When writing down this formula, its proof is not difficult, because it can be straightforwardly verified by induction on $k$. My question is where one can find it.

Dear Feng Qi,

although I cannot provide a reference I can give you a hint how to obtain the formula. Let D_k be the determinant under consideration, D_1=c, D_2=c^2-1 (and D_0:=1 by def.)

1. One  proves that D_k=cD_{k-1}-D_{k-2} expanding D_k .

2. Hence the generating function F(t)=1+D_1*t+D_2*t^2+...=1/(1-ct+t^2).

3. Consider the expansion 1/(1-ct+t^2)=1+t(c-t)+t^2(c-t)^2+...  (*)

Calculating the coefficient by t^k in the right-hand side of (*), I think, you get the desired formulas.

Best regards, Yuri

Dear Yuri,

I have no much knowledge about the theory of generating functions, so I do not understand Step 2: how can one derive the generating function $F(t)=1/(1-ct+t^2)$?

Best regards

Feng

Dear Feng,

Due to Alexei Uteshev, I suggest you to take a look at the following book:

Higher Algebra: Linear Algebra, Polynomials, General Algebra (International Series of Monographs in Pure and Applied Mathematics Mishina, A.P., and I.V. Proskuryakov

Pergamon Press. 1965. 

(section "Jacobi's determinants. Continuant and its relationship to Fibonacci's numbers")

http://www.biblio.com/book/higher-algebra-linear-algebra-polynomials-general/d/707682709

Best regards,

Wiwat Wanicharpichat

Dear Feng,

just note that  from the recurrent relation you have (t is an auxiliary variable)

D_k.t^k=(cD_{k-1}-D_{k-2})t^k=ct.D_{k-1}t^{k-1}-t^2.D_{k-2}t^{k-2}   (k>=2).

Formal summation of both sides from k=2 to \infty yields

F(t)-D_1t-D_0=ct.(F(t)-D_0)-t^2F(t), or

(1-ct+t^2)F(t)=1 in view of D_0=1, D_1=c..

Note also that F(t)=D_0+D_1t+D_2t^2+ ... is the Taylor expansion of 1/(1-ct+t^2).

More information: https://en.wikipedia.org/wiki/Generating_function

Yuri

I cann't get the book "Higher Algebra: Linear Algebra, Polynomials, General Algebra" by A.P. Mishina and I.V. Proskuryakov and the section "Jacobi's determinants. Continuant and its relationship to Fibonacci's numbers".

Now I see how the generating function $F(t)$ is derived.

I am grateful to you all.

My question in this RG: "Does anyone know about the explicit formula of determinant of tridiagonal matrix?"

You can see some useful information about your problem, which suggested by Prof. Alexei Uteshev.

Oh, you mean the website

https://www.researchgate.net/post/Does_anyone_know_about_the_explicit_formula_of_determinant_of_tridiagonal_matrix,

but, in which and related references therein I do not find an explicit formula (a closed form) like mine for tridiagonal determinants.

He said that

" In one old Russian textbook, I've found the following result: the determinant of the tridiagonal matrix of the form

  a_1 b_1 0 0 0 ...

-c_2 a_2 b_2 0 0 ...

0 - c_3 a_3 b_3 0 ...

...

of the order n equals the sum of its leading term a_1a_2...a_n and all the terms obtained from this product by replacement  one or several pairs of neighbour product a_ja_{j+1} by b_j c_{j+1}. So, for the order 5 it has the form:

a_1a_2a_3a_4a_5+b_1c_2a_3a_4a_5+a_1b_2c_3a_4a_5+a_1a_2b_3c_4a_5

+a_1a_2a_3b_4c_5+b_1c_2b_3c_4a_5+b_1c_2a_3b_4c_5+a_1b_2c_3b_4c_5"

By Alexei Uteshev's saying or describing, can one derive the closed form of mine? I can not immediately.

For the meanings of the so-called closed form, please click at https://en.wikipedia.org/wiki/Closed-form_expression.

This is an old result. Probably you will find it in books about Chebyshev polynomials. See also http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html.

Thank Professor Johann Cigler! I found at

http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html

the result showed by the picture below. This is important for me! However, can one say that the hypergeometric function 2F1 is a closed-form expression?

Please see the attachment in the following useful link:

http://www.phys.lsu.edu/~amarti9/adfaerf/241.%20Analytical%20Inversion%20of%20Symmetric%20Tridiagonal%20Matrices.pdf

Great! This is the accurate answer! However, just a sentence "After some direct algebra, we obtain" derived the explicit formula. Enough for me. Thank all.

Dear All

This afternoon I got a simpler expression for the tridiagonal determinant we discussed above. Is it simpler than others?

Best regards

Feng

Dear Feng,

the relation

D_k=cD_{k-1} - D_{k-2}  (*)

implies that

D_k=A\alpha^k+B\beta^k

where \alpha, \beta are roots of the associated with (*) characteristic equation

q^2=cq-1, i.e. your expressions for \alpha\beta (provided c\ne 2 or -2).

A, B are constants which can be determined from D_1=c, D_2=c^2-1.

Hence the result.

Yuri

Good idea! But I can not understand something. Why does the recurrence relation imply the formula "D_k=A\alpha^k+B\beta^k"? Why is the characteristic equation $q^2=cq-1$? Help me please.

Usually, one looks for the solution of the recurrence relation

a_k=C_1.a_{k-1}+...+C_m.a_{k-m} (*)

as a sum

a_k=b_1.r_1^k+...+b_m.r_m^k

of geometric progressions. A necessary condition on r_i is that r_i are roots of characteristic equation

q^m=C_1.q^{m-1}+...+C_m.(**)

The constants C_i can be determined from the initial conditions.

See also  https://en.wikipedia.org/wiki/Recurrence_relation

Yuri

Dear Yuri

I learned much from you, thank you a lot.

Best regards

Feng

Dear Professor Feng Qi,

The different formulae are  different expressions of Fibonacci or Chebyshev polynomials. As you probably know there is a journal “The Fibonacci Quarterly” which is exclusively devoted to such questions. The older articles can be freely downloaded. You can also find some of these elementary results in my paper “Some beautiful q-analogues of Fibonacci and Lucas polynomials” in arXiv:1104.2699.

Just a remark: Matrices of the given type occur in the so-called Hückel model for a linear chain. It is long known that the eigenvalue problem for such matrices is explicitly solvable (as is the  related one for rings). Essentially, this is nothing  but solving a three-term recurrence relation with certain boundary conditions to obtain the eigenvectors. Thus, one can also obtain the determinant explicitly in terms of the eigenvalues.

Dear Professor Johann Cigler

I am reading your paper at https://arxiv.org/abs/1104.2699 and I saw just now the so-called Binet's formula. Because their generating functions are very close, the Fibonacci, Chebysev, and other numbers and polynomials have similar properties. I believe this. I will cite your paper mentioned above. Has the arXiv preprint been formally published in some journal? Thank you very much for your valuable information.

Best regards

Feng Qi (F. Qi) 

Dear Herbert H H Homeier

Could you please list several references for showing me what you said above?

Best regards

Feng Qi (F. Qi)

Dear Professor Feng Qi

Due to Dr. Herbert H H Homeier. Take a look to the following useful link: (page 6.3)

http://www.chem.qmul.ac.uk/software/download/mo/6.pdf

Best regards

Wiwat Wanicharpichat

Thanks. However, how can one cite this reference? I do not find its title, author(s), publishing year, publishing city, press name, and the like. Thanks.

M.Bosson, C. Richard, A. Plet, S. Grudinin, and S. Redon, Interactive Quantum Chemistry: A Divide-and-Conquer

ASED-MO Method, Journal of Computational Chemistry, 2012. (p.7).

http://nano-d.inrialpes.fr/~redon/papers/JCC2012.pdf

Dear Professor Wiwat Wanicharpichat

Thank you for your information. I read page 7 in the paper. The origin of the results is the reference [56].

[56] C. D. Meyer, Soc. Ind.Appl. Math., Philadelphia, PA, USA. 2000, 1.

I am looking up this reference (a paper? a monograph? a conference proceedings?) by Meyer.

Thanks again.

Best regards

Feng

Just now I found the formula we are interesting appeared in the book: 

C. D. Meyer, Matrix Analysis and Applied Linear Algebra, With 1 CD-ROM and a solutions manual. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000; Available online at http://dx.doi.org/10.1137/1.9780898719512.

This monograph is just the reference [56] cited  in

M.Bosson, C. Richard, A. Plet, S. Grudinin, and S. Redon, Interactive Quantum Chemistry: A Divide-and-Conquer ASED-MO Method, Journal of Computational Chemistry, 2012.

Dear Feng Qi,

for the Hückel method, see https://en.wikipedia.org/wiki/H%C3%BCckel_method and references therein.

Best regards

Herbert

Okay, I noticed the website. Thanks.

I have just found a very simple formula, it is under unofficial reviewing. I´ll bring it as soon as possible.

Dear Viera

Just now I received your e-mail and will read carefully your attachment.

Feng

Where can I find the e-mail address for Yuri S. Semenov at Moscow State University of Railway Engineering? Who can provide me the e-mail address for Yuri S. Semenov at Moscow State University of Railway Engineering? Thanks.

I tried to find his e-mail on internet, including RG, but failed.

Dear Feng,

my e-mail:   [email protected]    

or,

[email protected]

Yuri

Dear Yuri

I recorded your e-mails. Soon I will contact you by e-mail.

Feng

Dear contributors, the question was Where can I find ...

My comment wanted be a response to another question of Feng Qi.

As I promised to publish the formula in this group, I am putting it here. 

Please check it.

Dear Feng, thank you  for your interesting question and provoking formulas in https://www.researchgate.net/post/Can_one_verify_a_guess_about_computation_of_the_general_tridiagonal_determinant.

Thanks also for the useful advice that you wrote me.

Hi Feng,

According to Wikipedia (https://en.wikipedia.org/wiki/Tridiagonal_matrix#cite_note-1), The determinant of a tridiagonal matrix is given by the continuant of its elements. 

The continuant is given in the following link: https://en.wikipedia.org/wiki/Continuant_(mathematics)

P.S: This is what @Yuri is saying btw.

Regards

Dear All

Several answers to this question 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

Dear Dr. Feng Qi, Dr. Viera Čerňanová and Dr. Yuri S Semenov,

The proposed problem is Problem 369 on page 44 in the book: I.V. Proskuryakov, Sbornik zadach po linejnoj algebre, 9th edition, BINOM, Laboratorij znaniji, Moscow, 2005 (Russian). Related formula (the expression from the Qi's question) is given in a hint of solution at the beginning of the page 369 of the mentioned book.

Best regards

Feng Qi, Viera Čerňanová, 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.

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.

someone gives me a code C++ to calculate determinant of a matrix?

Romeo Meštrović Viera Čerňanová Yuri S Semenov

Dear Romeo, please discuss my "kindly" respondence here. Thank you very much.

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.