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Riordan matrices in the reciprocation of quadratic polynomials

Luzón, Ana and Morón, Manuel A. (2009) Riordan matrices in the reciprocation of quadratic polynomials. Linear Algebra and its Applications, 430 (8-9). pp. 2254-2270. ISSN 0024-3795

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Abstract

We iterate contractive one-degree polynomials with coefficients in the ring K[[x]] of formal power series to calculate the reciprocal in K[[x]] of a quadratic polynomial. Doing this we meet thestructure of Riordan array. We interpret certain changes of variable as a Riordan array. We finish the paper by using our techniques to find new ways to get known formulas for the sum of powers of natural numbers involving Stirling and Eulerian numbers.

Item Type:Article
Uncontrolled Keywords:Inverse relations; arrays; involutions; sums; Banach's fixed point theorem; reciprocal of a quadratic polynomial; Riordan matrices; changes of variables
Subjects:Sciences > Mathematics > Algebra
ID Code:15132
References:

The Viewpoints 2000 Group, Proof without words: geometric series, Math. Mag. 74 (4) (2001) 320.

N.T. Cameron, A. Nkwanta, On some (pseudo) involutions in the Riordan Group, Journal of Integer Sequences 8 (2005), Article 05.3.7

T. Carroll, P. Hilton, J. Pedersen, Eulerian numbers, pseudo-Eulerian coefficients and weighted sums in Pascal’s triangle, Nieuw Arch.Wiskd. 9 (1) (1991) 41–64.

L. Comtet, Advances Combinatorics, Reidel, 1974.

G.-S. Cheon, S.-G. Hwang, S.-H. Rim, S.-Z. Song, Matrices determined by a linear recurrence relation among entries, Linear Algebra Appl. 373 (2003) 89–99.

G.-S. Cheon, H. Kim, Simple proofs of open problems about the structure of involutions in the Riordan group,Linear Algebra Appl. 428 (2008) 930–940.

G.-S. Cheon, H. Kim, L.W. Shapiro, Riordan group involutions, Linear Algebra Appl. 428 (2008) 941–952.

G.-S. Cheon, H. Kim, Simple proofs of open problems about the structure of involutions in the Riordan group,Linear Algebra Appl. 428 (2008) 930–940.

G.-S. Cheon, H. Kim, L.W. Shapiro, Riordan group involutions, Linear Algebra Appl. 428 (2008) 941–952.

J. Dugundji, A. Granas, Fixed Point Theory, Monografie Matematyczne, vol. I, Tom 61. PWN–Polish Scientific Publishers, 1982.

[9] G.P. Egorychev, Integral representation and the computation of combinatorial sums, Amer. Math. Soc. Transl. 59 (1984).

G.P. Egorychev, E.V. Zima, Decomposition and group theoretic characterization of pairs of inverse relations of the Riordan type, Acta Appl. Math. 85 (2005) 93–109.

R. Graham, D. Knuth, O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989.

P. Henrici, Applied and computational complex analysis, Wiley Classic Library, John Wiley and Sons, 1988.

P. Hilton, D. Holton, J. Pedersen, Mathematical Reflections: From A Room with ManyWindows, Springer, 2002.

I.C. Huang, Inverse relations and Schauder bases, J. Combin. Theory Ser. A 97 (2002) 203–224.

D. Knuth, Johann Faulhaber and sums of powers, Math. Comput. 61 (1993) 277–294.

A. Luzón, M.A. Morón, Ultrametrics, Banach’s fixed point theorem and the Riordan group,Discr. Appl. Math. 156 (14) (2008) 2620–2635.

D. Merlini, D.G. Rogers, R. Sprugnoli, M.C. Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math. 49 (2) (1997) 301–320.

D. Merlini, R. Sprugnoli, M.C. Verri, The method of coefficients, Amer. Math. Monthly 114 (2007) 40–57.

A. Nkwanta, N. Knox, A note on Riordan matrices, Contemp. Math. 252 (1999).

A. Nkwanta, A Riordan matrix approach to unifying a selected class of combinatorial arrays, Congr. Numer. 160 (2003) 33–45.

P. Peart, L.Woodson, Triple factorization of some riordan arrays, Fibonacci Quart. 31 (1993) 121–128.

J. Riordan, An Introduction to Combinatorial Analysis, Princeton University Press, 1958.

Alain M. Robert, Graduate Texts in Mathematics, Springer, NY, 2000.

D.G. Rogers, Pascal triangles, Catalan numbers and renewal arrays, Discrete Math. 22 (1978) 301–310.

S. Roman, G. Rota, The umbral calculus, Adv. Math. 27 (1978) 95–188.

L.W. Shapiro, S. Getu,W.J.Woan, L.Woodson, The Riordan group, Discrete Appl. Math. 34 (1991) 229–239.

L.W. Shapiro, Bijections and the Riordan group, Theoret. Comput. Sci. 307 (2003) 403–413.

L.W. Shapiro, A Catalan triangle, Discrete Math. 14 (1976) 83–90.

L.W. Shapiro, Some open question about random walks, involutions, limiting distributions and generating functions, Adv. Appl. Math. 27 (2001) 585–596.

R. Sprugnoli, Riordan arrays and combinatorial sums, Discrete Math. 132 (1994) 267–290.

R. Sprugnoli, Riordan arrays proofs of identities in Gould’s book, 2006. <http://www.dsi.unifi.it/resp/GouldBK.pdf>.

R. Sprugnoli, A bibliogaphy on Riordan arrays, 2008. <http://www.dsi.unifi.it/resp/BibRioMio.pdf>.

Richard P. Stanley, Enumerative Combinatorics, vol. I, Cambridge University Press, 1997.

X. Zhao, S. Ding, T.Wang, Some summation rules related to Riordan arrays, Discrete Math. 281 (2004) 295–307

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