A Linear Algebra Setting for the Rota-Mullin Theory of Polynomials of Binomial Type

Linear and Multilinear Algebra

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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

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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 polynomials 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 polynomials 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-,. {en)is a normalized shift basis for S. Furthermore, since Downloaded by [University of California, San Diego] at 11:21 28 June 2016 Nn+, fl Vo and Nn both have dimension n, S maps Nn,, n Vo isomorphically onto Nn. Hence, the choices are unique at each step, and the normalized shift basis for S is unique. Q.E.D. When speaking of normalized shift bases, we will tacitly assume that the conditions of (2.2) are satisfied. 2.3. Remark If T is a linear operator on V which commutes with a shift S on V, then T is a formal power series in S. For, let {en)be a shift basis for THE ROTA-MULLIN THEORY 71 S, and set e:, = Ten. Then Se:, = e:,-, for n > 1. The argument used in the proof of (2.2a) shows that e:, = Ye, where T' is a formal power series in S. T = T' since {en) is a basis for V. Note that {e;) will not be a basis for V unless T is invertible. Let S be a shift on V, and (en) a shift basis for S. If U is an invertible linear operator on V, it is clear that USU-I is a shift on V and {Ue,) is a shift basis for this shift. If Uis an operator for which USU-' admits an explicit description in terms of S, then this shift and its shift basis are immediately related by this description. This is the case for certain operators U,, g(u) a series in the umbral group G, which we now construct. Let F be the ring of formal power series in u with coefficients in F; let V be the F-module of formal power series in u with coeficients in V. An m element x vnunE V,vn E V, is multiplied by a series in F in the obvious way. n=O m OD Let g E G; for Z = x anunE F,define A,@) = an(g(u))"; similarly define n=O n=O A,@) for fi E V. We have immediately that A,@&) = Ag(Z)Ag(&)for 5, & E F and A,(%) = Ag(Z)A,(E) for Z E F and 6 E V. Furthermore, Ahog= AgAh for g, h E G. If T is any linear operator on V, extend T to VF-linearly ;that is, T is applied to an element of V by applying it to the coefficients of the element. Such an extension Tcommutes with the operators A,: A,T = TA,. m Let S be a shift on V, (en) a shift basis for S. Set 2 = 1 enunin 7? There n=O is a unique linear operator U, on V whose extension to Tin the above manner satisfies UgZ = A$ for this particular 2; for A,Z specifies what Ugenis to be for every n > 0. U, of course depends on the choice of shift basis for S, but we make this unique by choosing the normalized shift basis for S. Observe that Uh., = UhUgfor g, h E G; for Uh,,E = Ah,,Z = A,A,,Z = AgUhZ= UhA,Z = UhUgZ.That is, the U, constitute a representation of G. 2.4. DEFINITIONThe operators U,, g E G, constructed above, are called umbral operators ([2] p. 199; [I] Def. 2.2). Remark Using the above notions, we can give an alternate proof of Downloaded by [University of California, San Diego] at 11:21 28 June 2016 (2.2b). Let {en}be a shift basis for S for which ~(e,) Z 0 and set I = 'f enun. n=O Since &((l/rZ)Z)= 1, {e:,}, when (l/eZ)E = e:,un in is a normalized shift n=O basis for S. Uniqueness is similarly argued. 2.5. THEOREM([2] p. 200, Th. 6; [I], Th. 2.1 and Remark 2.1.) Let S be a shift on V. Then the umbral operators U,, g E G, satisfy U,SU;I = g-l(S). 72 J. P. FILLMORE AND S. G. WILLIAMSON

m Proof Let E = enun where {en)is any shift basis for S. When S is n=O extended to V, we have SE = uB, and consequently g-l(S)Z = g-'(u)E. If we apply A, to this last equation and observe that A, and g-'(S) commute, we obtain g-'(S) U,B = g-'(S)A,E = A,g-'(S)E = Ag(g-'(u)S) = A,@-'(u))A,(E) = u UgZ. If e; = Ugen,this equation tells us that g-l(S)e,: = e;-, for n 1. But UgSU;'e; = e;-, also. Since U, is invertible, (e;) is a basis for V. Since g-l(S) and UgSUilcoincide on this basis, they are equal. Q.E.D. The main application of this theorem will be the fact that, if {en)is a shift basis for S, (Ugen}is a shift basis for g-'(27). 2.6. Remark The proof of (2.5) did not require that {en)be the normalized shift basis for S, but, in fact, if {en)is the normalized shift basis for S, then {Ugen)is the normalized shift basis for g-'(S), with respect to the same normalization E. For, we may extend E F-linearly to K Normalized shift bases are characterized by &(Z) = 1. Since E and A, commute, &(UgB)= &(Ag.?)= Ag(&B)= Ag(l)= 1. When V is the vector space of polynomials in x with coefficients in F, multiplication by x carries xn/n!into (n + l)(xn+'/(n+ I)!). Certain operators constructed out of the operators D = d/dx and multiplication by x have a similar effect on pn(x)/n!,where {p,(x)) is a sequence of polynomials of binomial type. These operators correspond to anti-shifts.

2.7. DEFINITIONLet S be a shift on V. a) A linear operator M on V is called an anti-shift for S if SM - MS = I. b) The anti-shift M is called the definite anti-shift for S if the range of M is contained in the null-space of the evaluation 8. 2.8. PROPOSITIONa) If S is a shift on V, then an anti-shift M for S exists. If M and M' are two anti-shifts for S, then M' - M is a formal power series in S. b) IfS is a shift on V,then a de$nite anti-shift M for S exists and is unique. Proof a) Let {en)be any shift basis for S. Define Me, = (n + l)en+l for

Downloaded by [University of California, San Diego] at 11:21 28 June 2016 n 0 and extend linearly to V. SM - MS = I is clear. If M and M' are two anti-shifts for S, then M' - M commutes with S, so is a formal power series in S by (2.3). b) Let {en)be the normalized shift basis for S. Define M as before. Then &(Men)= 0 for n 0, so M is a definite anti-shift for S. Suppose M' is a second definite anti-shift for S. Then M' - M = T is a formal power m series T = anSnin S. Now an = &(Ten)= &(M1en)- &(Me,) = 0. Hence n=O T = 0. Q.E.D. THE ROTA-MULLIN THEORY 73

2.9. Remark Iff (u) = C a,,un is any formal power series in u, let n=O

denote its formal derivative. For any anti-shift M for S, one has f (S)M - Mf (S) = f '(0 One shows inductively that SnM- MSn = nSn-I for n 3 1. We note that, if T is a formal power series in the shift S, and M is an anti- shift for S, then the operator TM - MT corresponds, in the case of polynomials, to the Pincherle derivative of the operator T. ([2] p. 192, Remark following Prop. 1 ;[I], 1.6.) Pincherle's formula ([2], p. 192, Remark following Prop. 2) becomes, in our setting,

where T(")is the operator obtained from T by iterating the Pincherle derivative n times. This formula is easily proved using induction and T") = TM - MT. Remark Let S be a shift on the vector space V, let M be an anti-shift for S. Let V' be the vector space underlying the ring of all operators on V which are polynomials in M with coefficients in F. As in Remark (2.9), if

Define operators D' and M' on V' by D'T = ST - TS and M'T = MT for T in V'. Using SM - MS = I, one sees that D'M' - M'D' is the identity operator on V'. Hence we have recovered the original situation, where differentiation is the shift and multiplication by the variable is the anti-shift, on a vector space of polynomials. Let S be a shift on V, and M an anti-shift for S. If U is an invertible linear operator on V, then clearly UMU-I is an anti-shift for USU-l. Just as (2.5) allowed us to express USU-I in terms of S when U = U, was an umbra1 operator, we can express UgMU; in terms of S and M. U,, E

Downloaded by [University of California, San Diego] at 11:21 28 June 2016 2.10. Remark Let g G, be constructed using a normalized shift basis for S. Then eUg = E as linear functionals on V. This is immediate by Remark (2.6). If M is the definite anti-shift for S, then UgMU;l is the definite anti- shift for U,SU;l. For, EU~= 8 clearly implies that the range of UgMUil is contained in the null-space of E,if this is the case for M. 2.11. THEOREM(Rodrigues-type formula [2] p. 194, Th. 4(4); El], 3.6 and Th. 3.3) Let S be a shift on V, and M the definite anti-shift for S. Then UgMU; = M((g-l)'(S))-l for g E G. 74 J. P. FELMORE AND S. G. WILLIAMSON Remark The formal derivative of a series in G has non-zero constant term, so defines an invertible operator when the indeterminate u is replaced by S. It is the inverse of this operator, when the series is g-' E G, that appears on the right-hand side.

Since only g-' appears in the formula, we may replace g-I by g and prove

Now, by (2.9), g(S)M = Mg(S) + gt(S); so the left-hand side is (Mg(S) + g'(s))k7'(s))-i - M(g'(s))-lI~(S) which is the identity. 2) By (2.5), UgSU;l =g-'(S). Both UgMUil and M((g-')'(S))-I are anti-shifts for this shift. The former has its range contained in the null-space of E by (2.10) and the latter obviously does also. By (2.8b), these anti-shifts are equal. Q.E.D. The main application of this theorem will be the fact that M((g-')'(S))-I is the definite anti-shift for g-l(S). 2.12. COROLLARY(121 p. 193, Th. 4; [I], 3.6.) Assumptions as in (2.1 1). Let (en) be the normalized shift basis for S; set e:, = Ugen so that {el) is the normaIized shift basis for U,SU; = g-'(S). Then:

Remark The notation S/g-'(S) means that u is replaced by S in the series u/g-l(u). Similarly, we will use d/dS to denote taking the formal derivative of a series in u and replacing u by S. Proof a) Write (2.11) as UgM = ((Mg-l)'(S))U, and apply this to en-, a) now follows from en = M(e,-,In). b) We may replace g by g-I and prove b) in case e:, = U; 'en. We proceed by induction. The case n = 0 is true, since Downloaded by [University of California, San Diego] at 11:21 28 June 2016

if g(u) = alu + azuZ+ - . . implies eb = e,. We assume b) with n replaced by n - 1 and use a) to prove b) for n. We have en-, by (a) and induction ek = M--gt(S) - en-, = Q'(S) n (8;)). M(&>'- n hypothesis THE ROTA-MULLIN THEORY

1d sn = (&>'en - ,S;iS(a) en by behavior of M and S on {en)

Now one has the identity

([2] p. 194) which is obtained by computing

using the product rule on the first factor on the right. This identity shows that the last expression above is gr(S)(S/g(S))"+'enand completes the induction. Q.E.D. Remark Since g-'(S)(S/g-'(S))"fl = (S/g-'(S))"S, it is clear that b), even without the factor (g-')'(S) yields a shift basis for g-'(S). The role of the factor (g-')'(S) is to give the normalized shift basis, a fact which follows from e:, = Ueen. In the remainder of this section, we indicate how several other results of Rota and Mullin [2] appear in our setting. 2.13. PROPOSITION([2] p. 201, Th, 7; [I], Th. 2.4) Let S be a shift on V, {en) a normalized shift basisfor S, and U,, g E G, an umbra1 operator. Set el, = U,en. Then

Proof Using (2.1O), we have &(SVe:,)= E(S~U,~,)= E(U;~S~U,~,)= &((g(S))'e,).

Downloaded by [University of California, San Diego] at 11:21 28 June 2016 The assertion follows by setting v = e:, in v = x &(SVv)e,. Q.E.D. v=o This last proposition may be regarded as a version of Taylor's Theorem. Other versions, for vectors and operators, have already been used. 2.14. PROPOSITION([2] p. 208, Cor. 1 to Th. 8) Let S be a shift on V, and (en)a normalized shift basis for S. Given a sequence {c,) (n 2 1) of elements ' of F with c, # 0, there exists a unique shijlg-'(a, g E G, on V whose normalized sh$ basis {e;) satisJis &(Se;)= cnfor n 2 1. 76 J. P. FILLMORE AND S. G. WILLIAMSON Remark In the case of polynomials in x, this corresponds to prescribing the coefficients of x in the sequence of polynomials. Proof Set g(u) = clu + c,u2 + . . -;g(u) is a series in G. {e;), where e; = U,e,,, is the normalized shift basis for g-'(S). Like the calculation in the proof of (2.13), we have c(Se;) = E(U;~SU,~,) = e(g(S)en) = cn for n 2 1. Q.E.D. 2.15. PROPOSITION([2] p. 207, Th. 8) Let S be a shift on V. Let g-'(S), g E G, be a shift such that

where {e;) is the normalized shift basis for some shift h-'(S), h E G, not necessarilyg-'(S). But then, in fact, h-'(S) = g-'(S) and {en)is the normalized shift basis for g-'(S). Remark If h-'(S) = g-'(S), the conditions in the hypothesis clearly holds. These may be regarded as a set of recursive conditions to determine {e;) given by g E G. Proof We have e; = Uhen,where {en}is the normalized shift basis for S. The left hand side of the condition is &(Se;) = &(U,-'SU,e,) = c(h(S)en), while the right hand side is

These are equal for all n > 1 (and n = 0), so h(S) = h(S)S/g-'(h(S)). Cancelling h(u) from the corresponding equation in u, we obtain u = g-'(h(u)) and h(u) = g(u). Q.E.D. Finally, much of the exposition of Rota and Mullin is in terms of what we may call translation invariant operators on polynomials ([2] p. 179; [I] Def. 1.1). If p(x) is a polynomial in x, p(x + a) is its translation by a in F. The operator T is translation invariant if (Tp)(x + a) = T(p(x + a)) for every a in F and every polynomial p(x). By Taylor's Theorem for polynomials, the operator of translation by a is exp(aD) where D is the shift dldx. If S is any

Downloaded by [University of California, San Diego] at 11:21 28 June 2016 shift on V, the operator of translation by a is exp(aS). T translation invariant means that T commutes with exp(aS) for every a in I;, and hence T commutes with S. By (2.3) Tis just a formal power series in S.

3. EXAMPLES Let V be the vector space of polynomials in x with coefficients in F, S the shift operator D = dldx, and e the evaluation of a polynomial at 0: THE ROTA-MULLIN THEORY 77 E(P(x))= p(0). {en),where en = xn/n!,is the normalized shift basis for D. The definite anti-shift M for D is multiplication by x: (Mp)(x)= xp(x). Example 1 eD - 1 is the forward difference operator which sends p(x) into p(x + 1) - p(x). This operator may be written eD - I = g-'(D), where

is in G. We may determine the normalized shift basis {e;), e; = pn(x)/n!,for g-'(D) using (2.12a):

Since ((g-')'(Dl)-I = e-D is the operator sending p(x) to p(x - I), we obtain pn(x) = xpn-,(x). Starting with po(x) = 1, we have pl(x) = x, p2(x) = x(x - 1), . . ., pn(x) = x(x - l)(x - 2) . . . (x - n + 1). pn(x) is the lower factorial (x),. pn(x)/n!= Ug(xn/n!)may also be obtained directly from the definition (2.4) of U, or, equivalently, by generating functions as mentioned in 1 :

pn(x) = (x), is the nth partial derivative of (1 $ u)" with respect to u evaluated at u = 0. Example 2 (x)" = x(x + l)(x + 2) . . (x + n - 1) are the upper factorials. We may determine the shift g-'(D) for which {(x)"/n!}is the normalized shift basis as follows: Assume (x)"/n!= Ug(x"/n!)for some series g(u) = clu + c2u2 + . .. in G. By (2.14),

is the coefficient of x in (x)"/n!;hence cn = lln and g must be u2 u3 g(u)=u+-+-+...= -log(l - u). 2 3 Downloaded by [University of California, San Diego] at 11:21 28 June 2016 As in Example 1, one checks that this g does give (x)"/n!= Ug(xn/n!). g-l(D) = I - e-D is the backward difference operator which sends p(x) into P(X)- P(X - 1). Example 3 Write 78 J. P. FILLMORE AND S. G. WILLIAMSON where the S(n, k) are Stirling numbers of the second kind. The S(n, k) are the connection constants ([2]p. 169; [I] Th. 3.2) from {(x),) to {x");the doubly infinite matrix ((k!/n!)S(n,k)) is just the matrix of the change of basis in V given by xn/n! = U; l((x),/n!) with g(u) = log(1 + u); see Example 1.

defines a sequence of polynomials {cp,(x)) which is inverse to the sequence ((x),) in the sense that the connection constants from {x") to {cp,(x)) are the same as those from {(x),) to {x") (121 p. 201, Cor. to Th. 6). The polynomials cp,(x) may be identified as the exponential polynomials

by using g-'(u) = eU - 1 in the generating function:

Example 4 Let P be the field of real numbers. The operator

maps the space V of polynomials into itself. L is clearly translation invariant, so is a formal power series in the shift D = d/dx. To determine this series, note that L, as an operator, is m L = - [ e-'etDD dt. Since

L = 9-1(~)with g-l(u) = -u - u2 - u3 - . . . in G. We may combine (2.12) a) and b) to obtain Downloaded by [University of California, San Diego] at 11:21 28 June 2016

([2]p. 194, Th. 4(3)). Hence the normalized shift basis for the shift L may be obtained as LAX)-'( D xn-' >' or Ln(x) = x(D - I)"xn-' for n 2 1. n! nD/(D-I) (n-I)! THE ROTA-MULLIN THEORY 79 The polynomials Ln(x)may be identified as Laguerre polynomials: eXDe-" = D - I as operators, so (D - I)" = exDne-X,and Ln(x) = ~e"(d/dx)"(e-~x"-'1. Other classical identities concerning these poIynomiaIs may be obtained by these techniques ([2] p. 205 ff.). The substitution of D for u in an integral, as was done above, gives rise to may similar such operators. Example 5 (E. Rodemich) Let a 3 1 be an odd integer. Set

Then g(u) is an element of order two in G: g(g(u)) = u. This corresponds to the fact that, ifp,(x)/n! = U,(xn/n!) is the normalized shift basis for g-'(D), the polynomial sequence {pn(x))is self-inverse in the sense that the connection constants from {x,) to (pn(x)) are the same as those from {p,(x)) to {xn). This is the case for the Laguerre polynomials of Example 4 (or = 1) ([2] p. 206). Other examples of self-inverse sequences of polynomials may be obtained from other elements of order two in G. An element of order two in G may be obtained by conjugating a known element of order two by an element of G or by an element of a larger group containing G, as was done in this example. By the remarks in 1, the polynomial sequences of these examples are of binomial type.

4. GLOSSARY OF TERMS

Rota-Mullin Fillmore- Williamson 1. Space of polynomials p(x) 1. Vector space V 2. Evaluation at 0, p(0) 2. Evaluation 6 3. Sequence of polynomials {pn(x)) 3. Basis {en) of V with ~(e,) # 0 with pn(x) of degree n 4. Differentiation D = d/dx 4. Fixed shift S on V 5. Delta operator 5. Shift operator g-'(S), g E G

Downloaded by [University of California, San Diego] at 11:21 28 June 2016 6. Basic sequence of polynomials 6. Normalized shift basis (en), {pn(x) 1 n !en corresponding to pn(x) 7. Shift operator Ea 7. Translation operator PS 8. Shift invariant 8. Translation invariant 9. Indicator f (t) of T 9. T = f (S), f (u) a formal power series 10. Multiplication by x 10. Definite anti-shift M 11. Pincherle derivative 11. TM - MT 80 J. P. FILLMORE AND S. G. WILLIAMSON 12. Connection constants 12. Matrix of the change of basis e; = Uge, of V 13. Umbra1 notation p;(x) = p,Cg(x)) 13. e; = U; 'en, g € G 14. Inverse sequences {p,(x)), {qn(x)) 4. {}{Ugen), geG

References

[l] A. Garsia, An expos6 of the Mullin-Rota theory of polynomials of binomial type, J. Linear and Multilinear AIgebra (to appear). [2] G.-C. Rota and R. Mullin, On the foundations of combinatorial theory, Graph. Theory and its Applications, Academic Press (1970), pp. 167-213.

MOS numbers: OSAlO and 39A20 Downloaded by [University of California, San Diego] at 11:21 28 June 2016