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### Factorizations for q-Pascal matrices of two variables

Thomas Ernst
2015 Special Matrices
AbstractIn this second article on 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

Phuoc Si Nguyen
2016 Zenodo
Further, two 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

Naim TUĞLU, Fatma YEŞİL, Maciej DZIEMIAŃCZUK, E. Gökçen KOÇER
2016 Turkish Journal of Mathematics
Acknowledgment 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

Thomas Ernst
2018 Special Matrices
In the spirit of our earlier paper  and Zhang and Wang ,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]

Helmut Prodinger
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

Robert M. Yamaleev
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]

H.B. Benaoum
2009 arXiv   pre-print
In this paper, we introduce 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  .  ...

### Factorizations related to the reciprocal Pascal matrix

Helmut PRODINGER
2016 Turkish Journal of Mathematics
In 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

Halil Oruç
2007 Applied Mathematics Letters
Two applications 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 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

Helmut Prodinger
2015 Special Matrices
AbstractThe reciprocal super Catalan 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]

Arthur Szlam, Koray Kavukcuoglu, Yann LeCun
2010 arXiv   pre-print
Matching pursuit and K-SVD is demonstrated in 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

Luis Verde-Star
2011 Linear Algebra and its Applications
We use basic properties 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]

E. Burlachenko
2016 arXiv   pre-print
Generalized 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]

Said Amrouche, Hacène Belbachir
2020 arXiv   pre-print
We also give identities among which one equivalent to 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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