scholarly journals Identities of the Chebyshev Polynomials, the Inverse of a Triangular Matrix, and Identities of the Catalan Numbers

2017 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Chebyshev Polynomials ◽  
Triangular Matrix ◽  
Higher Order ◽  
Catalan Numbers ◽  
Lower Triangular Matrix ◽  
Fibonacci Polynomials ◽  
Delannoy Numbers ◽  
Lower Triangular

In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, deduce an explicit formula and an identities for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers respectively with the Chebyshev polynomials, the central Delannoy numbers, and the Fibonacci polynomials.

scholarly journals Identities of the Chebyshev Polynomials, the Inverse of a Triangular Matrix, and Identities of the Catalan Numbers

2017 ◽  
Author(s):  
Feng Qi ◽  
Qing Zou ◽  
Bai-Ni Guo
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Chebyshev Polynomials ◽  
Triangular Matrix ◽  
Higher Order ◽  
Catalan Numbers ◽  
Lower Triangular Matrix ◽  
Fibonacci Polynomials ◽  
Delannoy Numbers ◽  
Lower Triangular

In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, deduce an explicit formula and an identities for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers respectively with the Chebyshev polynomials, the central Delannoy numbers, and the Fibonacci polynomials.

scholarly journals The inverse of a triangular matrix and several identities of the Catalan numbers

2019 ◽  
Vol 13 (2) ◽  
pp. 518-541 ◽  
Author(s):  
Feng Qi ◽  
Qing Zou ◽  
Bai-Ni Guo
Keyword(s):  
Generating Function ◽  
Chebyshev Polynomials ◽  
Triangular Matrix ◽  
Higher Order ◽  
Catalan Numbers ◽  
Lower Triangular Matrix ◽  
Fibonacci Polynomials ◽  
Delannoy Numbers ◽  
Lower Triangular ◽  
Derivatives Of

In the paper, the authors establish two identities to express higher order derivatives and integer powers of the generating function of the Chebyshev polynomials of the second kind in terms of integer powers and higher order derivatives of the generating function of the Chebyshev polynomials of the second kind respectively, find an explicit formula and an identity for the Chebyshev polynomials of the second kind, conclude the inverse of an integer, unit, and lower triangular matrix, derive an inversion theorem, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers with the Chebyshev polynomials of the second kind, the central Delannoy numbers, and the Fibonacci polynomials respectively.

scholarly journals CONNECTEDNESS OF SELF-AFFINE SETS WITH PRODUCT DIGIT SETS

Fractals ◽  
2017 ◽  
Vol 25 (06) ◽  
pp. 1750053 ◽  
Author(s):  
JING-CHENG LIU ◽  
JUN JASON LUO ◽  
KE TANG
Keyword(s):  
Triangular Matrix ◽  
Sufficient Condition ◽  
Lower Triangular Matrix ◽  
Necessary And Sufficient Condition ◽  
Lower Triangular ◽  
Necessary And Sufficient ◽  
Affine Set

Let [Formula: see text] be an expanding lower triangular matrix and [Formula: see text]. Let [Formula: see text] be the associated self-affine set. In the paper, we generalize some connectedness results on self-affine tiles to self-affine sets and provide a necessary and sufficient condition for [Formula: see text] to be connected.

scholarly journals Alternative proofs of some formulas for two tridiagonal determinants

2018 ◽  
Vol 10 (2) ◽  
pp. 287-297 ◽  
Author(s):  
Feng Qi ◽  
Ai-Qi Liu
Keyword(s):  
Chebyshev Polynomials ◽  
Tridiagonal Matrix ◽  
Fibonacci Polynomials ◽  
Detailed Proof ◽  
Delannoy Numbers ◽  
Appl Math ◽  
Math Phys

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.

scholarly journals Matrix transformations of starshaped sequences

2001 ◽  
Vol 28 (4) ◽  
pp. 189-200
Author(s):  
Chikkanna R. Selvaraj ◽  
Suguna Selvaraj
Keyword(s):  
Sufficient Conditions ◽  
Triangular Matrix ◽  
Necessary And Sufficient Conditions ◽  
Matrix Transformations ◽  
Lower Triangular Matrix ◽  
Lower Triangular ◽  
Necessary And Sufficient

We deal with matrix transformations preserving the starshape of sequences. The main result gives the necessary and sufficient conditions for a lower triangular matrixAto preserve the starshape of sequences. Also, we discuss the nature of the mappings of starshaped sequences by some classical matrices.

scholarly journals The inverse along a lower triangular matrix

2012 ◽  
Vol 219 (3) ◽  
pp. 886-891 ◽  
Author(s):  
Xavier Mary ◽  
Pedro Patrício
Keyword(s):  
Triangular Matrix ◽  
Lower Triangular Matrix ◽  
Lower Triangular

scholarly journals Elementary triangular matrices and inverses of k-Hessenberg and triangular matrices

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Luis Verde-Star
Keyword(s):  
Triangular Matrix ◽  
Identity Matrix ◽  
Lower Triangular Matrix ◽  
Triangular Matrices ◽  
Hessenberg Matrices ◽  
Banded Matrices ◽  
Lower Triangular ◽  
And Inversion ◽  
Block Hessenberg Matrices ◽  
Explicit Expressions

AbstractWe use elementary triangular matrices to obtain some factorization, multiplication, and inversion properties of triangular matrices. We also obtain explicit expressions for the inverses of strict k-Hessenberg matrices and banded matrices. Our results can be extended to the cases of block triangular and block Hessenberg matrices. An n × n lower triangular matrix is called elementary if it is of the form I + C, where I is the identity matrix and C is lower triangular and has all of its nonzero entries in the k-th column,where 1 ≤ k ≤ n.

scholarly journals Degree of Approximation of Conjugate of Signals (Functions) by Lower Triangular Matrix Operator

2011 ◽  
Vol 02 (12) ◽  
pp. 1448-1452 ◽  
Author(s):  
Vishnu Narayan Mishra ◽  
Huzoor H. Khan ◽  
Kejal Khatri
Keyword(s):  
Triangular Matrix ◽  
Degree Of Approximation ◽  
Matrix Operator ◽  
Lower Triangular Matrix ◽  
Lower Triangular

A multi-parameter generalization of the symmetric algorithm

2018 ◽  
Vol 68 (4) ◽  
pp. 699-712
Author(s):  
José L. Ramírez ◽  
Mark Shattuck
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Fibonacci Numbers ◽  
Recursive Formula ◽  
Binomial Coefficient ◽  
Fibonacci Polynomials ◽  
Combinatorial Argument ◽  
Recurrent Sequences ◽  
Seidel Method ◽  
Multivariate Generalization

Abstract The symmetric algorithm is a variant of the well-known Euler-Seidel method which has proven useful in the study of linearly recurrent sequences. In this paper, we introduce a multivariate generalization of the symmetric algorithm which reduces to it when all parameters are unity. We derive a general explicit formula via a combinatorial argument and also an expression for the row generating function. Several applications of our algorithm to the q-Fibonacci and q-hyper-Fibonacci numbers are discussed. Among our results is an apparently new recursive formula for the Carlitz Fibonacci polynomials. Finally, a (p, q)-analogue of the algorithm is introduced and an explicit formula for it in terms of the (p, q)-binomial coefficient is found.

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