Half Riordan array sequences

2020 ◽  
Vol 604 ◽  
pp. 236-264
Author(s):  
Tian-Xiao He
Keyword(s):  
Riordan Array

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).

scholarly journals Half of a Riordan array and restricted lattice paths

2018 ◽  
Vol 537 ◽  
pp. 1-11 ◽  
Author(s):  
Sheng-Liang Yang ◽  
Yan-Ni Dong ◽  
Lin Yang ◽  
Juan Yin
Keyword(s):  
Lattice Paths ◽  
Riordan Array

scholarly journals $q$-Riordan array for $q$-Pascal matrix and its inverse matrix

2016 ◽  
Vol 40 ◽  
pp. 1038-1048 ◽  
Author(s):  
Naim TUĞLU ◽  
Fatma YEŞİL ◽  
Maciej DZIEMIAŃCZUK ◽  
E. Gökçen KOÇER
Keyword(s):  
Inverse Matrix ◽  
Pascal Matrix ◽  
Riordan Array

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 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 Some Inverse Relations Determined by Catalan Matrices

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Sheng-liang Yang
Keyword(s):  
Combinatorial Identities ◽  
Binomial Coefficients ◽  
Inverse Relation ◽  
Special Cases ◽  
Matrix Identities ◽  
Riordan Array

We use the A-sequence and Z-sequence of Riordan array to characterize the inverse relation associated with the Riordan array. We apply this result to prove some combinatorial identities involving Catalan matrices and binomial coefficients. Some matrix identities obtained by Shapiro and Radoux are all special cases of our identity. In addition, a unified form of Catalan matrices is introduced.

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 Generating functions for the area below some lattice paths

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Donatella Merlini
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Lattice Paths ◽  
The Matrix ◽  
International Audience ◽  
Riordan Array

International audience We study some lattice paths related to the concept ofgenerating trees. When the matrix associated to this kind of trees is a Riordan array $D=(d(t),h(t))$, we are able to find the generating function for the total area below these paths expressed in terms of the functions $d(t)$ and $h(t)$.

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