Tridiagonal Determinants Research Articles

Some determinantal representations of derangement polynomials of types B and D

Chow (2024) recently computed expressions of the types B and D derangement polynomials dnB(q)=∑σ∈DnBqfmaj(σ) and dnD(q)=∑σ∈DnDqmaj(σ) as tridiagonal and lower Hessenberg determinants of order n. Qi, Wang, and Guo (2016), based on a determinantal formula for the nth derivative of a quotient of two functions, derived an expression of the classical derangement number dn=n!∑k=0n(−1)k/k! as a tridiagonal determinant of order n+1. By q-extending the approach of Qi et al., we present in this work yet another determinantal expressions of dnB(q) and dnD(q) as determinants of order n+1.

Tricomi Continuants

In this paper, we introduce and study the Tricomi continuants, a family of tridiagonal determinants forming a Sheffer sequence closely related to the Tricomi polynomials and the Laguerre polynomials. Specifically, we obtain the main umbral operators associated with these continuants and establish some of their basic relations. Then, we obtain a Turan-like inequality, some congruences, some binomial identities (including a Carlitz-like identity), and some relations with the Cayley continuants. Furthermore, we show that the infinite Hankel matrix generated by the Tricomi continuants has an LDU-Sheffer factorization, while the infinite Hankel matrix generated by the shifted Tricomi continuants has an LTU-Sheffer factorization. Finally, by the first factorization, we obtain the linearization formula for the Tricomi continuants and its inverse.

New families of Horadam numbers associated with finite operators and their applications

In this paper, by applying finite operator to the Horadam sequence, we define the Horadam finite operator sequences. We not only give some special cases of this new sequences but also investigate some properties of our new sequences such as recurrent relation, Binet‐like formula, summation formula, and generating function, respectively. We also present closed formula for the Horadam finite operator numbers and their special cases in terms of tridiagonal determinants. Moreover, by using the tridiagonal determinant, we once again verify the recurrence relation of the Horadam finite operator numbers.

A block diagonalization based algorithm for the determinants of block k-tridiagonal matrices

In the current paper, we present a numerical algorithm for computing the determinants of block k-tridiagonal matrices. The algorithm is based on the use of a fast block diagonalization method and any algorithm for evaluating block tridiagonal determinants. Meanwhile, an explicit numerical formula for the block k-tridiagonal determinants is also derived, which is based on the combination of the proposed block diagonalization method and a two-term recurrence for block tridiagonal determinants. The experimental results of some representative numerical examples are provided to show the validity and effectiveness of the proposed algorithm and its competitiveness with MATLAB built-in function.

Computation of several Hessenberg determinants

Abstract In the paper, employing methods and techniques in analysis and linear algebra, the authors find a simple formula for computing an interesting Hessenberg determinant whose elements are products of binomial coefficients and falling factorials, derive explicit formulas for computing some special Hessenberg and tridiagonal determinants, and alternatively and simply recover some known results.

Horadam Sequences and Tridiagonal Determinants

We consider a family of particular tridiagonal matrix determinants which can represent the general second-order linear recurrence sequences. These determinants can be changed to symmetric or skew-symmetric tridiagonal determinants. To evaluate the complex factorizations of any Horadam sequence, we evaluate the eigenvalues of some special tridiagonal matrices and their corresponding eigenvectors. We also use these determinant representations to obtain some formulas in these sequences.

Retraction Note to: A Determinantal Expression for the Fibonacci Polynomials in Terms of a Tridiagonal Determinant

This article has been retracted. Please see the retraction notice for more detail: https://doi.org/10.1007/s41980-020-00472-9 .

RETRACTED ARTICLE: A Determinantal Expression for the Fibonacci Polynomials in Terms of a Tridiagonal Determinant

In the paper, after concisely reviewing and surveying some known results, the authors find a determinantal expression for the Fibonacci polynomials and, consequently, for the Fibonacci numbers, in terms of a tridiagonal determinant.

Lagrangian configurations and symplectic cross-ratios

We consider moduli spaces of cyclic configurations of $N$ lines in a $2n$-dimensional symplectic vector space, such that every set of $n$ consecutive lines generates a Lagrangian subspace. We study geometric and combinatorial problems related to these moduli spaces, and prove that they are isomorphic to quotients of spaces of symmetric linear difference operators with monodromy $-1$. The symplectic cross-ratio is an invariant of two pairs of $1$-dimensional subspaces of a symplectic vector space. For $N = 2n+2$, the moduli space of Lagrangian configurations is parametrized by $n+1$ symplectic cross-ratios. These cross-ratios satisfy a single remarkable relation, related to tridiagonal determinants and continuants, given by the Pfaffian of a Gram matrix.

A Closed Formula for the Horadam Polynomials in Terms of a Tridiagonal Determinant

In this paper, the authors present a closed formula for the Horadam polynomials in terms of a tridiagonal determinant and, as applications of the newly-established closed formula for the Horadam polynomials, derive closed formulas for the generalized Fibonacci polynomials, the Lucas polynomials, the Pell–Lucas polynomials, and the Chebyshev polynomials of the first kind in terms of tridiagonal determinants.

Alternative proofs of some formulas for two tridiagonal determinants

Abstract In the paper, the authors provide five alternative proofs of two formulas for a tridiagonal determinant, supply a detailed proof of the inverse of the corresponding tridiagonal matrix, and provide a proof for a formula of another tridiagonal determinant. This is a companion of the paper [F. Qi, V. Čerňanová,and Y. S. Semenov, Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 81 (2019), in press.

Rotundus: Triangulations, Chebyshev Polynomials, and Pfaffians

We introduce and study a cyclically invariant polynomial which is an analog of the classical tridiagonal determinant usually called the continuant. We prove that this polynomial can be calculated as the Pfaffian of a skew-symmetric matrix. We consider the corresponding Diophantine equation and prove an analog of a famous result due to Conway and Coxeter. We also observe that Chebyshev polynomials of the first kind arise as Pfaffians.

A representation for derangement numbers in terms of a tridiagonal determinant

In the paper, the authors discover a representation for the derangement numbers in terms of a tridiagonal determinant. From the determinantal representation, the authors recover several identities and recurrence relations of the derangement numbers.

Closed forms for derangement numbers in terms of the Hessenberg determinants

In the paper, the authors find closed forms for derangement numbers in terms of the Hessenberg determinants, discover a recurrence relation of derangement numbers, present a formula for any higher order derivative of the exponential generating function of derangement numbers, and compute some related Hessenberg and tridiagonal determinants.

On Period of Generalized Fibonacci Sequence Over Finite Ring and Tridiagonal Matrix

In this study, F n Fibonacci sequence was defined over an arbitrary ring and its some properties are investigated. The terms of this sequence are derivated by Tridiagonal determinant of the matrix. It was shown that this sequence is periodic and their period is obtained. It was shown that the sequence obtained by reducing modulo m coefficient and exponent of each Fibonacci sequence in arbitrary rings is periodic. It was seen that order of cyclic group generated with matrix R = a b 1 0 is equal to the period of this sequence where a , b are arbitrary elements of the ring. Also, the period of this sequence is compared with Wall number of Fibonacci sequence and it was shown that this period always was an even number.

A recovery of two determinantal representations for derangement numbers

In the paper, the authors recover, correct, and extend two representations for derangement numbers in terms of a tridiagonal determinant.

A determinantal representation for derangement numbers

In the note, the authors find a representation for derangement numbers in terms of a tridiagonal determinant whose elements are the first few natural numbers.

Janus configurations with SL(2, ℤ)-duality twists, strings on mapping tori and a tridiagonal determinant formula

We develop an equivalence between two Hilbert spaces: (i) the space of states of $U(1)^n$ Chern-Simons theory with a certain class of tridiagonal matrices of coupling constants (with corners) on $T^2$; and (ii) the space of ground states of strings on an associated mapping torus with $T^2$ fiber. The equivalence is deduced by studying the space of ground states of $SL(2,Z)$-twisted circle compactifications of $U(1)$ gauge theory, connected with a Janus configuration, and further compactified on $T^2$. The equality of dimensions of the two Hilbert spaces (i) and (ii) is equivalent to a known identity on determinants of tridiagonal matrices with corners. The equivalence of operator algebras acting on the two Hilbert spaces follows from a relation between the Smith normal form of the Chern-Simons coupling constant matrix and the isometry group of the mapping torus, as well as the torsion part of its first homology group.

Stationary and transient solution of Markovian queues - an alternate approach

This paper aims at presenting an alternate approach to derive the exact transient solution of Markovian queues. The continued fraction (Jacobi fraction) is expressed as power series and the power series coefficients are connected by a two-dimensional recurrence relation associated with infinite Stieltjes matrix equation. The recurrence relation is solved by using generating functions. A novel state-dependent birth-death queueing model is taken and expressed as continued fractions by employing integral transforms. The stationary probabilities of general state-dependent Markovian queues are obtained from the continued fractions and its associated tridiagonal determinants. As a special case the time-dependent system size probabilities and busy period distribution of a classical single server queue are deduced using continued fraction and its power series. Numerical illustrations are also presented.

On factorization of the Fibonacci and Lucas numbers using tridiagonal determinants

Abstract The aim of this paper is to give new results about factorizations of the Fibonacci numbers F n and the Lucas numbers L n. These numbers are defined by the second order recurrence relation a n+2 = a n+1+a n with the initial terms F 0 = 0, F 1 = 1 and L 0 = 2, L 1 = 1, respectively. Proofs of theorems are done with the help of connections between determinants of tridiagonal matrices and the Fibonacci and the Lucas numbers using the Chebyshev polynomials. This method extends the approach used in [CAHILL, N. D.—D’ERRICO, J. R.—SPENCE, J. P.: Complex factorizations of the Fibonacci and Lucas numbers, Fibonacci Quart. 41 (2003), 13–19], and CAHILL, N. D.—NARAYAN, D. A.: Fibonacci and Lucas numbers as tridiagonal matrix determinants, Fibonacci Quart. 42 (2004), 216–221].