1 2 Journal of Integer Sequences, Vol. 11 (2008), 3 Article 08.5.3 47 6 23 11 Riordan Arrays, Sheffer Sequences and “Orthogonal” Polynomials Giacomo Della Riccia Dept. of Math. and Comp. Science - Research Center Norbert Wiener University of Udine Via delle Scienze 206 33100 Udine Italy [email protected] Abstract Riordan group concepts are combined with the basic properties of convolution fam- ilies of polynomials and Sheffer sequences, to establish a duality law, canonical forms n m k k+1 x−k ρ(n, m) = m c Fn−m(m), c 6= 0, and extensions ρ(x,x − k) = (−1) x c Fk(x), where the Fk(x) are polynomials in x, holding for each ρ(n, m) in a Riordan array. Ex- n amples ρ(n, m)= m Sk(x) are given, in which the Sk(x) are “orthogonal” polynomials currently found in mathematical physics and combinatorial analysis. 1 Introduction We derive from basic principles in the theory of convolution families [10] and Sheffer se- n m quences [16] of polynomials, canonical forms ρ(n, m) = m c Fn−m(m), c =6 0, and exten- sions ρ(x, x − k) = xkcx−kF (x − k)/k!=(−1)kxk+1cx−kρ (x), holding for all elements ρ k k in the Riordan group. We show in Section 3 that the extensions are of polynomial type when c = 1. In Section 2, we define transformation rules and a duality law, that will greatly n simplify algebraic manipulations. In Section 4, we give examples ρ(n, m) = m Sn−m(m), in which S (x) are “orthogonal” polynomials currently found in mathematical physics and k combinatorial analysis. The concluding remarks are in Section 5. 1 For easy reference, we recall some definitions. The Riordan group is a set of invertible infinite lower triangular matrices M = {ρ(n, m)}n,m≥0, called Riordan matrices, with entries: un f(u)m n! f(u) m ρ(n, m)= g(u) = [un−m]g(u) , ρ(0, 0) = 1, (1) n! m! m! u un f(u)m equivalently, ρ(n, m) = g(u) (“exponential” Riordan group), n! m! n≥m≥0 X 2 n where g(u)=1+ g1u + g2u + . is invertible, with g(u) = 1/g(u)= n≥0 gnu , and f(u)= f u + f u2 + . is a delta series with compositional inverse f(u), f(f(u)) = f(f(u)) = u. 1 2 P A group element is denoted by ρ = (g(u), f(u))b and the group law ⋆ bis (l(u), h(u)) ⋆ (g(u), f(u)) = (l(u)g(h(u)), f(h(u)), with identity I = (1, u) and group inverse ρ−1 = (1/g(f(u)), f(u)). A group representation ρ = ρ1 ⋆ρ2, in matrix notation, is ρ(n, m) = i ρ1(n, i)ρ2(i,m). As we know, group representations are extensively used in the study of identities [13, 18, 14, 15] and they will play an essential role in this work. A sequence P k x ak(x), given by k ak(x)u = a(u) , is a convolution family of polynomials iff a(0) = 1 [10]. In a convolution family the polynomials ak(x) are multiples of x for k > 0, having P degree ≤ k, and xa0(x) ≡ 1. A Sheffer sequence Sk(x) for (g(u), f(u)) [16] is given by k xf(u) k≥0 Sk(x)u /k! = (1/g(f(u)))e , where (g(u), f(u)) are as in Riordan group theory; the Sheffer sequence for f = (1, f(u)) is the associated sequence for f(u), and the Sheffer sequenceP for (g(u), u) is the Appell sequence for g(u). 2 Transformation Rules and the Duality Law We first define some useful transformation rules, noting that transformation rules and Barry’s “transforms” [2] are different concepts. 1) The duality rule “∼” for a function f(n, m), n,m integers, is f(n, m) = f(−m, −n) and for a function fk(x), the rule is fk(x)= fk(k − x). When x = n and k = n − m, the two rules coincide. The following special cases will be used in the sequel, withoute reference. e ^ n n −m n − 1 = = =(−1)n−m , n ≥ m ≥ 0, m n − m n − m m − 1 g n! ] n! m = nn−m =(−m)n−m =(−1)n−m(n − 1)n−m =(−1)n−m , m! m! n f^ n! ^ n! = nn−m+1 =(−1)n−m+1 . (m − 1)! (m − 1)! The dual ρ(n, m) of ρ(n, m) and the dual of a group representation ρ = ρ1 ⋆ρ2 are n! m f(u) −n eρ(n, m)= ρ(−m, −n)=(−1)n−m [un−m]g(u) from (1), m! n u ρe(n, m)= ρ1(−m, −i)ρ2(−i, −n)= ρ2(n, i)ρ1(i,m), ρ = ρ2 ⋆ ρ1, i i X X e e e e e e 2 after a change i → −i; this change is legal because we are allowed to use summation i over an unbounded range. 2) For any number µ, the scaling rule “(µ)” is a group automorphism: P u f(µu) (µ)ρ = (1, µu) ⋆ ρ ⋆ 1, = g(µu), , lim(µ)(g(u), f(u)) = (1, u)= I, µ µ µ→0 n−m −1 −1 ((µ)ρ)(n, m)= µ ρ(n, m), ((µ)ρ) =(µ)ρ , (µ)(ρ1 ⋆ρ2)=(µ)ρ1 ⋆ (µ)ρ2. 3) The negation rule “(-)” is the special scaling µ = −1: (−)ρ =(g(−u), −f(−u)): ((−)ρ)(n, m)=(−1)n−mρ(n, m). Duality and negation applied to (1) and Lagrange’s inversion formula yield, respectively, n! m f(u) −n (−1)n−mf(n, m) = (−1)n−mf(−m, −n)= [un−m] , m! n u n! n! m f(u) −n f −e1(n, m) = [un]f(u)m = [un−m] . m! m! n u Combining the two formulas, we find a duality law in the associated subgroup {(1, f(u))}: f −1(n, m)=(−1)n−mf(n, m), f −1 =(−)f, (2) For g in the Appell subgroup {(g(u), u)} andeρ−1 = f −1 ⋆g−1 =(−e)f⋆g−1, (1) yields: n! n n g(n, m)= g = (n − m)!g , g−1(n, m)= e (n − m)!g , m! n−m m n−m m n−m n−m −1 i n−m−i b ρ (n, m)= (−1) f(n, n − i)(n − i) gn−m−i. i=0 X e b 3 Canonical Forms and Extensions Let us write the delta series f(u) in f = (1, f(u)) and the defining relation (1) as: f(u)= uca(u)= uc(1 + a1u + ··· ), c =6 0, n un−m f(u) m n ρ(n, m)= cm g(u) = cmF (m), (3) m (n − m)! cu m n−m n m and call c =6 0 the weight of f, {can} the f-reference sequence (f -refseq) and m c Fn−m(m) the canonical form of ρ(n, m). The remarkable structure of canonical forms, namely, a m binomial coefficient multiplied by a factor c Fn−m(m), implies the following. n m Proposition 1. A numerical array with entries ρ(n, m) = m c Fn−m(m) is a Riordan matrix iff the F (x) are polynomials forming a Sheffer sequence. k 3 Proof. If the ρ(n, m) are numbers related to an element ρ =(g(u), f(u)), then according to (3) we can write m f u Fn−m(m) n−m m ln ( ) f(u) u = g(u)e ( cu ) = g(u) , (n − m)! cu n−m X k x ln(f(u)/cu) thus the Fk(x) form a Sheffer sequence given by k Fk(x)u /k! = g(u)e . Con- versely, if the above Sheffer sequence is given, then, by letting x = m, we obtain (3), proving that the ρ(n, m) define a Riordan matrix. P Corollary 2. Each ρ(n, m) in a Riordan matrix has an extension (−1)k F (x − k) ρ(x, x − k)=(−1)kxk+1ρ (x), ρ (x)= cx−k k , (4) k k k! x − k which is of polynomial type when c = 1. Proof. Since Fn−m(m) is a polynomial in m, we can put in the canonical form n − m = k, n = x and m = x − k, thus obtaining immediately well-defined extensions (4), which are clearly of polynomial type when c = 1. k Since f(u)/cu = a(u) is invertible and a(0) = 1, the coefficients (−1) xfk(x + k) in the x x k k power series expansion of (f(u)/cu) , written as (f(u)/cu) = x k(−1) fk(x + k)u , form a convolution family. Similarly, (1) written for f −1(n, m)=(−1)n−mf(n, m), with u → cu P and m = −x, implies: m e cf(u) un−m = −m (−1)n−mf (−m) , u n−m cn−m n−m≥0 X −x −x f(cu) f(u)/c k k u = = x (−1) fk(x)u , cf f = u, u u ! c Xk≥0 k+1 x showing that the (−1) xfk(−x) form a convolution family for (f(u)/c/u) . Moreover, n! f(n, m) = (−1)n−m cmf (n), (m − 1)! n−m n! f −1(n, m) = (−1)n−mf(n, m)=(−1)n−m+1 c−nf (−m) (m − 1)! n−m x−k k k+1 −1 −x k+1 k+1 f(x, x − k) = c (−1)ex fk(x), f (x, x − k)= c (−1) x fk(k − x).