Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
Riordan array; central coefficient; central Riordan array; generating function; Fuss-Catalan number; Pascal matrix; Catalan matrix
Riordan array; central coefficient; central Riordan array; generating function; Fuss-Catalan number; Pascal matrix; Catalan matrix
Summary:
For integers $m > r \geq 0$, 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 $d(0), h'(0)\not = 0$, 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.
References:
[1] Andrews, G. H.: Some formulae for the Fibonacci sequence with generalizations. Fibonacci Q. 7 (1969), 113-130. MR 0242761 | Zbl 0176.32202
[2] Barry, P.: On integer-sequence-based constructions of generalized Pascal triangles. J. Integer Seq. 9 (2006), Article 06.2.4, 34 pages. MR 2217230 | Zbl 1178.11023
[3] Barry, P.: On the central coefficients of Bell matrices. J. Integer Seq. 14 (2011), Article 11.4.3, 10 pages. MR 2792159 | Zbl 1231.11029
[4] Barry, P.: On the central coefficients of Riordan matrices. J. Integer Seq. 16 (2013), Article 13.5.1, 12 pages. MR 3065330 | Zbl 1310.11032
[5] Brietzke, E. H. M.: An identity of Andrews and a new method for the Riordan array proof of combinatorial identities. Discrete Math. 308 (2008), 4246-4262. DOI 10.1016/j.disc.2007.08.050 | MR 2427755 | Zbl 1207.05010
[6] Cheon, G.-S., Jin, S.-T.: Structural properties of Riordan matrices and extending the matrices. Linear Algebra Appl. 435 (2011), 2019-2032. DOI 10.1016/j.laa.2011.04.001 | MR 2810643 | Zbl 1226.05021
[7] Cheon, G.-S., Kim, H., Shapiro, L. W.: Combinatorics of Riordan arrays with identical $A$ and $Z$ sequences. Discrete Math. 312 (2012), 2040-2049. DOI 10.1016/j.disc.2012.03.023 | MR 2920864 | Zbl 1243.05007
[8] Comtet, L.: Advanced Combinatorics. The Art of Finite and Infinite Expansions. D. Reidel Publishing, Dordrecht (1974). DOI 10.1007/978-94-010-2196-8 | MR 0460128 | Zbl 0283.05001
[9] Graham, R. L., Knuth, D. E., Patashnik, O.: Concrete Mathematics. A Foundation for Computer Science. Addison-Wesley Publishing Company, Reading (1989). MR 1001562 | Zbl 0668.00003
[10] He, T.-X.: Parametric Catalan numbers and Catalan triangles. Linear Algebra Appl. 438 (2013), 1467-1484. DOI 10.1016/j.laa.2012.10.001 | MR 2997825 | Zbl 1257.05003
[11] He, T.-X.: Matrix characterizations of Riordan arrays. Linear Algebra Appl. 465 (2015), 15-42. DOI 10.1016/j.laa.2014.09.008 | MR 3274660 | Zbl 1303.05007
[12] He, T.-X., Sprugnoli, R.: Sequence characterization of Riordan arrays. Discrete Math. 309 (2009), 3962-3974. DOI 10.1016/j.disc.2008.11.021 | MR 2537389 | Zbl 1228.05014
[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. MR 3005529 | Zbl 1292.05028
[14] Merlini, D., Rogers, D. G., Sprugnoli, R., Verri, M. C.: On some alternative characterizations of Riordan arrays. Can. J. Math. 49 (1997), 301-320. DOI 10.4153/CJM-1997-015-x | MR 1447493 | Zbl 0886.05013
[15] Merlini, D., Sprugnoli, R., Verri, M. C.: Lagrange inversion: when and how. Acta Appl. Math. 94 (2006), 233-249. DOI 10.1007/s10440-006-9077-7 | MR 2290868 | Zbl 1108.05008
[16] Młotkowski, W.: Fuss-Catalan numbers in noncommutative probability. Doc. Math., J. DMV 15 (2010), 939-955. MR 2745687 | Zbl 1213.44004
[17] Rogers, D. G.: Pascal triangles, Catalan numbers and renewal arrays. Discrete Math. 22 (1978), 301-310. DOI 10.1016/0012-365X(78)90063-8 | MR 0522725 | Zbl 0398.05007
[18] Shapiro, L. W.: A Catalan triangle. Discrete Math. 14 (1976), 83-90. DOI 10.1016/0012-365X(76)90009-1 | MR 0387069 | Zbl 0323.05004
[19] Shapiro, L. W., Getu, S., Woan, W.-J., Woodson, L. C.: The Riordan group. Discrete Appl. Math. 34 (1991), 229-239. DOI 10.1016/0166-218X(91)90088-E | MR 1137996 | Zbl 0754.05010
[20] Sprugnoli, R.: Riordan arrays and combinatorial sums. Discrete Math. 132 (1994), 267-290. DOI 10.1016/0012-365X(92)00570-H | MR 1297386 | Zbl 0814.05003
[21] Stanley, R. P.: Enumerative Combinatorics. Vol. 2. Cambridge Studies in Advanced Mathematics 62, Cambridge University Press, Cambridge (1999). DOI 10.1017/CBO9780511609589 | MR 1676282 | Zbl 0928.05001
[22] Yang, S.-L., Zheng, S.-N., Yuan, S.-P., He, T.-X.: Schröder matrix as inverse of Delannoy matrix. Linear Algebra Appl. 439 (2013), 3605-3614. DOI 10.1016/j.laa.2013.09.044 | MR 3119875 | Zbl 1283.15098
Search
Partner of
