scholarly journals Determinant Representations of Polynomial Sequences of Riordan Type

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Sheng-liang Yang ◽  
Sai-nan Zheng
Keyword(s):  
Recurrence Relation ◽  
Characteristic Polynomial ◽  
Chebyshev Polynomials ◽  
General Term ◽  
Polynomial Sequence ◽  
Polynomial Sequences ◽  
Principal Submatrix ◽  
Riordan Array

In this paper, using the production matrix of a Riordan array, we obtain a recurrence relation for polynomial sequence associated with the Riordan array, and we also show that the general term for the sequence can be expressed as the characteristic polynomial of the principal submatrix of the production matrix. As applications, a unified determinant expression for the four kinds of Chebyshev polynomials is given.

scholarly journals On Sequences of Numbers and Polynomials Defined by Linear Recurrence Relations of Order 2

2009 ◽  
Vol 2009 ◽  
pp. 1-21 ◽  
Author(s):  
Tian-Xiao He ◽  
Peter J.-S. Shiue
Keyword(s):  
Recurrence Relation ◽  
Chebyshev Polynomials ◽  
Recurrence Relations ◽  
Linear Recurrence ◽  
Lucas Numbers ◽  
Polynomial Sequences ◽  
Fibonacci And Lucas Numbers ◽  
Linear Recurrence Relation ◽  
Humbert Polynomials ◽  
General Method

Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a general method to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented.

The Dying Rabbit Problem Revisited

Integers ◽  
2009 ◽  
Vol 9 (2) ◽  
Author(s):  
Antonio M. Oller-Marcén
Keyword(s):  
Recurrence Relation ◽  
Characteristic Polynomial ◽  
Fibonacci Sequence ◽  
General Term ◽  
Technical Requirement ◽  
Positive Real ◽  
Real Roots

AbstractIn this paper we study a generalization of the Fibonacci sequence in which rabbits are mortal and take more that two months to become mature. In particular we give a general recurrence relation for these sequences (improving the work in [Hoggat and Lind, Fibonacci Quart. 7: 482–487, 1969]) and we calculate explicitly their general term (extending the work in [Miles, Amer. Math. Monthly 67: 745–752, 1960]). In passing, and as a technical requirement, we also study the behavior of the positive real roots of the characteristic polynomial of the considered sequences.

On factorization of the Fibonacci and Lucas numbers using tridiagonal determinants

2012 ◽  
Vol 62 (3) ◽  
Author(s):  
Jaroslav Seibert ◽  
Pavel Trojovský
Keyword(s):  
Recurrence Relation ◽  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Second Order ◽  
Tridiagonal Matrix ◽  
Lucas Numbers ◽  
Tridiagonal Matrices ◽  
Fibonacci And Lucas Numbers

AbstractThe 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].

scholarly journals On multivariable cumulant polynomial sequences with applications

2016 ◽  
Vol 7 (1) ◽  
Author(s):  
Elvira Di Nardo
Keyword(s):  
Main Tool ◽  
Multivariate Functions ◽  
Random Sums ◽  
Open Problems ◽  
Polynomial Sequence ◽  
Polynomial Sequences ◽  
Parameter Estimations ◽  
New Family ◽  
Random Index ◽  
Sheffer Polynomial

A new family of polynomials, called cumulant polynomial sequence, and its extension to the multivariate case is introduced relied on a purely symbolic combinatorial method. The coefficients are cumulants, but depending on what is plugged in the indeterminates, moment se- quences can be recovered as well. The main tool is a formal generalization of random sums,  when a not necessarily integer-valued multivariate random index is considered. Applications are given within parameter estimations, L\'evy processes and random matrices and, more generally, problems involving multivariate functions. The connection between exponential models and multivariable Sheffer polynomial sequences offers a different viewpoint in employing the method. Some open problems end the paper.

scholarly journals On Power Sums Involving Lucas Functions Sequences

2017 ◽  
Vol 2017 ◽  
pp. 1-11
Author(s):  
Stefano Barbero
Keyword(s):  
Chebyshev Polynomials ◽  
Power Sums ◽  
Polynomial Sequences ◽  
Divisibility Properties

We present some general formulas related to sum of powers, also with alternating sign, involving Lucas functions sequences. In particular, our formulas give a synthesis of various identities involving sum of powers of well-known polynomial sequences such as Fibonacci, Lucas, Pell, Jacobsthal, and Chebyshev polynomials. Finally, we point out some interesting divisibility properties between polynomials arising from our results.

scholarly journals On the Connection Coefficients of the Chebyshev-Boubaker Polynomials

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Paul Barry
Keyword(s):  
Orthogonal Polynomials ◽  
Closed Form ◽  
Chebyshev Polynomials ◽  
Recurrence Relations ◽  
Connection Coefficients ◽  
Riordan Arrays ◽  
Boubaker Polynomials ◽  
Riordan Array

The Chebyshev-Boubaker polynomials are the orthogonal polynomials whose coefficient arrays are defined by ordinary Riordan arrays. Examples include the Chebyshev polynomials of the second kind and the Boubaker polynomials. We study the connection coefficients of this class of orthogonal polynomials, indicating how Riordan array techniques can lead to closed-form expressions for these connection coefficients as well as recurrence relations that define them.

Identities connecting the Chebyshev polynomials

2016 ◽  
Vol 100 (549) ◽  
pp. 450-459 ◽  
Author(s):  
Jonny Griffiths
Keyword(s):  
Differential Equations ◽  
Weight Function ◽  
Orthogonal Polynomial ◽  
Recurrence Relation ◽  
Approximation Theory ◽  
Chebyshev Polynomials ◽  
Pell Equation ◽  
Fibonacci Polynomials ◽  
The Third ◽  
Polynomial Family

There are many families of polynomials in mathematics, and they often occur naturally in pairs. The Fibonacci polynomials and the Lucas polynomials, for example, are generated by the same recurrence relation but with different starting values, and there are many identities that link the two families [1]. The same is true for the Chebyshev polynomials of the first and second kinds, Tn (x) and Un (x) [2], respectively. There are two further polynomial families that are less well-known, the Chebyshev polynomials of the third and fourth kinds, Vn (x) and Wn (x) [3], respectively. Each of the four kinds is an example of an orthogonal polynomial family Pn (x), where for some appropriate weight function W (x), whenever n ≠ m. The families Tn (x) and Un (x) in particular are ubiquitous in their mathematical uses, in approximation theory, in differential equations, and in solving the Pell equation, to name but three. There are also many connections between Tn (x), Un (x), Vn (x) and Wn (x), some of which are explored here, and some of which we hope are new.

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