Riordan matrices in the reciprocation of quadratic polynomials. (English) Zbl 1175.41029
The paper deals with two arithmetical triangles: The first are remainders that appear in an iterative process while the second can be interpreted as change of variables. All elements in both type are invertible Riordan arrays (Riordan matrices). In Section 2, the authors develop the theory used in the following sections. In Section 3, the authors compare the remainder of a Taylor expansion with an element of the Riordan group. Finally, they get the Pascal triangle as product of certain matrices. All their remainders are 7-matrices (see (5) from References). In Section 4 the change of variables obtained in Section 3 is studied. In Section 5 some known formulas of the sums of powers of natural numbers are recovered.
Reviewer: Dan Bārbosu (Baia Mare)
MSC:
| 41A80 | Remainders in approximation formulas |
| 05E15 | Combinatorial aspects of groups and algebras (MSC2010) |
| 05A15 | Exact enumeration problems, generating functions |
Keywords:
Banach’s fixed point theorem; reciprocal of a quadratic polynomial; Riordan matrices; change of variablesOnline Encyclopedia of Integer Sequences:
Triangle read by rows: T(n,0) = n+1, T(n,k) = 2*T(n-1,k) - T(n-1,k-1), T(n,k) = 0 if k > n and if k < 0.Riordan array T((1-x)^(-2) | 2x-1) read by rows.
References:
| [1] | The Viewpoints 2000 Group, Proof without words: geometric series, Math. Mag., 74, 4, 320 (2001) |
| [2] | Cameron, N. T.; Nkwanta, A., On some (pseudo) involutions in the Riordan Group, Journal of Integer Sequences, 8 (2005), Article 05.3.7 · Zbl 1101.05005 |
| [3] | Carroll, T.; Hilton, P.; Pedersen, J., Eulerian numbers, pseudo-Eulerian coefficients and weighted sums in Pascal’s triangle, Nieuw Arch. Wiskd., 9, 1, 41-64 (1991) · Zbl 0766.11011 |
| [4] | Comtet, L., Advances Combinatorics (1974), Reidel |
| [5] | Cheon, G.-S.; Hwang, S.-G.; -H Rim, S.; Song, S.-Z., Matrices determined by a linear recurrence relation among entries, Linear Algebra Appl., 373, 89-99 (2003) · Zbl 1026.05003 |
| [6] | Cheon, G.-S.; Kim, H., Simple proofs of open problems about the structure of involutions in the Riordan group, Linear Algebra Appl., 428, 930-940 (2008) · Zbl 1131.05006 |
| [7] | Cheon, G.-S.; Kim, H.; Shapiro, L. W., Riordan group involutions, Linear Algebra Appl., 428, 941-952 (2008) · Zbl 1131.05012 |
| [8] | J. Dugundji, A. Granas, Fixed Point Theory, Monografie Matematyczne, vol. I, Tom 61. PWN-Polish Scientific Publishers, 1982.; J. Dugundji, A. Granas, Fixed Point Theory, Monografie Matematyczne, vol. I, Tom 61. PWN-Polish Scientific Publishers, 1982. · Zbl 0483.47038 |
| [9] | Egorychev, G. P., Integral representation and the computation of combinatorial sums, Amer. Math. Soc. Transl., 59 (1984) · Zbl 0524.05001 |
| [10] | Egorychev, G. P.; Zima, E. V., Decomposition and group theoretic characterization of pairs of inverse relations of the Riordan type, Acta Appl. Math., 85, 93-109 (2005) · Zbl 1074.05012 |
| [11] | Graham, R.; Knuth, D.; Patashnik, O., Concrete Mathematics (1989), Addison-Wesley · Zbl 0668.00003 |
| [12] | Henrici, P., Applied and computational complex Analysis, Vol. I (1988), Wiley Classic Library. John Wiley and Sons · Zbl 0635.30001 |
| [13] | Hilton, P.; Holton, D.; Pedersen, J., Mathematical Reflections: From A Room with Many Windows (2002), Springer · Zbl 1011.00001 |
| [14] | Huang, I. C., Inverse relations and schauder bases, J. Combin. Theory Ser. A, 97, 203-224 (2002) · Zbl 0998.05007 |
| [15] | Knuth, D., Johann Faulhaber and sums of powers, Math. Comput., 61, 277-294 (1993) · Zbl 0797.11026 |
| [16] | Luzón, A.; Morón, M. A., Ultrametrics, Banach’s fixed point theorem and the Riordan group, Discr. Appl. Math., 156, 14, 2620-2635 (2008) · Zbl 1152.54032 |
| [17] | Merlini, D.; Rogers, D. G.; Sprugnoli, R.; Verri, M. C., On some alternative characterizations of Riordan arrays, Canad. J. Math., 49, 2, 301-320 (1997) · Zbl 0886.05013 |
| [18] | Merlini, D.; Sprugnoli, R.; Verri, M. C., The method of coefficients, Amer. Math. Monthly, 114, 40-57 (2007) · Zbl 1191.05006 |
| [19] | Nkwanta, A.; Knox, N., A note on Riordan matrices, Contemp. Math., 252 (1999) · Zbl 0942.15010 |
| [20] | Nkwanta, A., A Riordan matrix approach to unifying a selected class of combinatorial arrays, Congr. Numer., 160, 33-45 (2003) · Zbl 1042.05005 |
| [21] | Peart, P.; Woodson, L., Triple factorization of some riordan arrays, Fibonacci Quart., 31, 121-128 (1993) · Zbl 0778.05005 |
| [22] | Riordan, J., An Introduction to Combinatorial Analysis (1958), Princeton University Press · Zbl 0078.00805 |
| [23] | Robert, Alain M., A course in p-adic analysis. A course in p-adic analysis, Graduate Texts in Mathematics, vol. 198 (2000), Springer: Springer NY · Zbl 0947.11035 |
| [24] | Rogers, D. G., Pascal triangles, Catalan numbers and renewal arrays, Discrete Math., 22, 301-310 (1978) · Zbl 0398.05007 |
| [25] | Roman, S.; Rota, G., The umbral calculus, Adv. Math., 27, 95-188 (1978) · Zbl 0375.05007 |
| [26] | Shapiro, L. W.; Getu, S.; Woan, W. J.; Woodson, L., The Riordan group, Discrete Appl. Math., 34, 229-239 (1991) · Zbl 0754.05010 |
| [27] | Shapiro, L. W., Bijections and the Riordan group, Theoret. Comput. Sci., 307, 403-413 (2003) · Zbl 1048.05008 |
| [28] | Shapiro, L. W., A Catalan triangle, Discrete Math., 14, 83-90 (1976) · Zbl 0323.05004 |
| [29] | Shapiro, L. W., Some open question about random walks, involutions, limiting distributions and generating functions, Adv. Appl. Math., 27, 585-596 (2001) · Zbl 0994.05008 |
| [30] | Sprugnoli, R., Riordan arrays and combinatorial sums, Discrete Math., 132, 267-290 (1994) · Zbl 0814.05003 |
| [31] | R. Sprugnoli, Riordan arrays proofs of identities in Gould’s book, 2006. <http://www.dsi.unifi.it/resp/GouldBK.pdf>.; R. Sprugnoli, Riordan arrays proofs of identities in Gould’s book, 2006. <http://www.dsi.unifi.it/resp/GouldBK.pdf>. |
| [32] | R. Sprugnoli, A bibliogaphy on Riordan arrays, 2008. <http://www.dsi.unifi.it/resp/BibRioMio.pdf>.; R. Sprugnoli, A bibliogaphy on Riordan arrays, 2008. <http://www.dsi.unifi.it/resp/BibRioMio.pdf>. |
| [33] | Stanley, Richard P., Enumerative Combinatorics, vol. I (1997), Cambridge University Press · Zbl 0889.05001 |
| [34] | Zhao, X.; Ding, S.; Wang, T., Some summation rules related to Riordan arrays, Discrete Math., 281, 295-307 (2004) · Zbl 1042.05009 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
