Ansatz in a Nutshell: A Comprehensive Step-by-Step Guide to Polynomial, C-finite, Holonomic, and C<sup>2</sup>-finite Sequences
Issued Date
2024-01-01
Resource Type
ISSN
21941009
eISSN
21941017
Scopus ID
2-s2.0-85210320227
Journal Title
Springer Proceedings in Mathematics and Statistics
Volume
471
Start Page
255
End Page
297
Rights Holder(s)
SCOPUS
Bibliographic Citation
Springer Proceedings in Mathematics and Statistics Vol.471 (2024) , 255-297
Suggested Citation
Krityakierne T., Thanatipanonda T.A. Ansatz in a Nutshell: A Comprehensive Step-by-Step Guide to Polynomial, C-finite, Holonomic, and C<sup>2</sup>-finite Sequences. Springer Proceedings in Mathematics and Statistics Vol.471 (2024) , 255-297. 297. doi:10.1007/978-3-031-69706-7_11 Retrieved from: https://repository.li.mahidol.ac.th/handle/20.500.14594/102256
Title
Ansatz in a Nutshell: A Comprehensive Step-by-Step Guide to Polynomial, C-finite, Holonomic, and C<sup>2</sup>-finite Sequences
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Abstract
Given a sequence 1, 1, 5, 23, 135, 925, 7285, 64755, 641075, 6993545, 83339745,…, how can we guess a formula for it? This article will quickly walk you through the concept of ansatz for classes of polynomial, C-finite, holonomic, and the most recent addition C2-finite sequences. For each of these classes, we discuss in detail various aspects of the guess and check, generating functions, closure properties, and closed-form solutions. Every theorem is presented with an accessible proof, followed by several examples intended to motivate the development of the theories. Each example is accompanied by a Maple program with the purpose of demonstrating use of the program in solving problems in this area. While this work aims to give a comprehensive review of existing ansatzes, we also systematically fill a research gap in the literature by providing theoretical and numerical results for the C2-finite sequences.