Taylor Expansions of q−Pochhammer Symbols via the Leverrier–Takeno Method
DOI:
https://doi.org/10.22105/kmisj.v2i1.88Keywords:
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 identityAbstract
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.
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