar, H. ., & López-Bonilla, J. L. (2025). Taylor Expansions of q−Pochhammer Symbols via the Leverrier–Takeno Method. Karshi Multidisciplinary International Scientific Journal, 2(1), 51-59. https://doi.org/10.22105/kmisj.v2i1.88

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