Discrete Applied Mathematics 159 (2011) 1290–1292 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Note Self-inverse Sheffer sequences and Riordan involutions Ana Luzón a,∗, Manuel A. Morón b a Departamento de Matemática Aplicada a los Recursos Naturales, E.T. Superior de Ingenieros de Montes, Universidad Politécnica de Madrid, 28040-Madrid, Spain b Departamento de Geometria y Topologia, Facultad de Matematicas, Universidad Complutense de Madrid, 28040- Madrid, Spain article info a b s t r a c t Article history: In this short note, we focus on self-inverse Sheffer sequences and involutions in the Riordan Received 23 October 2009 group. We translate the results of Brown and Kuczma on self-inverse sequences of Sheffer Received in revised form 7 December 2010 polynomials to describe all involutions in the Riordan group. Accepted 25 March 2011 ' 2011 Elsevier B.V. All rights reserved. Available online 8 May 2011 Keywords: Riordan group Involution Self-inverse Sheffer sequence Very recently, first in [7] and later in [17], see also [12], a very close relation has been established between the Sheffer group and the Riordan group; see [15,16]. In fact, there is a natural isomorphism between both groups. This means that the group properties can be translated from one group to the other equivalently. One of those properties is the structure of their finite subgroups. In this short note, we focus on self-inverse Sheffer sequences and involutions in the Riordan group. They determine the corresponding subgroups of order 2. In fact, we translate the results of Brown and Kuczma [2] on self-inverse sequences of Sheffer polynomials to describe all involutions in the Riordan group. Although the translation is almost automatic, we think that it is still interesting to point it out, because of the relations to some problems about involutions in the Riordan group posed in [14] that motivated the paper [3] and that have recently been solved in [5]; see also [6,4]. In [8], the authors give some properties of the so-called A and Z sequences of any Riordan involution. There are also some related results on pages 2264–2265 of [11]. In a sense, we want to point out that some aspects of Shapiro's problem were solved, even before it was posed, if we reinterpret it in terms of Sheffer sequences. In this paper, K always represents a field of characteristic zero and N is the set of natural numbers including 0. The notation used herein for Riordan arrays is that introduced in [10]; see also [11]. The natural translation from our notation to the classical notation is given by the formula f .t/ t T .f jg/ D ; (1) g.t/ g.t/ with g.0/ 6D 0. Up to the inconvenience that produces the fact that we call, following [1], a generalized Appell sequence associated D P n to Hadamard invertible series h.x/ n≥0 hnx simply the sequence obtained by multiplying by hn the n-term of the polynomial sequence named in the same way in [2], we have the following obvious result. The notation used below is that ∗ Corresponding author. Tel.: +34 913 366 399; fax: +34 915 439 557. E-mail addresses: [email protected] (A. Luzón), [email protected] (M.A. Morón). 0166-218X/$ – see front matter ' 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.dam.2011.03.025 A. Luzón, M.A. Morón / Discrete Applied Mathematics 159 (2011) 1290–1292 1291 x used in [12], the operation group ]h is the umbral composition as is also described in [7] for the particular case h.x/ D e , and ? represents the Hadamard product of series. D P n Proposition 1. Let N be a natural number. Suppose a polynomial sequence of Riordan type .pn.x//n2N and let h.x/ n≥0 hnx 6D 8 2 h D be a series with hn 0 n N. Consider the Hadamard h-weighted sequence .pn.x// .pn.x/? h.x//. Then, the N-fold umbral h N D composition, by means of ]h, of the sequence .pn.x// is the neutral element in .Rh;]h/ if and only if D I in the Riordan group, D D Pn k where D .dn;k/ is the Riordan matrix given by pn.x/ kD0 dn;kx . n Remark 2. Note that the sequence en.x/ D hnx is the neutral element in .Rh;]h/. We now translate the results in [2] on self-inverse Sheffer sequences into Riordan involutions. In particular, using the results in Section 3 of [2], we can give a procedure to compute all the elements T .f j g/ of the Riordan group such that T 2.f j g/ D T .1 j 1/ D I. First, if we impose g D 1, then T 2.f j g/ D T .1 j 1/ if and only if f D 1 or f D −1. To construct the remaining cases we proceed in the following way. D P n 6D j Choose any series φ n≥0 φnx with φ0 0. Consider the Riordan matrix T .1 φ/. As a consequence of the results in [9], we get that T .1 j A/ D T −1.1 j φ/, where A is the so-called A-sequence of T .1 j φ/; see [16,13]. It is clear that x x D x x D x, that is, x is the inverse for the composition of the series x . Following [2], we take A φ φ A A φ x g D : (2) x − x A φ D P 2k−1 Choose now any odd power series u k≥1 u2k−1x , and take ±x x u φ f D e : (3) x − x A φ Consequently, we have the following. Proposition 3. Any Riordan involution different from the identity I D T .1 j 1/ and from −I D T .−1 j 1/ is of the form T .f j g/ for f and g satisfying (3) and (2), respectively. We want to point out that the pair of series f and g above is far from being univocally determined by φ and u. For example, consider the following. Proposition 4. Let D P xn be a series with 6D 0. Suppose that x is the compositional inverse of x (equivalently that φ n≥0 φn φ0 A φ x A is the A-sequence of T .1 j φ/). Suppose also that g D . Then g D −1 if and only if φ.x/ D φ.−x/ (i.e., φ is even). x − x A φ D − x Proof. Note that φ.x/ φ. x/ if and only if the series φ is odd. If is even, then x is odd. Since x is the compositional inverse of an odd power series, x is odd itself. Consequently, φ φ A A − g D −1. On the other hand, if g D −1, then x − x D −x. Composing the right-hand side by x , we have x −x D x D A φ A A . / A.−x/ − x . This implies that x is odd, and then is even. A.x/ φ φ As we proved on page 2265 of [11], the Riordan matrices T .±1 j αx − 1/ are involutions for any α 6D 0. 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 6D 0, and take αx φ.x/ D : log.1 − αx/ Consequently, αx A.x/ D : 1 − eαx This implies that x g D D αx − 1: x − x A φ We know that, if u 1 log.1−αx/ f D ±(αx − 1/e α ; 1292 A. Luzón, M.A. Morón / Discrete Applied Mathematics 159 (2011) 1290–1292 then T .f j g/ is an involution when u is an odd series. In particular, if u.x/ D −αx, then we obtain that T .±1 j αx − 1/ is a Riordan involution. From this point of view we have the following. The fact that the Pascal triangle is a pseudo-involution (see [3]) is equivalent to the fact that the classical family of Laguerre polynomials is self-inverse (see [2]). Remark 5. Theorem 4.3 in [8] gives the expressions of the A-sequence and the Z-sequence of any Riordan involution. Using this theorem and the description of the involutions given in Proposition 3 above, we get that, if T .fQjgQ/ is an involution and AQ u x Q x Q ±xe φ Q x and Z are the corresponding A-sequence and Z-sequence, respectively, then gQ.x/ D , f .x/ D , A.x/ D x − x x − x x − x A φ A φ A φ u x Q δ±e φ and Z.x/ D , where φ is a series with φ.0/ 6D 0, A is the A-sequence of T .1jφ/, and u is any odd power series. So the x − x A φ u x non-trivial Riordan involutions in the classical notation are ±e φ x − x , or equivalently ±eu.!.x// −1 − x , ; A φ . ;! . !. /// where ! is any power series invertible for the composition operation. Moreover, using Theorem 4.3 in [8], we obtain − u.!.x// AQ.x/ D x and ZQ.x/ D ± 1 e . !−1.−!.x// !−1.−!.x// The results in Theorem 4.3 in [8] are also reflected in the formula T −1.f j g/ D T g0 .A − xZ/ A , which appears f0 in [9].