New determinant expressions of Bernoulli, Euler &amp; Tribonacci polynomials

Takao Komatsu; Fatih Yilmaz; Takao Komatsu; Fatih Yilmaz

doi:10.3934/math.2026044

AIMS Mathematics

2026, Volume 11, Issue 1: 1021-1035. doi: 10.3934/math.2026044

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New determinant expressions of Bernoulli, Euler & Tribonacci polynomials

Takao Komatsu ^1,2,
Fatih Yilmaz ^{3
,
,}

1.
Institute of Mathematics, Henan Academy of Sciences, Zhengzhou, 450046, China
2.
Department of Mathematics, Institute of Science, Tokyo, 152-8551, Japan
3.
Department of Mathematics, Ankara Hacı Bayram Veli University, 06900, Ankara, Turkey

Received: 11 August 2025 Revised: 18 December 2025 Accepted: 24 December 2025 Published: 13 January 2026

In 1875, Glaisher systematically found several interesting determinant expressions of numbers, including Bernoulli, Cauchy, and Euler numbers. In this paper, we identify several determinants that express Euler polynomials. Goy and Shattuck presented several determinantal expressions of some families of Toeplitz–Hessenberg matrices with Tribonacci number entries. However, a determinant expression of Tribonacci numbers has not been studied much. By using a similar form of determinants to Euler's, we also give some determinant representations of generalized Tribonacci numbers.
- Euler polynomials,
- Tribonacci numbers,
- determinants
Citation: Takao Komatsu, Fatih Yilmaz. New determinant expressions of Bernoulli, Euler & Tribonacci polynomials[J]. AIMS Mathematics, 2026, 11(1): 1021-1035. doi: 10.3934/math.2026044

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Abstract

In 1875, Glaisher systematically found several interesting determinant expressions of numbers, including Bernoulli, Cauchy, and Euler numbers. In this paper, we identify several determinants that express Euler polynomials. Goy and Shattuck presented several determinantal expressions of some families of Toeplitz–Hessenberg matrices with Tribonacci number entries. However, a determinant expression of Tribonacci numbers has not been studied much. By using a similar form of determinants to Euler's, we also give some determinant representations of generalized Tribonacci numbers.

References

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