( m , r ) -central Riordan arrays and their applications

Sheng-Liang Yang; Yan-Xue Xu; Tian-Xiao He

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 4, page 919-936
  • ISSN: 0011-4642

Abstract

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For integers m > r ≥ 0 , Brietzke (2008) defined the ( m , r ) -central coefficients of an infinite lower triangular matrix G = ( d , h ) = ( d n , k ) n , k ∈ ℕ as d m n + r , ( m - 1 ) n + r , with n = 0 , 1 , 2 , ⋯ , and the ( m , r ) -central coefficient triangle of G as G ( m , r ) = ( d m n + r , ( m - 1 ) n + k + r ) n , k ∈ ℕ . It is known that the ( m , r ) -central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array G = ( d , h ) with h ( 0 ) = 0 and d ( 0 ) , h ' ( 0 ) ≠ 0 , we obtain the generating function of its ( m , r ) -central coefficients and give an explicit representation for the ( m , r ) -central Riordan array G ( m , r ) in terms of the Riordan array G . Meanwhile, the algebraic structures of the ( m , r ) -central Riordan arrays are also investigated, such as their decompositions, their inverses, and their recessive expressions in terms of m and r . As applications, we determine the ( m , r ) -central Riordan arrays of the Pascal matrix and other Riordan arrays, from which numerous identities are constructed by a uniform approach.

How to cite

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Yang, Sheng-Liang, Xu, Yan-Xue, and He, Tian-Xiao. "$(m,r)$-central Riordan arrays and their applications." Czechoslovak Mathematical Journal 67.4 (2017): 919-936. <http://eudml.org/doc/294691>.

@article{Yang2017,
abstract = {For integers $m > r \ge 0$, Brietzke (2008) defined the $(m,r)$-central coefficients of an infinite lower triangular matrix $G=(d, h)=(d_\{n,k\})_\{n,k \in \mathbb \{N\}\}$ as $ d_\{mn+r,(m-1)n+r\}$, with $n=0,1,2,\cdots $, and the $(m,r)$-central coefficient triangle of $G$ as \[ G^\{(m,r)\} = (d\_\{mn+r,(m-1)n+k+r\})\_\{n,k \in \mathbb \{N\}\}. \] It is known that the $(m,r)$-central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array $G=(d,h)$ with $h(0)=0$ and $d(0), h^\{\prime \}(0)\ne 0$, we obtain the generating function of its $(m,r)$-central coefficients and give an explicit representation for the $(m,r)$-central Riordan array $G^\{(m,r)\}$ in terms of the Riordan array $G$. Meanwhile, the algebraic structures of the $(m,r)$-central Riordan arrays are also investigated, such as their decompositions, their inverses, and their recessive expressions in terms of $m$ and $r$. As applications, we determine the $(m,r)$-central Riordan arrays of the Pascal matrix and other Riordan arrays, from which numerous identities are constructed by a uniform approach.},
author = {Yang, Sheng-Liang, Xu, Yan-Xue, He, Tian-Xiao},
journal = {Czechoslovak Mathematical Journal},
keywords = {Riordan array; central coefficient; central Riordan array; generating function; Fuss-Catalan number; Pascal matrix; Catalan matrix},
language = {eng},
number = {4},
pages = {919-936},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$(m,r)$-central Riordan arrays and their applications},
url = {http://eudml.org/doc/294691},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Yang, Sheng-Liang
AU - Xu, Yan-Xue
AU - He, Tian-Xiao
TI - $(m,r)$-central Riordan arrays and their applications
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 4
SP - 919
EP - 936
AB - For integers $m > r \ge 0$, Brietzke (2008) defined the $(m,r)$-central coefficients of an infinite lower triangular matrix $G=(d, h)=(d_{n,k})_{n,k \in \mathbb {N}}$ as $ d_{mn+r,(m-1)n+r}$, with $n=0,1,2,\cdots $, and the $(m,r)$-central coefficient triangle of $G$ as \[ G^{(m,r)} = (d_{mn+r,(m-1)n+k+r})_{n,k \in \mathbb {N}}. \] It is known that the $(m,r)$-central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array $G=(d,h)$ with $h(0)=0$ and $d(0), h^{\prime }(0)\ne 0$, we obtain the generating function of its $(m,r)$-central coefficients and give an explicit representation for the $(m,r)$-central Riordan array $G^{(m,r)}$ in terms of the Riordan array $G$. Meanwhile, the algebraic structures of the $(m,r)$-central Riordan arrays are also investigated, such as their decompositions, their inverses, and their recessive expressions in terms of $m$ and $r$. As applications, we determine the $(m,r)$-central Riordan arrays of the Pascal matrix and other Riordan arrays, from which numerous identities are constructed by a uniform approach.
LA - eng
KW - Riordan array; central coefficient; central Riordan array; generating function; Fuss-Catalan number; Pascal matrix; Catalan matrix
UR - http://eudml.org/doc/294691
ER -

References

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  1. Andrews, G. H., Some formulae for the Fibonacci sequence with generalizations, Fibonacci Q. 7 (1969), 113-130. (1969) Zbl0176.32202MR0242761
  2. Barry, P., On integer-sequence-based constructions of generalized Pascal triangles, J. Integer Seq. 9 (2006), Article 06.2.4, 34 pages. (2006) Zbl1178.11023MR2217230
  3. Barry, P., On the central coefficients of Bell matrices, J. Integer Seq. 14 (2011), Article 11.4.3, 10 pages. (2011) Zbl1231.11029MR2792159
  4. Barry, P., On the central coefficients of Riordan matrices, J. Integer Seq. 16 (2013), Article 13.5.1, 12 pages. (2013) Zbl1310.11032MR3065330
  5. Brietzke, E. H. M., 10.1016/j.disc.2007.08.050, Discrete Math. 308 (2008), 4246-4262. (2008) Zbl1207.05010MR2427755DOI10.1016/j.disc.2007.08.050
  6. Cheon, G.-S., Jin, S.-T., 10.1016/j.laa.2011.04.001, Linear Algebra Appl. 435 (2011), 2019-2032. (2011) Zbl1226.05021MR2810643DOI10.1016/j.laa.2011.04.001
  7. Cheon, G.-S., Kim, H., Shapiro, L. W., 10.1016/j.disc.2012.03.023, Discrete Math. 312 (2012), 2040-2049. (2012) Zbl1243.05007MR2920864DOI10.1016/j.disc.2012.03.023
  8. Comtet, L., 10.1007/978-94-010-2196-8, D. Reidel Publishing, Dordrecht (1974). (1974) Zbl0283.05001MR0460128DOI10.1007/978-94-010-2196-8
  9. Graham, R. L., Knuth, D. E., Patashnik, O., Concrete Mathematics. A Foundation for Computer Science, Addison-Wesley Publishing Company, Reading (1989). (1989) Zbl0668.00003MR1001562
  10. He, T.-X., 10.1016/j.laa.2012.10.001, Linear Algebra Appl. 438 (2013), 1467-1484. (2013) Zbl1257.05003MR2997825DOI10.1016/j.laa.2012.10.001
  11. He, T.-X., 10.1016/j.laa.2014.09.008, Linear Algebra Appl. 465 (2015), 15-42. (2015) Zbl1303.05007MR3274660DOI10.1016/j.laa.2014.09.008
  12. He, T.-X., Sprugnoli, R., 10.1016/j.disc.2008.11.021, Discrete Math. 309 (2009), 3962-3974. (2009) Zbl1228.05014MR2537389DOI10.1016/j.disc.2008.11.021
  13. Kruchinin, D., Kruchinin, V., A method for obtaining generating functions for central coefficients of triangles, J. Integer Seq. 15 (2012), Article 12.9.3, 10 pages. (2012) Zbl1292.05028MR3005529
  14. Merlini, D., Rogers, D. G., Sprugnoli, R., Verri, M. C., 10.4153/CJM-1997-015-x, Can. J. Math. 49 (1997), 301-320. (1997) Zbl0886.05013MR1447493DOI10.4153/CJM-1997-015-x
  15. Merlini, D., Sprugnoli, R., Verri, M. C., 10.1007/s10440-006-9077-7, Acta Appl. Math. 94 (2006), 233-249. (2006) Zbl1108.05008MR2290868DOI10.1007/s10440-006-9077-7
  16. Młotkowski, W., Fuss-Catalan numbers in noncommutative probability, Doc. Math., J. DMV 15 (2010), 939-955. (2010) Zbl1213.44004MR2745687
  17. Rogers, D. G., 10.1016/0012-365X(78)90063-8, Discrete Math. 22 (1978), 301-310. (1978) Zbl0398.05007MR0522725DOI10.1016/0012-365X(78)90063-8
  18. Shapiro, L. W., 10.1016/0012-365X(76)90009-1, Discrete Math. 14 (1976), 83-90. (1976) Zbl0323.05004MR0387069DOI10.1016/0012-365X(76)90009-1
  19. Shapiro, L. W., Getu, S., Woan, W.-J., Woodson, L. C., 10.1016/0166-218X(91)90088-E, Discrete Appl. Math. 34 (1991), 229-239. (1991) Zbl0754.05010MR1137996DOI10.1016/0166-218X(91)90088-E
  20. Sprugnoli, R., 10.1016/0012-365X(92)00570-H, Discrete Math. 132 (1994), 267-290. (1994) Zbl0814.05003MR1297386DOI10.1016/0012-365X(92)00570-H
  21. Stanley, R. P., 10.1017/CBO9780511609589, Cambridge Studies in Advanced Mathematics 62, Cambridge University Press, Cambridge (1999). (1999) Zbl0928.05001MR1676282DOI10.1017/CBO9780511609589
  22. Yang, S.-L., Zheng, S.-N., Yuan, S.-P., He, T.-X., 10.1016/j.laa.2013.09.044, Linear Algebra Appl. 439 (2013), 3605-3614. (2013) Zbl1283.15098MR3119875DOI10.1016/j.laa.2013.09.044

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