Linear and Multilinear Algebra ISSN: 0308-1087 (Print) 1563-5139 (Online) Journal homepage: http://www.tandfonline.com/loi/glma20 A linear algebra setting for the rota-mullin theory of polynomials of binomial type Jay P. Fillmore & S.G. Williamson To cite this article: Jay P. Fillmore & S.G. Williamson (1973) A linear algebra setting for the rota- mullin theory of polynomials of binomial type, Linear and Multilinear Algebra, 1:1, 67-80, DOI: 10.1080/03081087308817006 To link to this article: http://dx.doi.org/10.1080/03081087308817006 Published online: 03 Apr 2008. Submit your article to this journal Article views: 16 View related articles Citing articles: 18 View citing articles Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=glma20 Download by: [University of California, San Diego] Date: 28 June 2016, At: 11:21 Linear and Multilinear Algebra, 1973, Vol. 1, pp. 67-80 0 Gordon and Breach Science Publishers Ltd. Printed in Great Britain A Linear Algebra Setting for the Rota-Mullin Theory of Polynomials of Binomial Type JAY P. FILLMORE and S. G. WILLIAMSONI- University of California, San Diego (Received February 28,1972) The linear algebra and combinatorial aspects of the Rota-Mullin theory of poly- nomials of binomial type are separated and the former is developed in terms of shift operators on in6nite dimensional vector spaces with a view towards application in the calculus of finite differences. 1. INTRODUCTION A sequence of polynomials {p,(x)), the nth polynomial of which has degree n and coefficients in a field F of characteristic zero, is said to be of binomial type if for every n > 0, where x and y are indeterminates over F and (9 is the binomial coefficient. ([2] p. 169; [I].) Assuming that po(x) is a nonzero Downloaded by [University of California, San Diego] at 11:21 28 June 2016 constant, one has that po(x) = 1 and pn(0) = 0 for every n > 1. Sequences of polynomials of binomial type are characterized by the fact that they are obtained from generating functions of the form exp(xg(u)), where u is an indeterminate and g(u) is a formal power series in u whose constant term is zero and whose coefficient of u is not zero; namely, one writes t Second author supported by AFOSR grant 71-2089. 67 68 J. P. FILLMORE AND S. G. WILLIAMSON in the ring of formal power series in x and u with coefficients in P ([2] p. 189, Cor. 2 to Th. 3; [I], Th. 4.1). Indeed, if {pn(x))is defined by such a relation, then exp((x -I-y)g(u)) = exp(xg(u))exp(yg(u)) implies that it is of binomial type. Conversely, if a sequence of polynomials is of binomial type, and if the series it defines as in the left-hand side of the above equation is written as the exponential of a series in x and u, then this last series is easily shown to be of the form xg(u) with g(u) as previously described. Let G be the set of all formal power series in u with coefficients in P whose constant term is zero and whose coefficient of u is not zero. Under com- position of series, (h o g)(u) = h(g(u)), G becomes a group which we call the umbra1 group. The inverse of an element g of G is denoted by g-l, so that g-l(g(u)) = u and g(g-l(u)) = u. The characterization of sequences of poly- nomials of binomial type in terms of generating functions exp(xg(u)) tells us that each such sequence of polynomials {p,(x)} corresponds to a unique series g(u) in G; namely, one rewrites Some examples of sequences of polynomials of binomial type are : ij {xn}, obtained using g(u) = u in the generating function exp(xg(u)). ii) {(x),}, where (x), = x(x - l)(x - 2) - - (x - n + 1) are the lower factorials, obtained using iii) {L,(x)), where Ln(x) = ~e"(d/dx)~(e-~x"-~)are Laguerre polynomials, obtained using g(u) = u/(u - 1) = -u - u2 - u3 - . .. Let D denote the operator d/dx on polynomials and I the identity operator. One observes in these examples that: and g-'(u) = u is the inverse of g(u) = u in G. Downloaded by [University of California, San Diego] at 11:21 28 June 2016 ii) (eD - I) = -(X)n-l and g-'(u) = eu - 1 is the inverse of (0") (n - I)! D Ln-~(x) u iii> -("""') - = -and g-'(u) = -is the inverse of D-I n! (n-I)! u - 1 THE ROTA-MULLIN THEORY 69 In each case one sees that, if {pn(x))is the sequence of polynomials of binomial type which corresponds to the series g(u) in G, the effect of the operator g-'(D) is to shift the sequence of polynomials {pn(x)/n!): for all n 2 1. The sequence of polynomials {pn(x)/n!)is a basis for the inlinite dimensional vector space over F of polynomials in x with coefficients in F. From these examples it is clear that an appropriate setting for the theory of polynomials of binomial type is the study of shift operators on infinite dimensional vector spaces. Indeed, in Section 2 of this paper we develop a theory of shift operators and show how the results of the Rota-Mullin theory can be proved efficiently from this view-point. In Section 3, we discuss several examples, including those above. Finally, in Section 4, we provide a glossary connecting our terminology with that of Rota and Mullin [12]. We refer the reader to Garsia [I] for an excellent exposition of the work of Rota and Mullin and for the connection between "operator theoretic" and "generating function" techniques in this theory. 2. SHIFTS ON VECTOR SPACES Let F be a field of characteristic zero, and V be a vector space of countably infinite dimension over F. 2.1. DEFINITION a) A linear operator S on V is called a shift on V if the dimension of the null-space of Sn is n for every n > 0 and if the union of all these null-spaces is V. b) A sequence of vectors {en} (0 < n < m) is called a shift basis for S if Sen = en-, for n 2 1 and Se, = 0. By So we understand the identity operator I on V. A shift basis for S is in fact a basis for the vector space V; e,, el, . ., en is a basis for the null- , Downloaded by [University of California, San Diego] at 11:21 28 June 2016 space of Snfl. Let E be an evaluation on V, that is, a nonzero linear functional on V with values in Fa 2.1. DE~NITION(continued) c) A shift basis {en}for S is said to be normalized, with respect to the evaluation E, ife(en)= 0 for n 2 1 and ~(e,)= 1. For a normalized shift basis for S to exist, it is necessary that 8 not vanish on the null-space of S. When V is the vector space of polynomials in x with coefficients in F, 70 J. P. FILLMORE AND S. G. WILLIAMSON certain formal power series in the operator D = d/dx correspond to shifts, the evaluation E corresponds to evaluating a polynomial at x = 0, and normalized shift bases are sequences of polynomials {pn(x)/n!)where (pn(x)) is a sequence of polynomials of binomial type. CQ If anun is any formal power series in the indeterminate u, then the n= 0 m operator anSn is well-defined; for, every vector of V is annihilated by n=O some power of S. We will say that such an operator is a formal power series in S with constant term ao. 2.2. PROPOSITIONa) If S is a shift on V, then a shift basis {en)for S exists. If {en)and {e;) are two shift bases for S, then e:, = Tenfor every n 2 0, where T is a formal power series in S with non-zero constant term. b) IfS is a shift on V such that the evaluation E does not vanish on the null-space of S, then a normalized shift basis {en)for S exists and is unique. Remark Trivially, if Tis a formal power series in S with nonzero constant term and if {en)is a shift basis for S then so also is {Ten). Proof a) Let Nnbe the null-space of Sn.Clearly SsendsN,,, into Nn E N,,,. S, when viewed as a map of N,,, into itself, has nullity one, so in fact maps N,,, onto Nn. Choose eo # 0 arbitrarily in N, and, inductively, en in N,,, so that Sen = en-,. {en)is a shift basis for S. Suppose that {e;) is a second shift basis for S. N1 is one-dimensional and spanned by eo, so eb = aoeowith a. $: 0 in F. Now S(ei - aoe,) = eb - aoeo = 0, so ei - aoel is in N, and equals aleo with a, in F. Proceed inductively to obtain e; - aoen - alen-, - . - - ~,,-~e,= aneo. Then b) Let Vo be the null space of E. Then V = Vo + Nl and, since Nl c N,,,, Nn+, = (N,,, fl Vo)+ N,. Since S maps Nl to zero, it maps N,,, fl V, onto Nn. Choose eo in N, such that ~(e,)= 1 and, inductively, en in N,,, fl Vo so that Sen = en-,.