Can somebody verify or deny the guess stated above? Thanks.

In [15] the authors refer to the reccurent formula:

J_n = a_n*J_{n_1} + b_{n-1}*c_n*J_{n-2},

which is obtained by the last row expansion.

To the best of my knowledge, it is not easy to derive an explicit or closed formula from a recursive relation.

A related result is showed by the PDF files uploaded with this message.

Dear All

Several answers to this problem are contained in the following preprint:

Feng Qi, Viera Cernanova, and Yuri S. Semenov, On tridiagonal determinants and the Cauchy product of central Delannoy numbers, ResearchGate Working Paper (2016), available online at http://dx.doi.org/10.13140/RG.2.1.3772.6967.

Feng

Due to L.G. Molinari, I just have a determinant formula of tridiagonal matrix, please see the attachment page 2.

L.G. Molinari, Determinants of block tridiagonal matrices, Linear Algebra and its Applications 429 (2008) 2221-2226, (p.2222).

Please see:

M.E.A. El-Mikkawy, F. Atlan, A novel algorithm for inverting a general k-tridiagonal matrix, Appl. Math. Lett. 32 (2014) 41–47.

The following formally papers are related to this question:

[1] Feng Qi, Viera Cernanova, and Yuri S. Semenov, Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials, University Politehnica of Bucharest Scientific Bulletin Series A---Applied Mathematics and Physics 81 (2019), no. 1, 123--136.

[2] Feng Qi and Ai-Qi Liu, Alternative proofs of some formulas for two tridiagonal determinants, Acta Universitatis Sapientiae Mathematica 10 (2018), no. 2, 287--297; available online at https://doi.org/10.2478/ausm-2018-0022

[3] Feng Qi, Wen Wang, Dongkyu Lim, and Bai-Ni Guo, Several explicit and recurrent formulas for determinants of tridiagonal matrices via generalized continued fractions, Nonlinear Analysis: Problems, Applications and Computational Methods, Editors: Zakia Hammouch, Hemen Dutta, Said Melliani, Michael Ruzhansky; Springer Book Series Lecture Notes in Networks and Systems. The 6th International Congress of the Moroccan Society of Applied Mathematics (SM2A 2019) organized by Sultan Moulay Slimane University, Faculte des sciences et techniques, BP 523, Beni-Mellal, Morocco, during 7-9 November, 2019.