Bibliographic Information: Petkova, I. (2024). Powers of the operator $u(z)\frac{d}{dz}$ and their connection with some combinatorial numbers. arXiv:2311.18776v3 [math.CO]
Research Objective: This paper aims to find an explicit representation for the powers of the generalized Euler-Cauchy operator, denoted as A = u(z) d/dz, where u(z) is an entire or meromorphic function. The paper investigates the connection between these powers and various combinatorial numbers, including Stirling numbers.
Methodology: The paper employs operator theory and combinatorial analysis. It derives recurrence relations for coefficients in the expansion of A^k in terms of the powers of the differential operator D = d/dz. These coefficients are then linked to known combinatorial numbers like Stirling numbers. Specific cases of the operator A are studied, including u(z) = z, u(z) = e^z, and u(z) = 1/z, highlighting their connections to known mathematical concepts like Bessel functions.
Key Findings: The paper successfully derives an explicit representation for the powers of the generalized Euler-Cauchy operator. It proves that these powers act as generating operators for different combinatorial numbers. Furthermore, the paper establishes new identities and orthogonality relations involving Stirling numbers and other combinatorial sequences.
Main Conclusions: The study reveals a deep connection between the generalized Euler-Cauchy operator and combinatorial numbers. The derived explicit representations and identities provide new insights into the properties of this operator and its applications in various mathematical fields.
Significance: This research significantly contributes to the field of operator theory by providing a deeper understanding of the generalized Euler-Cauchy operator and its connection to combinatorics. The established identities and relations involving combinatorial numbers are valuable findings with potential applications in other areas of mathematics and related fields.
Limitations and Future Research: The paper primarily focuses on specific forms of the function u(z). Future research could explore the properties of the operator with more general functions u(z) and investigate potential applications of the derived results in fields like differential equations and special function theory.
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by Ioana Petkov... at arxiv.org 10-15-2024
https://arxiv.org/pdf/2311.18776.pdfDeeper Inquiries