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Polynomial Calculus with D-Like Operators Author(s): J. W. Burgmeier and R. E. Prather Source: The American Mathematical Monthly, Vol. 82, No. 7 (Aug. - Sep., 1975), pp. 730-737 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2318730 . Accessed: 29/09/2013 09:23 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. http://www.jstor.org This content downloaded from 128.228.173.41 on Sun, 29 Sep 2013 09:23:45 AM All use subject to JSTOR Terms and Conditions

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Page 1: Polynomial Calculus with D-Like Operators

Polynomial Calculus with D-Like OperatorsAuthor(s): J. W. Burgmeier and R. E. PratherSource: The American Mathematical Monthly, Vol. 82, No. 7 (Aug. - Sep., 1975), pp. 730-737Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2318730 .

Accessed: 29/09/2013 09:23

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access toThe American Mathematical Monthly.

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Page 2: Polynomial Calculus with D-Like Operators

730 J. W. BURGMEIER AND R. E. PRATHER [Aug.-Sept.

16. Marston Morse, John von Neumann and Luther P. Eisenhart, Report to the President of the National Academy of Sciences, 28 May 1948.

17. Laurence F. Schmeckebier, Congressional Apportionment, The Brookings Institution, Washington, D. C. 1941.

18. George Washington, The Writings of George Washington, Vol. 32 (March 10, 1792-June 30, 1793), John C. Fitzpatrick, editor, United States Government Printing Office, Washington, D. C. 1939, pp. 16-17.

19. Daniel Webster, The Writings and Speeches of Daniel Webster, vol. VI, National Edition, Little, Brown & Company, Boston Mass., 1903, pp. 102-123.

20. Walter F. Willcox, The Apportionment of Representatives, The American Economic Review, vol. VI, No. 1, Supplement (March 1916) 3-16.

21. - , Last Words on the Apportionment Problem, Legislative Reapportionment, vol. 17, No. 2 (1952) issue of Law and Contemporary Problems, pp. 290-302.

22. Comment, Apportionment of the House of Representatives, Yale Law Journal, 58 (1949) 1360-1386.

GRADUATE CENTER, THE CITY UNIVERSITY OF NEW YORK, 33 WEST 42 STREET, NEW YORK, N. Y. 10036.

POLYNOMIAL CALCULUS WITH D-LIKE OPERATORS

J. W. BURGMEIER AND R. E. PRATHER

1. Introduction. The newcomer to numerical analysis is usually impressed on finding an unexpected formal analogy [1] between the Taylor and Newton series expansions:

f(x) - E (Dkf)(a)(x a) k =

f(X) A (Akf)(a) (X - a)(k

We say unexpected because the difference operator

(f)(x) =f(x + l)-f(x)

and its associated factorial polynomials

x"' x(x -l) (x-j+l j! =j!

seem somewhat removed from their counterparts in the differential calculus. True, the formal identity

D k A = e D _ I = D + D +, . D +... 2! k

and the Stirling numbers s (i,j) defined by x'j) = 2=5s (i,j)x' do help to provide a connection. And yet, the question surely arises as to whether these two "expansion systems" are merely isolated curiosities, or instead, singular but typical examples from a family of such systems. Naturally, we wish to infer that it is the latter.

In order to establish an appropriate setting for the investigation, we first seek to extract the common features of the linear operators D and A (and their associated polynomials xj/j!, xWp/j!, resp.) on the space P' of all polynomials. In both cases, the associated polynomials form a simple basis (there being just one polynomial for each degree) and the operators are linear and strictly unit-degree-decreasing (abbreviated u.d.d. with linearity understood) over P'. And so, we begin with a brief analysis of these u.d.d. or "derivative-like" operators and their expansion capability relative to various simple bases. We are then led to impose a succession of familiar differential properties, leading ultimately to a characterization of the derivative among all u.d.d. operators.

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Page 3: Polynomial Calculus with D-Like Operators

1975] POLYNOMIAL CALCULUS WITH D-LIKE OPERATORS 731

Though some of our results might have been anticipated, we feel that the investigation does provide new insights as to the essential uniqueness of the classical differential calculus. Assuming this is so, we have tried to make our presentation self-contained and reasonably inclusive.

2. Expansion systems. In any linear function space, a sequence (Li) of linear functionals and a companion sequence (gi) of functions are said to be bi-orthonormal [2] if

Lig1 = Sij.

Perhaps the most common situations of this kind occur in the inner-product spaces where the functionals Li take the form Lif = (f, g). Then the bi-orthonormality accounts for the familiar orthogonal expansions. But the same is seen to be true for the generalized Taylor-Newton expansions:

THEOREM 1. Let L be a u.d.d. operator and (g1) a simple basis in P'. Then every f E P' has a representation

f(x) = >j(Lkf)(a)gk(x) k =O

iff the functionals Lif = (Lif)(a) and the polynomials gj(x) are bi-orthonormal.

Proof. Assuming these representations exist, we would have in particular 00

g1(x) = >(LLgj)(a)gi(x). i =O

And because (g1) is a basis it would follow that

Ligj = (L1g1)(a) = 5ii.

Conversely, if we assume the bi-orthonormality and write 00

f(X) = E akgk (X) k =O

for any f E P", then applying the functional Lj yields 00

(L f)(a) = E ak(Ljgk)(a) = aj k =O

so that f has the representation claimed. O As a consequence, L and (gj) are said to constitute an expansion system (at a) provided that

(L gj)(a) = Sij. An expansion system (at 0) will be called a Maclaurin system. A more explicit characterization of these expansion systems is useful in the sequel, namely

THEOREM 2. L and (gj) constitute an expansion system (at a) iff (i) go= 1 (ii) gj (a)=O (j > 0). (iii) Lgj = gj-, (j> 0).

Proof. First suppose that all these conditions are met. Then

[gj_i(a) = < (i<j), (L1gj)(a) = go(a) = 1 (i =) = ij.

L(L i go)(a) - 0 (i >P J Conversely, if L and (gj) form an expansion system (at a) we have (i) go = go(a) = (Logo)(a) = Soo 1, (ii) gj (a) = (Logj)(a) = 50=O (J > O),

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Page 4: Polynomial Calculus with D-Like Operators

732 J. W. BURGMEIER AND R. E. PRATHER [Aug.-Sept.

and finally, considering Theorem 1

(*ii 00 (i1i) (Lgj)(x) = >_(L k (Lgj ))(a )gk (x) k=O

00

= ,(L k+gj)(a)gk(x) = gj,(x). O k=O

COROLLARY. Let (g1) be a simple basis satisfying (i), (ii). Then there is a unique u.d.d. operator L such that L and (gj) constitute an expansion system (at a). Conversely, to each u.d.d. operator L and each real number a, there corresponds a unique simple basis (gj) for which L and (gj) is an expansion system (at a).

In particular, each u.d.d. operator L may be identified with that (unique) simple basis for which the pair constitute a Maclaurin system. And in considering the resulting infinite matrix representa- tion, it should be clear that any such L will have the same "Jordan canonical form" as the derivative. We obtain somewhat more information in this same vein if we define the Stirling transformations S: P" -> P" to be those linear operators with the following properties:

(I) S(1) = 1, (II) Po and each pn are S-invariant subspaces, (III) S is invertible. Here, pn is the space of all polynomials of degree at most n, and PO represents the subspace of

polynomials which vanish at the origin. Then, just as the factorial polynomials may be viewed as a change of basis via the matrix s(i,j) of Stirling numbers, we have the following generalization:

THEOREM 3. Let M and (hj) be a Maclaurin system and S a Stirling transformation. If

g = Sh L = SMS1

then L and (gj) comprise another Maclaurin system. Conversely, if the two Maclaurin systems are given, there exists a Stirling transformation S such that

g = Sh L = SMS1.

Proof. First we observe that L is u.d.d. and gn = Shn has degree -' n by (II). But if gn had degree < n we would contradict (III). So (g,) is again a simple basis, and moreover,

(i) go = Sho = S(1) = 1 using (I), (ii) g1(O)=Sh,(O)=O by (II) (>O). (iii) Lg1 = SMS-lg1 = SMhj = Sh1-l = gj-l,

so that according to Theorem 2, L and (gj) constitute a new Maclaurin system. Conversely, if L, (g1) and M, (hj) satisfy the conditions of Theorem 2, we have only to define S

by the change of basis g = Sh. Then the invertibility of S and the invariance of each Pn is assured, and furthermore

(I) S(1) =Sho = go = 1, (II) f(O) = 0

=> (Sf)(0) = Ej(Mkf)(O)(Shk)(O) k =O

= (M?f)(O)go(O) = f(0) = 0

so that S is a Stirling transformation. Finally, for any f E P" we use its Maclaurin representation relative to M, (hj) to obtain

(S-1LS)(f) = Y,(Mkf)(O)(S-lLS)hk = ,(Mkf)(O)hk-l = M(f). C1 k k

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Page 5: Polynomial Calculus with D-Like Operators

1975] POLYNOMIAL CALCULUS WITH D-LIKE OPERATORS 733

COROLLARY. The only Maclaurin systems L, (gn) are

gn -S(xn/n !) L = SDS-'

for Stirling transformations S.

3. Taylor operators. We have seen that each u.d.d. operator L may be identified with a unique simple basis (g1) so that the pair comprise a Maclaurin system. As we move to another point of expansion, say a 7 0, there will again exist a unique simple basis resulting in an expansion system (at a). But these new polynomials, in general, will bear little relationship to the sequence (g,). In order to achieve the desired universality for the expansion basis, we find that the commutativity property of the following definition is needed.

The u.d.d. operator L is called a Taylor operator if

LTa = TaL

for every real number a. Here Ta is the translation

(Taf)(x) = f(x - a).

Considering the universality requirement as exemplified by the Taylor-Newton expansion, our above definition is justified by the following result.

THEOREM 4. Let L be a u.d.d. operator and (gj) the unique simple basis for which L, (gj) constitute a Maclaurin system. Then L, (gf(x - a)) is an expansion system (at each a) if L is a Taylor operator.

Proof. First suppose that LTa = TaL. Then considering the simple basis (gj(x - a)) one has (i) go(x - a ) = go(x) -1 (ii) g,(x - a)(a) = gj(0) =0 ( > 0), (iii) Lg1(x -a) = LTag(x) = TaLgj(x) = Tagj-i(x) =gi-(x -a)

so that L, (gj (x - a)) is an expansion system (at a), by virtue of Theorem 2. On the other hand, if the usual properties (i), (ii), (iii) hold relative to L and (gj(x - a)), then for

any f E P' we may consider its Maclaurin expansion

f(x) = >j(Lkf)(O)gk(x) k =O

and use (iii) to obtain 00

(L Taf)(x) = > (Lkf)(0)(LTagk)(x) k =O

00

= J(Lkf)(0)gk-I(x - a) k =O

= Ta>(Lkf)(0)gk 1(x) = (TaLf)(X). E k =O

Note that this result establishes a relationship for Taylor operators which is analogous to the connection between Maclaurin and Taylor series for D. Consequently, it seems that we have come a step closer to the derivative by our introduction of the Taylor operators (but we still admit the difference operator A). As for seeing just how close, it remains for us to provide an explicit characterization of the Taylor operators. This could perhaps be accomplished by several means. One could try to examine the nature of their associated Maclaurin polynomials (gj). But instead, we have found a direct characterization of the operators themselves to be more illuminating. For this purpose, we need the following lemma.

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Page 6: Polynomial Calculus with D-Like Operators

734 J. W. BURGMEIER AND R. E. PRATHER [Aug.-Sept.

LEMMA. Let L be a u.d.d. operator having the matrix (Ais) relative to the basis (xj/j!). Then L is a Taylor operator iff

An,n+k = AO,k (k - 0).

Proof. Let all matrices be expressed relative to the basis (x'/j !). Then, because L is u.d.d., the matrix (Ai;) is upper-triangular. The same is true for the matrix (ij) = (in(a)) of the translation Ta. In fact,

[0 (i <j) fTiJ= (a)i-(' j

as found from the binomial theorem. It follows that both of the product matrices

(Ai;)(Tij) = (pi1) (1ij)(Aij) = (air)

are upper-triangular, and in addition, have zero diagonal. Otherwise, the entries in these product matrices are given by the finite sums:

I j-i

Pi; =E ikTki 0Jij = ITikAki k=i+l k=i

for i <j. Now if L is a Taylor operator, these product matrices must agree for every a. But this gives the

identity

,Aik (a)' = a

l( ) (=i-lk)! >=k=ik (k -iT

Equating like powers of a, we obtain Ai,j_r = A i+r,j, (i < j). And this means that the entries along all super-diagonals are constant, as claimed. Since our arguments are obviously reversible, the result follows. LI

THEOREM 5. L is a Taylor operator iff it has the form

00

L = E>AkDk k = I

for a sequence (Ak) with A1 7 0.

Proof. First we observe that the infinite sum when applied to any f E P' is actually finite, so that questions of convergence do not arise. Now assume that L is any Taylor operator and use the ordinary Taylor expansion

f(x) = _(Dif)(0) xj

for f E P'. Applying the operator L and using the notation of the lemma, we have 00j

(Lf)(x) = (Djf)(O)L . 1=0 J1

= (Dif)(0)Aij . i=1 1=0 l.

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Page 7: Polynomial Calculus with D-Like Operators

1975] POLYNOMIAL CALCULUS WITH D-LIKE OPERATORS 735

= .(Djf)(O)ZAoj

co co X -k

= O l?kJ(D f)(0) X

= Aok(D f)(x) = (YAOkD f)(x). D1 k =I k=1

Of course, if we take A, = 1 and all other Ak = 0 we obtain the derivative, whereas the choice Ak = 1/k! gives the difference operator. But more importantly, one sees in general the intimate connection of these Taylor operators with the derivative.

4. Rolle operators. Surely one of the most important results in the differential calculus is the mean value theorem, or equivalently, Rolle's theorem. We now propose to study the more general u.d.d. operators having this same property. Our original interest in this property was motivated by considerations which are explained in the following concluding section.

We will say that the u.d.d. operator L is a Rolle operator if we have

f(a) = f(b) = 0 => (Lf)(6) = 0 (some 6 E (a, b))

for every f E P'. Of course D is a Rolle operator. And one would hope to discover other such operators and to characterize them as was done for the Taylor operators in the previous section. But the following theorem shows that D is (essentially) the only Rolle operator!

THEOREM 6. Every Rolle operator is a non-zero multiple of D.

Proof. Let L be a Rolle operator. First we show that the unique sequence (g1) making L, (g1) a Maclaurin system must have the form g1 = a1x'. Certainly this is true for j = 0, 1. But suppose, on the contrary, that not all gi are of this form, and let n be the least integer for which

gn(X) = X kp (X)

with p (O) Z 0 and i ? k < n. Then we set

f(x) = gn(X) -xkp(a)

with a as yet unspecified. We have f(0) = f(a) = 0, whereas

X) = Xk _,n-k _ak-I (Lf)(x ) ̀=x (an-i X -_ p (a)).

Since p (0) , 0, we- can now choose a k 0 so that both factors in (Lf)(x) are of one sign for all x between 0 and a, thus contradicting that L is a Rolle operator. We must therefore conclude that gn(x) = anXn for all n.

We know that ao = 1. And it is clear that L, (g1) is a Maclaurin system with L Rolle if and only if the same is true for cL, (gj/cj). Thus we may continue as if a, = 1 also. Introducing the quotients Pk = ak-Ilak we have

LXk =fPkX.

This makes I3S = 1 and we intend to show that 3m = m for all m ' 1. Proceeding inductively, we suppose that Pj = j for j = 1, 2, , m - 1 and set

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Page 8: Polynomial Calculus with D-Like Operators

736 J. W. BURGMEIER AND R. E. PRATHER [Aug.-Sept.

Since L is a Rolle operator and h is zero m times on [a,, am], Lm -'h should be zero once on (a,, qm). But our inductive assumption gives

(L m -'h)(x) = /3m (m - 1) !x - (a,I + a, + ***+ am )(m - 1)!

and this is zero only at a i+ a2+' * * *am

xo = - 3r

But we can have xo E (a,, am) for all choices of a,, a2, ,am only if (m = m. Thus, pf =j for all i > 0, and the gi for this (normalized) L are

gj = x'Ij!

Consequently, in the general case, L = cD. C1 Another important property of differentiation is the so-called chain rule. In view of its

importance in applications of the calculus, and in the sake of completeness, we now wish to investigate the class of operators satisfying such a rule. Thus, we shall say that the u.d.d. operator L is a chain operator provided that

(L (f o g))(x) = (Lf)(x))(Lg)(x)

for all f, g E P'. As with Rolle operators, the "class" of chain operators turns out to be rather limited. In fact, as the next theorem shows, the chain rule completely characterizes differentiation among u.d.d. operators. But first, we need a lemma.

LEMMA. The only Taylor-chain operator is D.

Proof. Considering the representation of Theorem 5, we may write

L = >AiDL

with A1 0 0. In fact, AX = 1, for if we take f(x) = g(x) = x in the chain rule, we find that Al = A . Now suppose that there is some other non-zero coefficient Ak so that

L =D+AkDk + .. (k_ 2).

We have only to take f(x) = x2 and g(x) - Xk in the chain rule to arrive at a contradiction:

(L(fog))(x) = 2kx2k-1 + Ak (2k)! xk +

(Lf)(g(x))(Lg)(x) = 2kx2k-l + 2Akk !Xk +* ..

(When k = 2 the last line must be modified slightly, but the argument is still the same.) Thus we conclude that Ak = 0 for k > 1, and finally, L = D. D1

THEOREM 7. The only chain operator (among u.d.d. operator) is D.

Proof. We need only verify that every chain operator L is a Taylor operator. Let (gj) be the sequence of polynomials for which L, (gj) is a Maclaurin system. Then go = 1 and g1(x) = cx. But choosing f = g = g, in the chain rule gives c = 1. It following that

(LTaf)(x) = Lf(x - a) = (Lf)(x - a)L(x - a)

= (Lf)(x - a) = (TaLf)(x)

for any f E P'. So L is a Taylor operator, and by the lemma, L = D. C1

5. Concluding remarks. Having begun with rather weak assumptions for linear operators on P' sharing certain features of differentiation, we have imposed a sequence of "derivative-like"

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Page 9: Polynomial Calculus with D-Like Operators

1975] ADDENDUM TO: "EULER AND THE ZETA FUNCTION" 737

properties, leading ultimately to D itself. We believe that the intermediate characterizations of the Taylor and Rolle operators are quite interesting in their own right. Overall, the investigation seems to provide for a new appreciation of the essential uniqueness of the classical differential calculus.

The question of possible extensions of these operators to larger spaces must naturally arise, particularly in connection with the concept of an expansion system. The usual Taylor theorem is phrased in the space C' and introduces the idea of a remainder. And so, a suitable generalization of the Taylor remainders would be needed for the u.d.d. operators. But our result concerning Rolle operators suggests that the class of operators having "derivative-like" remainders on C" may be rather limited, thus accounting for our restriction to polynomials in this study.

Added in proof. Since this article was accepted for publication, it has come to the authors' attention that certain of these ideas have appeared in a paper by G. C. Rota, D. Kahaner and 0. Odlyzko, On the Foundations of Combinatorial Theory, VIII, Finite Operator Calculus, Jour. of Math. Anal. and Appl., 42 (1973).

References

1. F. B. Hildebrand, Introduction to Numerical Analysis, McGraw-Hill, New York, 1956. 2. P. J. Davis, Interpolation and Approximation, Blaisdell, New York, 1963.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF VERMONT, BURLINGTON, VT 05401. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF DENVER, DENVER, CO 80210.

ADDENDUM TO:

"EULER AND THE ZETA FUNCTION"

A. G. HoWSON

Readers of Raymond Ayoub's article (this MONTHLY, 81 (1974) 1067-1085) may be amused to learn that one of the questions set to candidates for the first London University Matriculation Examination (in 1838), an examination for students of 19 years or under who wished to enter the university, was

"Find the sums to infinity of the series

12 + 2+3+ S12+ 2. 3 j ~1.2 2.3 3.4

There is no indication how the examiner intended the question to be solved; the examination syllabus, which did not include the calculus, referred only to "arithmetical and geometrical progressions" and "arithmetic and algebra." It can, however, be inferred from a previous question

"Prove that

Nap'logx =(x-x')-1/2(x2-x-2)+1/3(x3-x-3)-"

that a cavalier treatment of infinite series was not only tolerated but actively encouraged.

DEPARTMENT OF MATHEMATICS, SOUTHAMPTON UNIVERSITY, SOUTHAMPTON, S09 5NH, ENGLAND.

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