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Factorizations for q-Pascal matrices of two variables

2015
*
Special Matrices
*

AbstractIn this second article on

doi:10.1515/spma-2015-0020
fatcat:pdvhnaio6vhenprcmnv4rgyrje
*q*-*Pascal*matrices, we show how*the*previous factorizations by*the*summation matrices and*the*so-called*q*-unit matrices extend in a natural way to produce*q*-*analogues**of*...*Pascal*matrices*of*two variables by Z. ... 5 y A*q*-*analogue**of**the**matrix*Ψn[x, y] in[13, p.175]. ...##
###
Frequency Transformation With Pascal Matrix Equations

2016
*
Zenodo
*

Further, two

doi:10.5281/zenodo.1338617
fatcat:4ytx52aww5egbaioax7dnuluoq
*Pascal**matrix*equations are derived: an*analogue*to digital filter*Pascal**matrix*equation and a digital to digital filter*Pascal**matrix*equation. ... Frequency transformation with*Pascal**matrix*equations is a method for transforming an electronic filter (*analogue*or digital) into another filter. ... With*the*support*of**Pascal**matrix*equations,*the**analogue*low pass prototype to digital filter*Pascal**matrix*equation and*the*digital low pass to digital filter*Pascal**matrix*equation can be implemented ...##
###
$q$-Riordan array for $q$-Pascal matrix and its inverse matrix

2016
*
Turkish Journal of Mathematics
*

Acknowledgment

doi:10.3906/mat-1506-56
fatcat:b3s4mfdddfhsnffentrxzbkabi
*The*authors would like to thank*the*anonymous reviewer for his/her valuable comments and suggestions to improve*the*quality*of**the*paper. ... Then we obtain*the**q*-*analogue**of**the*fundamental theorem*of*Riordan arrays. As a result*of*this theorem we get a representation for*the**q*-*Pascal**matrix*. ... In conclusion,*the**q*-*analogue**of**the*Riordan representation*of*P is ( 1 (1 − x)*q*, x (1 − x)*q*)*q*. 2 Corollary 3.4*The*generic element p ij*of**the**q*-*Pascal**matrix*P = ( 1 (1 − x)*q*, x (1 − x)*q*...##
###
On the q-Lie group of q-Appell polynomial matrices and related factorizations

2018
*
Special Matrices
*

In

doi:10.1515/spma-2018-0009
fatcat:fbbkbjhgifee7lne5wwaw23faa
*the*spirit*of*our earlier paper [10] and Zhang and Wang [16],we introduce*the**matrix**of*multiplicative*q*-Appell polynomials*of*order M ∈ ℤ. ... In*the*limit*q*→ 1 we obtain corresponding formulas for Appell polynomial matrices.We conclude by presenting*the*commutative ring*of*generalized*q*-*Pascal*functional matrices,which operates on all functions ... A*q*-*analogue**of*[15, p. 54 ]. ...##
###
Factorizations related to the reciprocal Pascal matrix
[article]

2015
*
arXiv
*
pre-print

*The*reciprocal

*Pascal*

*matrix*has entries i+jj^-1. Explicit formullae for its LU-decomposition,

*the*LU-decomposition

*of*its inverse, and some related matrices are obtained. ... For all results,

*q*-

*analogues*are also presented. ... Introduction Recently, there has been some interest in

*the*reciprocal

*Pascal*

*matrix*M, defined by M i,j = i + j j −1 ;

*the*indices start here for convenience with 0, 0, and

*the*

*matrix*is either infinite ...

##
###
Pascal Matrix Representation of Evolution of Polynomials

2015
*
International Journal of Applied and Computational Mathematics
*

*Pascal*

*matrix*is an adjoint operator

*of*

*the*differential operator

*of*translation. ... This feature

*of*

*the*

*Pascal*

*matrix*is used in order to construct evolution equations for coefficients

*of*polynomials induced by shifts

*of*

*the*roots. ... An

*analogue*

*of*

*the*classical inverse transformation K = E − V (r ), (73) obviously, is given by

*the*formula [K ] = exp − V (r )A up [E]. (74) Geometrical Interpretation

*of*

*Pascal*

*Matrix*Transformation ...

##
###
h-analogue of Fibonacci Numbers
[article]

2009
*
arXiv
*
pre-print

In this paper, we introduce

arXiv:0910.0038v1
fatcat:cfgrec7ccjftpkivkgcdixwlae
*the*h-*analogue**of*Fibonacci numbers for non-commutative h-plane. For h h'= 1 and h = 0, these are just*the*usual Fibonacci numbers as it should be. ... We also derive a collection*of*identities for these numbers. Furthermore, h-Binet's formula for*the*h-Fibonacci numbers is found and*the*generating function that generates these numbers is obtained. ... It is possible to introduce*the*(*q*− h)-*analogue**of*Finonacci numbers by using*the*(*q*− h)-*analogue**of*binomial coefficients which was introduced in [5] . ...##
###
Factorizations related to the reciprocal Pascal matrix

2016
*
Turkish Journal of Mathematics
*

In

doi:10.3906/mat-1504-30
fatcat:vvzhmkuzj5cs3e7iktfnqas5pa
*the*following section, we provide*q*-*analogues**of*these results. ... Introduction Recently, there has been some interest in*the*reciprocal*Pascal**matrix*M , defined by M i,j = ( i + j j ) −1 ;*the*indices start here for convenience with 0, 0 , and*the**matrix*is either infinite ... We only cite*the*results; justifications are in*the*same style as in*the*previous instances. There are also*q*-*analogues*for*the**matrix*. ...##
###
LU factorization of the Vandermonde matrix and its applications

2007
*
Applied Mathematics Letters
*

Two applications

doi:10.1016/j.aml.2006.10.003
fatcat:jngtp4k7j5hlpopm73guqpyc3u
*of**the**q*-*Pascal**matrix*resulting from*the*factorization*of**the*Vandermonde*matrix*at*the**q*-integer nodes are introduced. ... A scaled version*of**the*lower and*the*upper triangular factors*of**the*inverse*of**the*Vandermonde*matrix*is given. ... It follows from (3.6) that*the**matrix*E*q*(H t) has entries*of**the**q*-*Pascal**matrix*E*q*(H t) i j = t i− j i j , i j 0. ...##
###
Euler Matrices and their Algebraic Properties Revisited

2020
*
Applied Mathematics & Information Sciences
*

We establish some explicit expressions for

doi:10.18576/amis/140407
fatcat:zs4lfm7awnfzlotklg3joythse
*the*Euler polynomial*matrix*E (x), which involves*the*generalized*Pascal*, Fibonacci and Lucas matrices, respectively. ... Taking into account some properties*of*Euler polynomials and numbers, we deduce product formulae for E (α) (x) and define*the*inverse*matrix**of*E . ...*The*authors are so grateful to*the*anonymous reviewers for their careful checking*of**the*details and*the*helpful comments and suggestions that improved this paper. ...##
###
The reciprocal super Catalan matrix

2015
*
Special Matrices
*

AbstractThe reciprocal super Catalan

doi:10.1515/spma-2015-0010
fatcat:wo736dueyfdcdiqxtlcvfxg7xu
*matrix*has entries ... Acknowledgement:*The*author was supported by an incentive grant*of**the*National Research Foundation*of*South Africa. ...*q*-*analogues**of**the*decomposition*of*M We assume that M has entries (︀ (*q*) 2i (*q*) 2j 2j , (*q*) i (*q*) i+j (*q*) j )︀ −1 . ...##
###
Convolutional Matching Pursuit and Dictionary Training
[article]

2010
*
arXiv
*
pre-print

Matching pursuit and K-SVD is demonstrated in

arXiv:1010.0422v1
fatcat:poqukr7crfczpj6pcqahxrydw4
*the*translation invariant setting ... In this model, we try to represent a given d × n data*matrix*X*of*n points in R d written as columns via a solution to*the*problem {W * , Z * } = {W * (K, X,*q*), Z * (K, X,*q*)} = arg min 1) or its Z ... In this short note we work with greedy algorithms for solving*the*convolutional*analogues**of*1.1. ...##
###
Infinite triangular matrices, q-Pascal matrices, and determinantal representations

2011
*
Linear Algebra and its Applications
*

We use basic properties

doi:10.1016/j.laa.2010.08.022
fatcat:5ephchervvhafpveq74vrvrtfa
*of*infinite lower triangular matrices and*the*connections*of*Toeplitz matrices with generating-functions to obtain inversion formulas for several types*of**q*-*Pascal*matrices, determinantal ... representations for polynomial sequences, and identities involving*the**q*-Gaussian coefficients. ... (4.4) This means that*the*determinant*of**the*Hessenberg*matrix*hP x (k + n, k) is equal to its lower-left entry. We consider next some*q*-*analogues**of**the**Pascal*matrices. ...##
###
Generalized Riordan arrays and zero generalized Pascal matrices
[article]

2016
*
arXiv
*
pre-print

Generalized

arXiv:1612.07657v1
fatcat:s4uxp36cyrgkzbwg5xmenor7lq
*Pascal**matrix*whose elements are generalized binomial coefficients is included in*the*group*of*generalized Riordan arrays. ... It is shown that every such*matrix*included in*the**matrix*group similar to*the*generalized Riordan group. ... Each nonzero generalized*Pascal**matrix*is*the*Hadamard product*of**the*matrices ϕ,*q*P . ...##
###
Asymmetric extension of Pascal-Dellanoy triangles
[article]

2020
*
arXiv
*
pre-print

We also give identities among which one equivalent to

arXiv:2001.11665v1
fatcat:hvybgwa55jgbter22mewttrs3u
*the*de Moivre sum and establish a*q*-*analogue**of**the*coefficient*of**the*quasi s-*Pascal*triangle. ... We give a generalization*of**the**Pascal*triangle called*the*quasi s-*Pascal*triangle where*the*sum*of**the*elements crossing*the*diagonal rays produce*the*s-bonacci sequence. ...*q*-*analogue**of**the*quasi s-*Pascal*triangle In this section we define*the**q*-*analogue**of**the*quasi s-*Pascal*triangle, we denote by n k [s] these coefficients; for that, we give an explicit formula and generating ...
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