Self-inverse Sheffer sequences and Riordan involutions
aa r X i v : . [ m a t h . C O ] J u l Title: ”Self-inverse Sheffer sequences and Riordan involutions.”
Authors:
Ana Luz´on* and Manuel A. Mor´on**
Address: *Departamento de Matem´atica Aplicada a los Recursos Naturales. E.T.S. I.Montes. Universidad Polit´ecnica de Madrid. 28040-Madrid, SPAIN.**Departamento de Geometria y Topologia. Facultad de Matematicas. UniversidadComplutense de Madrid. 28040- Madrid, SPAIN. e-mail: *[email protected]**[email protected]
All correspondence should be sent to:
Ana Maria Luz´on Corderoe-mail: [email protected]:Departamento de Matem´atica Aplicada a los Recursos Naturales.E.T.S.I. Montes.Universidad Polit´ecnica de Madrid.28040-Madrid, SPAINPhone number: 34 913366399Fax number: 34 915439557
Abbreviated title: ”Self-inverse sequences and involutions”.
Keywords:
Riordan group, involution, self-inverse Sheffer sequence.
MSC: ELF-INVERSE SHEFFER SEQUENCES AND RIORDANINVOLUTIONS
ANA LUZ ´ON* AND MANUEL A. MOR ´ON** *Departamento de Matem´atica Aplicada a los Recursos Naturales. E.T. Superior deIngenieros de Montes. Universidad Polit´ecnica de Madrid. 28040-Madrid, [email protected]**Departamento de Geometria y Topologia. Facultad de Matematicas. UniversidadComplutense de Madrid. 28040- Madrid, [email protected]
Abstract.
In this short note we focus on self-inverse Sheffer sequences and involutionsin the Riordan group. We translate the results of Brown and Kuczma on self-inversesequences of Sheffer polynomials to describe all involutions in the Riordan group.
Keywords: Riordan group, involution, self-inverse Sheffer sequence.MSC: 05A15, 33A70.Very recently, first in [7] later in [16], see also [11], it has been established a very closerelation between the Sheffer group and the Riordan group, see [13], [15]. In fact, there isa natural isomorphism between both groups. This means that the group properties can e translated from one group to the other equivalently. One of those properties is just thestructure of their finite subgroups.In this short note we focus on self-inverse Sheffer sequences and involutions in the Riordangroup. They determine the corresponding subgroups of order two. In fact, we translatethe results of Brown and Kuczma, [2], on self-inverse sequences of Sheffer polynomials todescribe all involutions in the Riordan group. Although the translation is almost automaticwe think that it is still interesting to point it out, because of the relations to some problemsabout involutions in the Riordan group posed in [14] that motivated the paper [3] and thathas been recently solved see [4], [5], [6]. Also in pages 2264-2265 of [10] we have somerelated results.In some sense we want to point out that some aspects of Shapiro’s problem was solved,even before it was posed, if we reinterpret this in terms of Sheffer sequences.In this paper K always represents a field of characteristic zero and N is the set of naturalnumbers including 0. The notation used herein for Riordan arrays is that introduced in[9], see also [10].Up to the inconvenience that produce the fact that we call, following [1], a generalizedAppell sequence associated to Hadamard invertible series h ( x ) = X n ≥ h n x n just to the se-quence obtained by multiplying by h n the n -term of the polynomial sequence named bythe same way in [2], we have the following obvious result. The notation used below is justthat used in [11], the operation group ♯ h is the umbral composition as it is also describedin [7] for the particular case h ( x ) = e x , and ⋆ represents the Hadamard product of series. roposition 1. Let N be a natural number. Suppose a polynomial sequence of Rior-dan type ( p n ( x )) n ∈ N and h ( x ) = X n ≥ h n x n be a series with h n = 0 ∀ n ∈ N . Consider theHadamard h -weighted sequence ( p hn ( x )) = ( p n ( x ) ⋆ h ( x )) . Then, the N -fold umbral com-position, by means of ♯ h , of the sequence ( p hn ( x )) is the neutral element in ( R h , ♯ h ) if andonly if D N = I in the Riordan group where D = ( d n,k ) is the Riordan matrix given by p n ( x ) = n X k =0 d n,k x k . Remark 2.
Note that the sequence e n ( x ) = h n x n is the neutral element in ( R h , ♯ h ).We now translate the result in [2] on Self-inverse Sheffer sequences into Riordan involu-tions. In particular, using the results of Section 3 in [2] we can give a procedure to computeall the elements T ( f | g ) of the Riordan group such that T ( f | g ) = T (1 |
1) = I . First ifwe impose g = 1, then T ( f | g ) = T (1 |
1) if and only if f = 1 or f = −
1. To constructthe remaining cases we proceed in the following way:Choose any series φ = X n ≥ φ n x n with φ = 0. Consider the Riordan matrix T (1 | φ ) . Asa consequence of the results in [8] we get that T (1 | A ) = T − (1 | φ ) where A is the socalled, [15], [12], the A -sequence of T (1 | φ ). It is clear that xA (cid:18) xφ (cid:19) = xφ (cid:16) xA (cid:17) = x , that is xA is the inverse, for the composition of the series xφ . Following [2] we take(1) g = x xA (cid:16) − xφ (cid:17) Choose now any odd power series u = X k ≥ u k − x k − , finally take(2) f = ± x xA (cid:16) − xφ (cid:17) e u ( xφ )Consequently we have roposition 3. Any Riordan involution different from the identity I = T (1 | and − I = T ( − | is of the form T ( f | g ) for f and g satisfying (2) and (1) respectively. We want to point out that the pair of series ( f, g ) above is far from being univocallydetermined by φ and u . For example Proposition 4.
Let φ = X n ≥ φ n x n be a series with φ = 0 . Suppose that xA is thecompositional inverse of xφ (equivalently A is the A -sequence of T (1 | φ ) ). Suppose alsothat g = x xA (cid:16) − xφ (cid:17) . Then g = − if and only if φ ( x ) = φ ( − x ) (i.e. φ is even).Proof. Note that φ ( x ) = φ ( − x ) if and only if the series xφ is odd.If φ is even then xφ is odd and since xA is the compositional inverse of an odd powerseries then xA is odd itself. Consequently g = −
1. On the other hand if g = − xA (cid:18) − xφ (cid:19) = − x . Composing by the right by xA we have xA ( − x ) = − xA ( − x ) = − xA ( x ) . Thisimplies that xφ is odd and the φ is even. (cid:3) In [10] we proved that for any α = 0, the Riordan matrices T ( ± | αx −
1) are involutions,see page 2265 in [10]. Now we are going to recover this result using the construction above.In fact we will get a more general class of Riordan involutions:Let α = 0 and take φ ( x ) = αx log(1 − αx )consequently A ( x ) = αx − e αx his implies that g = x xA (cid:16) − xφ (cid:17) = αx − f = ± ( αx − e u ( α log(1 − αx ) )then T ( f | g ) is an involution when u is an odd series. In particular if u ( x ) = − αx thenwe obtain that T ( ± | αx −
1) is a Riordan involution.From this point of view the fact that Pascal triangle is a pseudo-involution, [3], is equiv-alent to the fact that the classical Laguerre polynomials are self-inverse see [2].
Acknowledgment:
The first author was partially supported by DGES grant MICINN-FIS2008-04921-C02-02. The second author was partially supported by DGES grant MTM-2006-0825.
References [1] R.P. Boas, R.C. Buck.
Polynomial expansions of analytic functions.
Springer-Verlag(1964.)[2] J.W. Brown, M. Kuczma.
Self-inverse Sheffer sequences.
SIAM J. Math. Anal. Vol. 7,No 3 October 1976 (723-726).[3] N. T. Cameron. and A. Nkwanta.
On some (pseudo) involutions in the Riordan Group.
Journal of Integer Sequences. Vol. 8 (2005) Article 05.3.7.[4] G.-S. Cheon, H. Kim.
Simple proofs of open problems about the structure of involutionsin the Riordan group.
Linear Algebra Appl. 428 (2008) 930-940.
5] G.-S. Cheon, H. Kim, L.W. Shapiro.
Riordan group involutions.
Linear Algebra Appl.428 (2008) 941-952.[6] G.-S. Cheon, S.-T. Jin, H. Kim, L.W. Shapiro.
Riordan group involutions and the ∆ -sequence. Discrete Applied Mathematics. 157 (8) (2009) 1696-1701 .[7] T-X. He, L.C. Hsu, P.J-S. Shiue.
The Sheffer group and the Riordan group.
DiscreteApplied Mathematics. 155 2007 (1895-1909).[8] A. Luz´on.
Iterative processes related to Riordan arrays: Thereciprocation and the inversion of power series.
Preprint(http://matematicas.montes.upm.es/ana/articulos/2ways.pdf).[9] A. Luz´on and M. A. Mor´on.
Ultrametrics, Banach’s fixed point theorem and the Riordangroup.
Discrete Appl. Math. 156 (2008) 2620-2635.[10] A. Luz´on and M. A. Mor´on.
Riordan matrices in the reciprocation of quadratic poly-nomials.
Linear Algebra Appl. 430 (2009) 2254-2270.[11] A. Luz´on and M. A. Mor´on.
Recurrence relations for polynomial sequences via Riordanmatrices.
Preprint (ArXiv: 0904.2672) .[12] D.G. Rogers.
Pascal triangles, Catalan numbers and renewal arrays.
Discrete Math.22 (1978) 301-310.[13] L. W. Shapiro, S. Getu, W.J. Woan and L. Woodson.
The Riordan group.
DiscreteAppl. Math. 34 (1991) 229-239.[14] L. W. Shapiro.
Some open question about random walks, involutions, limiting distri-butions, and generating functions.
Adv. Appl. Math. 27 (2001) 585-596.
15] R. Sprugnoli.
Riordan arrays and combinatorial sums.
Discrete Math. 132 (1994)267-290.[16] W. Wang, T. Wang.
Generalized Riordan arrays.