Continued fractions, determinant expressions, and identities

Takao Komatsu; Nikita Kalinin; Takao Komatsu; Nikita Kalinin

doi:10.3934/math.2026402

AIMS Mathematics

2026, Volume 11, Issue 4: 9712-9735. doi: 10.3934/math.2026402

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Research article Special Issues

Continued fractions, determinant expressions, and identities

Takao Komatsu ^1,2,
Nikita Kalinin ^{3
,
,}

1.
Institute of Mathematics, Henan Academy of Sciences, Zhengzhou 450046, China
2.
Department of Mathematics, Institute of Science Tokyo, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan
3.
Mathematics and Computer Science Department, Guangdong Technion-Israel Institute of Technology, Shantou 515603, China
In memory of Narakorn Rompurk Kanasri^*
^*Ms Narakorn Rompurk Kanasri in ^[1] passed away in a traffic accident on January 3, 2026. Rest in peace.

Received: 09 December 2025 Revised: 02 April 2026 Accepted: 07 April 2026 Published: 13 April 2026
MSC : 05A15, 05A19, 11A55, 11B39, 11B75, 15A15

In this paper, we clarified the relationship between continued fractions, determinants, and identities, making it easier to apply these methods systematically in other settings. In particular, we studied finite continued fractions from the perspective of incomplete numbers (restricted or associated numbers) and also explored their relationships with determinant representations and identities. Most of the new results in this paper concern $ q $-analogues of special numbers, whereas the classical cases mainly serve to illustrate and unify the general framework. The framework developed here is flexible and allows one to derive continued fractions, determinant formulas, and coefficient identities in a uniform way for several new $ q $-families, and it is expected to be applicable to other families of special numbers, as well.
- determinants,
- continued fractions,
- identities,
- $ q $-analogues
Citation: Takao Komatsu, Nikita Kalinin. Continued fractions, determinant expressions, and identities[J]. AIMS Mathematics, 2026, 11(4): 9712-9735. doi: 10.3934/math.2026402

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Abstract

In this paper, we clarified the relationship between continued fractions, determinants, and identities, making it easier to apply these methods systematically in other settings. In particular, we studied finite continued fractions from the perspective of incomplete numbers (restricted or associated numbers) and also explored their relationships with determinant representations and identities. Most of the new results in this paper concern $ q $-analogues of special numbers, whereas the classical cases mainly serve to illustrate and unify the general framework. The framework developed here is flexible and allows one to derive continued fractions, determinant formulas, and coefficient identities in a uniform way for several new $ q $-families, and it is expected to be applicable to other families of special numbers, as well.

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