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Journal articles on the topic 'Riordan arrays'

Author: Grafiati

Published: 10 March 2023

Last updated: 27 July 2025

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1

Barry, Paul. "Extensions of Riordan Arrays and Their Applications." Mathematics 13, no. 2 (2025): 242. https://doi.org/10.3390/math13020242.

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The Riordan group of Riordan arrays was first described in 1991, and since then, it has provided useful tools for the study of areas such as combinatorial identities, polynomial sequences (including families of orthogonal polynomials), lattice path enumeration, and linear recurrences. Useful extensions of the idea of a Riordan array have included almost Riordan arrays, double Riordan arrays, and their generalizations. After giving a brief overview of the Riordan group, we define two further extensions of the notion of Riordan arrays, and we give a number of applications for these extensions. T

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2

Barry, Paul. "Embedding Structures Associated with Riordan Arrays and Moment Matrices." International Journal of Combinatorics 2014 (March 17, 2014): 1–7. http://dx.doi.org/10.1155/2014/301394.

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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.

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3

Wang, Weiping, and Tianming Wang. "Generalized Riordan arrays." Discrete Mathematics 308, no. 24 (2008): 6466–500. http://dx.doi.org/10.1016/j.disc.2007.12.037.

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4

Luzón, Ana, Donatella Merlini, Manuel A. Morón, and Renzo Sprugnoli. "Complementary Riordan arrays." Discrete Applied Mathematics 172 (July 2014): 75–87. http://dx.doi.org/10.1016/j.dam.2014.03.005.

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5

Barry, Paul. "On the Connection Coefficients of the Chebyshev-Boubaker Polynomials." Scientific World Journal 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/657806.

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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.

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6

Merlini, Donatella, Douglas G. Rogers, Renzo Sprugnoli, and M. Cecilia Verri. "On Some Alternative Characterizations of Riordan Arrays." Canadian Journal of Mathematics 49, no. 2 (1997): 301–20. http://dx.doi.org/10.4153/cjm-1997-015-x.

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AbstractWe give several new characterizations of Riordan Arrays, the most important of which is: if {dn,k}n,k∈N is a lower triangular arraywhose generic element dn,k linearly depends on the elements in a well-defined though large area of the array, then {dn,k}n,k∈N is Riordan. We also provide some applications of these characterizations to the lattice path theory.

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7

O'Farrell, Anthony G. "Riordan Groups in Higher Dimensions." Mathematical Proceedings of the Royal Irish Academy 123A, no. 2 (2023): 95–124. http://dx.doi.org/10.1353/mpr.2023.a909312.

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Abstract: The classical Riordan groups associated to a given commutative ring are groups of infinite matrices (called Riordan arrays) associated to pairs of formal power series in one variable. The Fundamental Theorem of Riordan Arrays relates matrix multiplication to two group actions on such series, namely formal (convolution) multiplication and formal composition. We define the analogous Riordan groups involving formal power series in several variables, and establish the analogue of the Fundamental Theorem in that context. We discuss related groups of Laurent series and pose some questions.

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8

Lee, GwangYeon, and Mustafa Asci. "Some Properties of the(p,q)-Fibonacci and(p,q)-Lucas Polynomials." Journal of Applied Mathematics 2012 (2012): 1–18. http://dx.doi.org/10.1155/2012/264842.

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Riordan arrays are useful for solving the combinatorial sums by the help of generating functions. Many theorems can be easily proved by Riordan arrays. In this paper we consider the Pascal matrix and define a new generalization of Fibonacci polynomials called(p,q)-Fibonacci polynomials. We obtain combinatorial identities and by using Riordan method we get factorizations of Pascal matrix involving(p,q)-Fibonacci polynomials.

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9

Zhao, Xiqiang, and Shuangshuang Dings. "Sequences Related to Riordan Arrays." Fibonacci Quarterly 40, no. 3 (2002): 247–52. http://dx.doi.org/10.1080/00150517.2002.12428651.

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10

Luzón, Ana, Donatella Merlini, Manuel A. Morón, and Renzo Sprugnoli. "Identities induced by Riordan arrays." Linear Algebra and its Applications 436, no. 3 (2012): 631–47. http://dx.doi.org/10.1016/j.laa.2011.08.007.

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11

He, Tian-Xiao. "Matrix characterizations of Riordan arrays." Linear Algebra and its Applications 465 (January 2015): 15–42. http://dx.doi.org/10.1016/j.laa.2014.09.008.

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12

Krelifa, Ali, and Ebtissem Zerouki. "Riordan arrays and d-orthogonality." Linear Algebra and its Applications 515 (February 2017): 331–53. http://dx.doi.org/10.1016/j.laa.2016.11.039.

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13

Sprugnoli, Renzo. "Riordan arrays and combinatorial sums." Discrete Mathematics 132, no. 1-3 (1994): 267–90. http://dx.doi.org/10.1016/0012-365x(92)00570-h.

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14

Deutsch, Emeric, Luca Ferrari, and Simone Rinaldi. "Production Matrices and Riordan Arrays." Annals of Combinatorics 13, no. 1 (2009): 65–85. http://dx.doi.org/10.1007/s00026-009-0013-1.

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15

He, Tian-Xiao, and Renzo Sprugnoli. "Sequence characterization of Riordan arrays." Discrete Mathematics 309, no. 12 (2009): 3962–74. http://dx.doi.org/10.1016/j.disc.2008.11.021.

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16

Chen, Xi, Huyile Liang, and Yi Wang. "Total positivity of Riordan arrays." European Journal of Combinatorics 46 (May 2015): 68–74. http://dx.doi.org/10.1016/j.ejc.2014.11.009.

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17

Sprugnoli, Renzo. "Combinatorial sums through Riordan arrays." Journal of Geometry 101, no. 1-2 (2011): 195–210. http://dx.doi.org/10.1007/s00022-011-0090-2.

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18

Merlini, Donatella. "C-Finite Sequences and Riordan Arrays." Mathematics 12, no. 23 (2024): 3671. http://dx.doi.org/10.3390/math12233671.

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Many prominent combinatorial sequences, such as the Fibonacci, Lucas, Pell, Jacobsthal and Tribonacci sequences, are defined by homogeneous linear recurrence relations with constant coefficients. These sequences are often referred to as C-finite sequences, and a variety of representations have been employed throughout the literature, largely influenced by the author’s background and the specific application under consideration. Beyond the representation through recurrence relations, other approaches include those based on generating functions, explicit formulas, matrix exponentiation, the meth

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19

Yang, Sheng-Liang, and Yuan-Yuan Gao. "The Pascal Rhombus and Riordan Arrays." Fibonacci Quarterly 56, no. 4 (2018): 337–47. http://dx.doi.org/10.1080/00150517.2018.12427683.

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20

Agapito, José, Ângela Mestre, Pasquale Petrullo, and Maria M. Torres. "A symbolic treatment of Riordan arrays." Linear Algebra and its Applications 439, no. 7 (2013): 1700–1715. http://dx.doi.org/10.1016/j.laa.2013.05.007.

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21

Mu, Lili, Jianxi Mao, and Yi Wang. "Row polynomial matrices of Riordan arrays." Linear Algebra and its Applications 522 (June 2017): 1–14. http://dx.doi.org/10.1016/j.laa.2017.02.006.

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22

Wang, Weiping, and Chenlu Zhang. "Riordan arrays and related polynomial sequences." Linear Algebra and its Applications 580 (November 2019): 262–91. http://dx.doi.org/10.1016/j.laa.2019.06.008.

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23

Cheon, Gi-Sang, and M. E. A. El-Mikkawy. "Generalized harmonic numbers with Riordan arrays." Journal of Number Theory 128, no. 2 (2008): 413–25. http://dx.doi.org/10.1016/j.jnt.2007.08.011.

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24

Wang, Weiping. "Riordan arrays and harmonic number identities." Computers & Mathematics with Applications 60, no. 5 (2010): 1494–509. http://dx.doi.org/10.1016/j.camwa.2010.06.031.

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25

Merlini, Donatella, and M. Cecilia Verri. "Generating trees and proper Riordan Arrays." Discrete Mathematics 218, no. 1-3 (2000): 167–83. http://dx.doi.org/10.1016/s0012-365x(99)00343-x.

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26

Burlachenko, E. V. "Riordan arrays and generalized Lagrange series." Mathematical Notes 100, no. 3-4 (2016): 531–39. http://dx.doi.org/10.1134/s0001434616090248.

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27

Merlini, Donatella, Renzo Sprugnoli, and Maria Cecilia Verri. "Combinatorial sums and implicit Riordan arrays." Discrete Mathematics 309, no. 2 (2009): 475–86. http://dx.doi.org/10.1016/j.disc.2007.12.039.

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28

Merlini, Donatella, Renzo Sprugnoli, and Maria Cecilia Verri. "Combinatorial inversions and implicit Riordan arrays." Electronic Notes in Discrete Mathematics 26 (September 2006): 103–10. http://dx.doi.org/10.1016/j.endm.2006.08.019.

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29

Alp, Yasemin, and E. Gökçen Koçer. "Bivariate Leonardo polynomials and Riordan arrays." Notes on Number Theory and Discrete Mathematics 31, no. 2 (2025): 236–50. https://doi.org/10.7546/nntdm.2025.31.2.236-250.

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In this paper, bivariate Leonardo polynomials are defined, which are closely related to bivariate Fibonacci polynomials. Bivariate Leonardo polynomials are generalizations of the Leonardo polynomials and Leonardo numbers. Some properties and identities (Cassini, Catalan, Honsberger, d’Ocagne) for the bivariate Leonardo polynomials are obtained. Then, the Riordan arrays are defined by using bivariate Leonardo polynomials.

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30

Xi, Gao Wen, Lan Long, Xue Quan Tian, and Zhao Hui Chen. "Inverse Generalized Harmonic Numbers with Riordan Arrays." Advanced Materials Research 842 (November 2013): 750–53. http://dx.doi.org/10.4028/www.scientific.net/amr.842.750.

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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.

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31

Yang, Lin, and Sheng-Liang Yang. "Riordan arrays, Łukasiewicz paths and Narayana polynomials." Linear Algebra and its Applications 622 (August 2021): 1–18. http://dx.doi.org/10.1016/j.laa.2021.03.012.

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32

Ma, Qianqian, and Weiping Wang. "Riordan arrays and r-Stirling number identities." Discrete Mathematics 346, no. 1 (2023): 113211. http://dx.doi.org/10.1016/j.disc.2022.113211.

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33

Baccherini, D., D. Merlini, and R. Sprugnoli. "Level generating trees and proper Riordan arrays." Applicable Analysis and Discrete Mathematics 2, no. 1 (2008): 69–91. http://dx.doi.org/10.2298/aadm0801069b.

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34

Zhu, Bao-Xuan. "Total Positivity from the Exponential Riordan Arrays." SIAM Journal on Discrete Mathematics 35, no. 4 (2021): 2971–3003. http://dx.doi.org/10.1137/20m1379952.

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35

Cheon, Gi-Sang, and Sung-Tae Jin. "The group of multi-dimensional Riordan arrays." Linear Algebra and its Applications 524 (July 2017): 263–77. http://dx.doi.org/10.1016/j.laa.2017.03.010.

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36

Sprugnoli, Renzo. "Riordan arrays and the Abel-Gould identity." Discrete Mathematics 142, no. 1-3 (1995): 213–33. http://dx.doi.org/10.1016/0012-365x(93)e0220-x.

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37

Merlini, Donatella, and Renzo Sprugnoli. "Arithmetic into geometric progressions through Riordan arrays." Discrete Mathematics 340, no. 2 (2017): 160–74. http://dx.doi.org/10.1016/j.disc.2016.08.017.

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38

Xi, Gao Wen, and Zheng Ping Zhang. "Summations of Inverse Generalized Harmonic Numbers with Riordan Arrays." Applied Mechanics and Materials 687-691 (November 2014): 1394–98. http://dx.doi.org/10.4028/www.scientific.net/amm.687-691.1394.

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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.

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39

Yang, Sheng-Liang, Yan-Xue Xu, and Tian-Xiao He. "$(m,r)$-central Riordan arrays and their applications." Czechoslovak Mathematical Journal 67, no. 4 (2017): 919–36. http://dx.doi.org/10.21136/cmj.2017.0165-16.

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40

Barry, Paul. "Riordan arrays, generalized Narayana triangles, and series reversion." Linear Algebra and its Applications 491 (February 2016): 343–85. http://dx.doi.org/10.1016/j.laa.2015.10.032.

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41

He, Tian-Xiao, and Louis W. Shapiro. "Row sums and alternating sums of Riordan arrays." Linear Algebra and its Applications 507 (October 2016): 77–95. http://dx.doi.org/10.1016/j.laa.2016.05.035.

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42

Chen, Xi, and Yi Wang. "Notes on the total positivity of Riordan arrays." Linear Algebra and its Applications 569 (May 2019): 156–61. http://dx.doi.org/10.1016/j.laa.2019.01.015.

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43

Słowik, R. "Some (counter)examples on totally positive Riordan arrays." Linear Algebra and its Applications 594 (June 2020): 117–23. http://dx.doi.org/10.1016/j.laa.2020.02.021.

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44

Zhao, Xiqiang, Shuangshuang Ding, and Tingming Wang. "Some summation rules related to the Riordan arrays." Discrete Mathematics 281, no. 1-3 (2004): 295–307. http://dx.doi.org/10.1016/j.disc.2003.08.007.

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45

Petrullo, P. "Palindromic Riordan arrays, classical orthogonal polynomials and Catalan triangles." Linear Algebra and its Applications 618 (June 2021): 158–82. http://dx.doi.org/10.1016/j.laa.2021.02.007.

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46

He, Tian-Xiao. "Sequence characterizations of double Riordan arrays and their compressions." Linear Algebra and its Applications 549 (July 2018): 176–202. http://dx.doi.org/10.1016/j.laa.2018.03.029.

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47

Cheon, Gi-Sang, and Minho Song. "A new aspect of Riordan arrays via Krylov matrices." Linear Algebra and its Applications 554 (October 2018): 329–41. http://dx.doi.org/10.1016/j.laa.2018.05.028.

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48

Baccherini, D., D. Merlini, and R. Sprugnoli. "Binary words excluding a pattern and proper Riordan arrays." Discrete Mathematics 307, no. 9-10 (2007): 1021–37. http://dx.doi.org/10.1016/j.disc.2006.07.023.

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49

Chandragiri, S. "Riordan Arrays and Difference Equations of Subdiagonal Lattice Paths." Siberian Mathematical Journal 65, no. 2 (2024): 411–25. http://dx.doi.org/10.1134/s0037446624020149.

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50

Mező, István, Victor H. Moll, José Ramírez, and Diego Villamizar. "On the \(r\)-derangements of type B." Online Journal of Analytic Combinatorics, no. 16 (December 31, 2021): 1–21. https://doi.org/10.61091/ojac-1605.

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Extensions of a set partition obtained by imposing bounds on the size of the parts and the coloring of some of the elements are examined. Combinatorial properties and the generating functions of some counting sequences associated with these partitions are established. Connections with Riordan arrays are presented.

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