scholarly journals Embedding Structures Associated with Riordan Arrays and Moment Matrices

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Paul Barry
Keyword(s):  
Orthogonal Polynomials ◽  
Riordan Arrays ◽  
Riordan Array

Every ordinary Riordan array contains two naturally embedded Riordan arrays. We explore this phenomenon, and we compare it to the situation for certain moment matrices of families of orthogonal polynomials.

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.

scholarly journals On identities involving generalized harmonic, hyperharmonic and special numbers with Riordan arrays

2021 ◽  
Vol 9 (1) ◽  
pp. 22-30
Author(s):  
Sibel Koparal ◽  
Neşe Ömür ◽  
Ömer Duran
Keyword(s):  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Riordan Arrays ◽  
Riordan Array

Abstract In this paper, by means of the summation property to the Riordan array, we derive some identities involving generalized harmonic, hyperharmonic and special numbers. For example, for n ≥ 0, ∑ k = 0 n B k k ! H ( n . k , α ) = α H ( n + 1 , 1 , α ) - H ( n , 1 , α ) , \sum\limits_{k = 0}^n {{{{B_k}} \over {k!}}H\left( {n.k,\alpha } \right) = \alpha H\left( {n + 1,1,\alpha } \right) - H\left( {n,1,\alpha } \right)} , and for n > r ≥ 0, ∑ k = r n - 1 ( - 1 ) k s ( k , r ) r ! α k k ! H n - k ( α ) = ( - 1 ) r H ( n , r , α ) , \sum\limits_{k = r}^{n - 1} {{{\left( { - 1} \right)}^k}{{s\left( {k,r} \right)r!} \over {{\alpha ^k}k!}}{H_{n - k}}\left( \alpha \right) = {{\left( { - 1} \right)}^r}H\left( {n,r,\alpha } \right)} , where Bernoulli numbers Bn and Stirling numbers of the first kind s (n, r).

Summations of Inverse Generalized Harmonic Numbers with Riordan Arrays

2014 ◽  
Vol 687-691 ◽  
pp. 1394-1398
Author(s):  
Gao Wen Xi ◽  
Zheng Ping Zhang
Keyword(s):  
Stirling Numbers ◽  
Harmonic Number ◽  
Harmonic Numbers ◽  
Riordan Arrays ◽  
Combinatorial Sums ◽  
Generalized Harmonic Numbers ◽  
Riordan Array

By observing that the infinite triangle obtained from some generalized harmonic numbers follows a Riordan array, we using connections between the Stirling numbers of both kinds and other inverse generalized harmonic numbers. we proved some combinatorial sums and inverse generalized harmonic number identities.

Inverse Generalized Harmonic Numbers with Riordan Arrays

2013 ◽  
Vol 842 ◽  
pp. 750-753
Author(s):  
Gao Wen Xi ◽  
Lan Long ◽  
Xue Quan Tian ◽  
Zhao Hui Chen
Keyword(s):  
Stirling Numbers ◽  
Harmonic Number ◽  
Harmonic Numbers ◽  
Riordan Arrays ◽  
Combinatorial Sums ◽  
Generalized Harmonic Numbers ◽  
Riordan Array

In this paper, By observing that the infinite triangle obtained from some generalized harmonic numbers follows a Riordan array, we obtain connections between the Stirling numbers of both kinds and other inverse generalized harmonic numbers. Further, we proved some combinatorial sums and inverse generalized harmonic number identities.

scholarly journals Jordan Canonical Forms of Riordan Arrays

2021 ◽  
Vol 76 (2) ◽  
Author(s):  
Roksana Słowik
Keyword(s):  
Transition Matrix ◽  
Canonical Forms ◽  
Riordan Arrays ◽  
Riordan Group ◽  
The Given ◽  
Riordan Array

AbstractWe consider the possible Jordan canonical forms of Riordan arrays. We prove that there are, in fact, only two such forms. Moreover, the transition matrix is in the Riordan group only in the case when the given Riordan array has one of some three specific forms.

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