JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 199, 39᎐58Ž. 1996 ARTICLE NO. 0125

Sequences of with Persistent Roots

A. Di Bucchianico*

Department of Mathematics and Computing Science, Eindho¨en Uni¨ersity of Technology, P.O. Box 513, 5600 MB Eindho¨en, The Netherlands

and

View metadata, citation and similar papers at core.ac.uk brought to you by CORE D. E. Loeb† provided by Elsevier - Publisher Connector

LaBRI, Uni¨ersite´ de Bordeaux I, 33405 Talence, France Submitted by George Gasper

Received March 22, 1995

Ž. Ž Ž. We find all sequences of pnnG0with persistent roots i.e., pxn s Ž.Ž.Ž.. cxn yrx12yrиии x y rnthat are of binomial type in Viskov’s generalization of Rota’s to generalized Appell polynomials. We show that such sequences only exist in the classical umbral calculus, the divided difference umbral 2 calculus, and the new ‘‘hyperbolic’’ umbral calculus generated by Ž.drdx' .In each of these three umbral calculi, we also find all Sheffer sequences with persistent roots. ᮊ 1996 Academic Press, Inc.

1. INTRODUCTION

Most sequences of polynomials that interest mathematicians fall into one of the following three categories or their generalizations:

ⅷ sequences of binomial typewx 13᎐15 ,

ⅷ sequences with persistent rootswx 4, 5 and

ⅷ sequences of orthogonal polynomialswx 3 . These theories are very rich; powerful formulas allow the computation of one such sequence of polynomials in terms of another of the same type.

*Author supported by NATO CRG 930554. WWW URL http:rrwww.win.tue.nlrwinr mathrbsrstatisticsrbucchianico. E-mail address: [email protected]. †Author partially supported by URA CNRS 1304, EC Grant CHRX-CT93-0400, the PRC Maths-Info, and NATO CRG 930554. WWW URL http:rrwww.labri.u-bordeaux.frr;loeb. E-mail address: [email protected].

39

0022-247Xr96 $18.00 Copyright ᮊ 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. 40 DI BUCCHIANICO AND LOEB

However, to go from, for example, an arbitraryŽ. generalized sequence to an arbitrary sequence with persistent roots, it is necessary to n pass through a sequence that is both, for example, x rn!. This gives rise to the following three questions:

What are the con¨olution sequences of ? This first question is treated bywx 1, 3, 10, 16 . What are the persistent sequences of orthogonal polynomials? Since persistent sequences obey a two-term recurrence and orthogonal se- quences obey a three-term recurrenceŽ. Favard’s theorem , there are no persistent sequences of orthogonal polynomials. What are the persistent sequences of binomial type? The objective of this paper is the complete resolution of this last question. Up to rescaling, all persistent convolution sequences can be found in Table I. Ž. A sequence of polynomials pnnG0is said to be of binomial type if it obeys the identity

n n Epy Ž. x pxpyŽ. Ž.,1Ž. nnsÝž/k ykk ks0

y where EpxŽ.spx Žqy .. Well-known examples include the powers of x ŽŽ..in which case Eq. 1 is called the , the Abel polynomi- n1 als xxŽ.yna y, the , and so onwx 13, 14 .

TABLE I Complete List of Persistent Convolution Sequences

Umbral calculus Roots Polynomials

Name bnn⌽ rfŽD b .Ž.px qnŽ.x

xn any any any 0, 0, 0, 0, . . . Dpxb Ž. bn 1 xpŽ. x p Ž.1 divided y n 1 1 0,1,1,1,... xxŽ.y1 y difference 1 y txy1 Ž.Ž. x classical n! exp 0, 1, 2, 3, . . . pxq1ypx ž/n 2 2 xxŽ.y1иии Ž.x y Ž.n y 1 Ž. 1 hyperbolic 2n ! cosh 0, 1, 4, 9, . . . 2px'q1q Ž.wx Ž.2n! 2 pxŽ.wx'y1y2pxŽ. SEQUENCES WITH PERSISTENT ROOTS 41

Ž. Ž. If pnnG0 is of binomial type, then the sister sequence qnnG0s Ž. pnnrn!G0 obeys the identity n y EqnnŽ. x sÝqxqyykk Ž. Ž.. ks0

It is often more convenient to consider the sister con¨olution sequence rather than the sequence of binomial type itself. An important generalization of the umbral calculus involves replacing n the derivative D with a homogeneous linear operator of the form Dxb s ny1 Ž. y bxnnrby1where bn/ 0 for all n provided that the shift E y⌽Ž.⌽ Ž.Ýϱ n is replaced with Ebbs yD where t s ns0t rbnwx11, 12, 20, 21 . The resulting polynomials are sometimes called generalized Appell poly- nomials. However, umbral calculus does not include all important sequences of polynomials even as generalized above. For that reason, we now turn to sequences with persistent roots. Ž. A sequence pnnG0 is said to have persistent roots or be persistent if Ž. Ž. pxnndivides pxq1for all n. That is to say, all of the roots in pnpersist in pnq1. Ž. In Section 3, we consider the classical umbral calculus bns n!. All persistent convolution sequences have their roots in an arithmetic pro- gression

pxnŽ.sxx Žya .иии Ž.x y Žn y 1 .a . Moreover, if we allow Sheffer sequences, then we are no longer limited to progressions which start at zero. We then consider two other umbral calculi and characterize all of their convolution and Sheffer sequences with persistent roots. In Section 4, we Ž. consider the divided difference umbral calculus bn s 18,17wx᎐19 which is important in interpolation theory. In Section 5, we consider a new ŽŽ.. umbral calculus bn s 2n ! which we call the hyperbolic umbral calculus since ⌽Ž.t s cosh Ž't .. Surprisingly, these three examples give essentially all persistent convolu- Ž Ž .. Ž nn . tion sequences. Obviously, any sequence qxnnG0scxrbnnG0 has persistent rootsŽ. all zero , and is the basic convolution sequence for the delta operator Dbrc. However, in Section 6, we show that, other than these trivial examples, all persistent convolution sequences belong to the classical, divided difference, or hyperbolic umbral calculus. We have thus explicitly characterized all persistent divided difference se- quences. Our work was assisted by extensive use of the computer algebra system Maple and our umbral calculus packagewx 2 . 42 DI BUCCHIANICO AND LOEB

2. GENERATING FUNCTIONS

We show in this section that in any umbral calculus, there is, up to vertical and horizontal scaling, at most one nontrivial persistent convolu- tion sequence. These results are superseded by the explicit examples in Sections 3᎐5, and the completeness results in Section 6 which follow as consequences of these uniqueness results.

2.1. Umbral Calculus Modern umbral calculus employs powerful operator methods to derive results on polynomialswx 13᎐15 . Different commutation classes of operators give yield to analogous results concerning different sequences of polynomialswx 9, 21 . The classical umbral calculus concerns the family of operators that commute with the derivative D. These operators can all be expressed as formal in the derivative. Thus, they all commute with each y other. In particular, they commute with the shift operator E s expŽ.yD . Hence, these operators are usually called shift-invariant operators. The situation in other umbral calculi is analogous. A generalized deri¨a- ti¨e is defined by a non-zero generalized factorial function bn as

bn ¡ x ny1 if n ) 0, and Dxn b s~ bny1 ¢0ifns0.

y Ž. Ž. The generalized shift operator is then Ebbs ⌽ yD where ⌽ t s Ýϱn ns0 trbn so that we have the following generalized binomial theorem wx13, 20

n bn yn knyk Exb sÝ xy . bb ks0 knyk

2.2. Con¨olution Sequences Ž. The sequences of polynomials pnnG0under consideration in the um- Ž. bral calculus are required to satisfy deg pn s n. These sequences are studied in terms of the operators which define them. For example, let f be a delta series, i.e., a formal power series with zero Ž. Ž . Ž. Ž. constant term. Then any sequence qnnG0satisfying fDbn q xsqx ny1 Ž. will be said to be Sheffer. If further we have qn 0 s 0 for n ) 0, and Ž. q0 '1, then qnnG0is said to be a convolution sequence. SEQUENCES WITH PERSISTENT ROOTS 43

In the classical umbral calculus, convolution sequences qn have generat- ing functions of the form ϱ n Žy1. ÝqxtnŽ. sexpŽ.xfŽ. t , ns0 where f Žy1. denotes the compositional inverse of f.

In general, convolution sequences qn are characterized by generating functions of the form ϱ n Žy1. ÝqxtnŽ. s⌽Ž.xfŽ. t .2Ž. ns0

The delta operator of qxnbŽ.is then fD Ž .. Ž. Thus, each convolution sequence qnnG0is determined uniquely by an umbral calculus b and a delta series f. Conversely, every convolution sequence belongs to essentially only one umbral calculus b Ž.Lemma 6.1 . Ž. However, if we require that qnnG0have persistent roots, then we can Ž. say much more. In this case, qnnG0is determined by the umbral calculus band the polynomial q2 alone.

ROPOSITION Ž. P 2.1. Let qnnG0 be a persistent con¨olution sequence in the umbral calculus b with respect to the delta operator fŽ. Dbn with roots r . Then either:

nn ⅷ Ž. Ž. The sequence is tri¨ial, that is, qxnnscxrb,fDbbsDrc,and rn s0for all n G 1, or else r ⅷ Ž. Ž2 . Ž Ž. r12s0, r / 0, and f Dbbs cE y1rr22. For c s r s 1, fDb is called the forward difference operator ⌬ b.. We will give two proofs of this important result. The first one uses operator methods, and the second one uses generating functions.

r 2 Ž. Proof 1. Case r2/ 0. By the generalized binomial theorem, Eqbn xs Ýn Ž.Ž. Ž. Ž. ks0qxqrnykk2. However, qrk21s0 for k G 2, and qxsxrc. r2Ž. Ž. Ž . Ž. Ž. Ž r2 . Thus, Eqbn xsqx n qr2rcqny1 x. Hence, qxny1 scEbny1qr Žr2. r2, and Q s cEb y1rr2.

Case r2 s 0. Assume Q s Dbrc q R where R is an operator of order n G 2. The constant term of pny1 s Qpnbnns DprcqRp is zero, so the constant term of Rpn is zero. This contradicts the fact that R is an operator of order n in which case Rpn should be a non-zero constant. Ž. Proof 2. Case r22/ 0. Evaluating 2 at x s r , annihilates all but the first two terms on the left hand side leaving only

Žy1. 1 q ct s ⌽Ž.rf2Ž.t . 44 DI BUCCHIANICO AND LOEB

Ž. Substituting t s fDb yields

1 q rfD2Ž.bs⌽ ŽrD2b .

fDŽ.bsŽ.⌽ ŽrD2b .y1rr2.

Žy1.Ž. Ž. Case r2 s 0. Suppose ftsct q Rt where R is a formal power series of order at least two. The coefficient of x on the left hand side ofŽ. 2 Ž. Ž. is ct and on the right hand side ct q Rtrb1. Thus, Rt s0. Proposition 2.1 gives a necessary condition for persistent convolution sequences. As we will see, this condition is not sufficient. In fact, this Ž. condition only guarantees that the first two roots persist. That is, xxyr2 Ž. divides qxn for n G 2.

2.3. Sheffer Sequences Ž. Given a convolution sequence qnnG0, we say that a polynomial se- Ž. Ž. quence snnG0is Sheffer relative to qnnG0if n y EsbnŽ. x sÝqxsynykk Ž. Ž., ks0 or equivalently if there is an invertible operator A such that Askks q for all k. The for sn is of the form ϱ n Žy1.Žy1. ÝsxtnŽ. s⌽Ž.Ž.xfŽ. t raf Ž. t ,3Ž. ns0 Ž.Ž . where f and ⌽ are as above and A s aDb seewx 20 . Examples here include the and the generalized Laguerre polynomi- als. Proposition 2.1 has a succinct Sheffer generalization. A Sheffer se- Ž. quence snnG0is determined uniquely by an umbral calculus b and delta Ž. series f and a. However, if snnG0is known to have persistent roots, then Ž. Ž. Ž. snnG012is determined by the umbral calculus b, qx, and qxalone. ROPOSITION Ž. P 2.2. Let snnG0 be a persistent in the Ž. umbral calculus b with respect to the delta operator f Db and in¨ertible operator aŽ. Dbn with roots r .

ⅷ If r12s r , then

r1 aDŽ.bbsErk and

kDb⌽ЈŽ.rD1b fDŽ.bs . c⌽Ž.rD1b SEQUENCES WITH PERSISTENT ROOTS 45

ⅷ If r12/ r , then

r1 aDŽ.bbskE rk and

r 2 Eb fDŽ.k 1 crŽ.r . bs r1y 21y ž/Eb Ž. Note that if r1 s 0, then snnG0 is actually a convolution sequence. Thus, a ' 1, and we may apply Proposition 2.1. Ž. Ž. Proof. Case r12/ r . Substituting x s r1and t s fu into 3 , we obtain

auŽ.s⌽ Žru1 .rk,4Ž. Ž. Ž . Ž. where k s s020 . Now, substitute x s r and t s fu into Eq. 3 which yields k k q cfŽ.Ž u r21y r .s ⌽Ž.ru2. ⌽Ž.ru1 Solving for fuŽ., we conclude

⌽Ž.ru2 fuŽ.sk y1 crŽ.21yr . ž/⌽Ž.ru1

Ž. Case r12s r . Shift the generating function 3 by r1, ϱ n ÝsxnŽ.qrt11s⌽Ž. Ž.Ž.Ž.xqrgtragtŽ., ns0 Ž 1. where gtŽ.sfty Ž.. Compare the coefficients of x on both sides,

232 gtŽ.2rgt11 Ž.3rgt Ž. ct s agtŽ.Ž. qqqиии ž/bb12 b 3

s⌽ЈŽ.rgt1Ž. gtagt Ž.Ž. Ž.. EquationŽ. 4 now implies that

⌽Ž.rgt11Ž. ctrks⌽Ј Ž.rgtŽ. ft Ž.

⌽Ž.Ž.r11 u cf u s k⌽Ј Ž.ruu

fuŽ.sku⌽Ј Ž.ru11rc⌽ Ž.ru. 46 DI BUCCHIANICO AND LOEB

Note that Proposition 2.2 gives a necessary condition for persistent Sheffer sequences. As we will see, this condition is not sufficient. In fact, this condition only guarantees that the first two roots persist. That is, Ž.Ž. Ž. xyrx12yrdivides sxnfor n G 2.

3. CLASSICAL UMBRAL CALCULUS

3.1. Umbral Calculus What are the persistent convolution sequences in the classical umbral calculus?

In the classical umbral calculus, bnbs n! is the usual factorial, D s D is yŽ. Ž . y the usual derivative, Epxb spxqy is the usual shift operator E , and ⌽Ž.t sexp Ž.t .

3.2. Con¨olution Sequences Up to scalingŽ. vertical and horizontal the only persistent convolution n Ž.x sequences in the classical umbral calculus are x rn! andn .

ROPOSITION Ž. P 3.1. Let qnnG0 be a classical persistent con¨olution se- quence. Then either:

nn ⅷ Ž. qxn scxrn!with ᎏdelta operator fŽ. D s Drc, and ᎏroots rn s 0 for all n G 1, or else

nn ⅷŽ. Ž . Ž Ž . . Ž .Žxrr2 . qxn scxxyr222иии x y n y 1 r rn!s crn with Ž. Žr2 . ᎏdelta operator f D s E y 1 rcr2 , and Ž. ᎏroots rn s n y 1 r2 for all n G 1. Ž. The classical forward difference operator is thus given by ⌬ bpxs pxŽ.Ž.q1ypx.

1 Proof. fŽ. D must be as given by Proposition 2.1. Note that ŽE y 1. Ž.xxx Žxq1 . Ž. Ž .. nnsysnny1

3.3. Sheffer Sequences The result for Sheffer sequences is surprisingly similar. Up to scaling Ž.vertical and horizontal the only persistent Sheffer sequences in the nnŽ. Ž.xya classical umbral calculus are of the form x rn!, x y 1 rn!, orn . SEQUENCES WITH PERSISTENT ROOTS 47

ROPOSITION Ž. P 3.2. Let snnG0 be a classical persistent Sheffer sequence. Then either:

nn ⅷ Ž. Ž . sxn scxyr1 rkn!, which is Sheffer Ž. nn ᎏrelati¨etoqn x scxrkn!, ᎏwith delta operator fŽ. D s Drc, r ᎏin¨ertible operator aŽ. D s E 1rk, and ᎏwith roots rn s r1 for all n, or else

nn ⅷ Ž. Ž .ŽŽ.Ž.xyr121rryr . sxn scr21yr n rk,which is Sheffer nn Ž. Ž .ŽxrŽ.r21yr . ᎏrelati¨etoqn x scr21yr n rk, Ž. Žr2yr1 .Ž . ᎏwith delta operator f D s E y 1 rcr21yr , r ᎏin¨ertible operator aŽ. D s E 1rk, and Ž.Ž. ᎏwith roots rn s n y 1 r21y r q r 1. r 1 Proof. By Proposition 2.2, we have aDŽ.sE1rk. Now, apply aDŽ.y r skEy1 to the sequences of polynomials mentioned in Proposition 3.1.

4. DIVIDED DIFFERENCE UMBRAL CALCULUS

4.1. Umbral Calculus

n ny1 In the divided difference umbral calculus, bnbs 1. Thus, Dxsx 0 Ž. ŽŽ. Ž.. for n ) 0, and Dxbbs0. Thus, by linearity, Dpxs pxyp0rx. y n n ny1 n The divided difference shift is given by Eb x s x q yx q иии qy s Ž nq1 nq1.Ž . y Ž. Ž Ž. Ž..Ž . x yy rxyy. Thus, Epxb s xp x y yp y r x y y , and y Ž. Ž.Ž. Eb s1r1yyb , or equivalently, ⌽ t s 1r 1 y t . Note however, that y yy Ž. the inverse of E is not Eb s 1r 1 q yb , but rather

y y1 Ž.Ebbs 1 y yD

yy1 Ž.Epxb Ž.sŽ. Žxyypx .Ž.qyp Ž.0 rx.

4.2. Con¨olution Sequences Up to scalingŽ. vertical and horizontal , the only persistent convolu- tion sequences in the divided difference umbral calculus are x n and n1 xxŽ.y1.y

ROPOSITION Ž. P 4.1. Let qnnG0 be a di¨ided difference persistent con¨olu- tion sequence. Then either:

nn ⅷ Ž. qxn sc x with Ž. ᎏdelta operator f Dbbs D rc, and ᎏroots rn s 0 for all n G 1, or else 48 DI BUCCHIANICO AND LOEB

nn1 ⅷ ŁyŽ. qniscs02xyir with Ž. Ž . Ž.Ž. ᎏdelta operator f Dbbs D rc 1 y rD2bb gi¨en by f D p x s ŽŽ. Ž ..Ž . pxypr22rcxyr ,and ᎏroots rn s r2 for n G 2. The divided difference forward difference operator is thus the operator

pxŽ.yp Ž.1 ⌬bpxŽ.s . xy1 Ž. Ž . Proof. By Proposition 2.1, fDbbsDrc1yrD2b. Now,

r2 EbpxŽ.ypx Ž. fDŽ.Ž.b pxs cr2

xpŽ. x y rpr22 Ž . px Ž. sy cr22Ž. x y rcr 2

xpŽ. x y rpr22 Ž .yxp Ž. x q rpx2 Ž. s cr22Ž. x y r

pxŽ.ypr Ž2 . s . cxŽ.yr2 Ž. Ž. Ž. In particular, for pxsqxn where n G 2, pr2 s0, so

ny1 ny2 fDŽ.Ž.bn q xscxx Žyr2 .sqxny1 Ž.. Ž. Ž. Moreover, fDbb cxs1 and fD 1s0.

4.3. Sheffer Sequences The result for Sheffer sequences is surprisingly similar. Other than those mentioned above, up to scalingŽ. vertical and horizontal the only persistent Sheffer sequences in the divided difference umbral calculus are of the n form Ž.Ž.x y axy1.

ROPOSITION Ž. P 4.2. Let snnG0 be a di¨ided persistent Sheffer sequence. Ž. nŽ.Ž.ny1 Then sn x s cxyrx12yr rk,which is Sheffer nn1 ⅷ Ž. Ž .y relati¨etoqn x scxxyr2 , 1 ⅷ Ž. Ž . with delta operator f Dbbs D rc 1 y rD2b,

1 Ž. Ž. Note that in this case fDb does not depend on r1, and fDbbsimplifies to D rc if furthermore r2 s 0. SEQUENCES WITH PERSISTENT ROOTS 49

ⅷ Ž. Ž .Ž Ž.Ž. with in¨ertible operator a Db s 1rk 1 yr1 Dbb so that a D p x s Ž Ž . Ž .. Ž .. xp x y rpr11rxyr 1,and

ⅷ with roots rn s r1 for all n G 1.

Proof. Case r1 s 0. This specializes to Proposition 4.1.

Case r1/ 0. It suffices to check that the operators fDŽ.bband aD Ž. Ž. are indeed those required by Proposition 2.2, and that fDbn sss ny1. For n)1,

sxnnŽ.ysr Ž2 . fDŽ.Ž.bn s xs cxŽ.yr2

n ny1 cxŽ.Ž.yrx12yr y0 s cxŽ.yr2

ssxny1Ž.. n Remark 1. The sequence Ž.x y a is of binomial type in both the n classical and the divided difference umbral calculus. However, Ž.x y a rn! n ŽŽ.resp. x y a .Žis a convolution sequence in the classical resp. divided difference. umbral calculus, but not the other. As we will see in Lemma 6.1, convolution sequences characterize in some sense their umbral calcu- lus. For this reason in part, we have chosen to work with convolution sequences instead of sequences of binomial type.

n Remark 2. The sequence Ž.x y a is Sheffer on the divided difference n1 n umbral calculus with respect to xxŽ.ya yand not x as its classical counterpart might seem to suggest.

5. HYPERBOLIC UMBRAL CALCULUS

5.1. Umbral Calculus Ž. Consider the hyperbolic factorials bn s 2n !. The hyperbolic derivative is given by Ž.2n ! nny1 Dxb s x . Ž.2ny2! Let z s 'x . 2n 2ny2 Dzbs2nŽ.2ny1z d2z2n s. dz 2 50 DI BUCCHIANICO AND LOEB

Thus, the hyperbolic derivative is simply the second derivative with respect to z s 'x . As a linear operatorwx 6 , the hyperbolic derivative can be expressed in terms of the ordinary derivative with respect to x, namely 2 Db sDx q D wx7, Proposition 2.4.2.1 . The hyperbolic shift is given by

ϱ ynn EbbsÝyDrŽ.2n!scoshž/'yDb, ns0 justifying the designation ‘‘hyperbolic.’’

PROPOSITION 5.1. The hyperbolic shift is gi¨en explicitly by

y''22 Epb Ž. x s pxž/ž/q''y qpxyy 2.

n Proof. By linearity it suffices to consider pxŽ.sx.

n Ž.2n! yn knyk Exb sÝ xy Ž.Ž2k!2n 2k .! ks0 y n 2n 2k 2n2k 'xy'y sÝž/2k ks0 12n 2n k 2n kk2nyk 2nk sÝ ''xy''yqyŽ.1 xyy 2 ž/k ž/ ks0

1 2n 2n sŽ.Ž.''xq''yqxyy. 2ž/

yyyŽ. Note, however, that the inverse of Ebbis not E s cosh iyD' b, but

ϱ yy1 n Ž.Ebbssechž/'yD s ÝE2nbŽ.Ž.aD r 2n !, ns0 where E2 n are the Euler numbers.

5.2. Con¨olution Sequences Up to scalingŽ. vertical and horizontal the only hyperbolic persistent n convolution sequences are x rŽ.2n ! and xx Žy1 .Žxy4 .Žxy9 .иии Žx y 2 Ž.ny1 .Žr2n .!. SEQUENCES WITH PERSISTENT ROOTS 51

ROPOSITION Ž. P 5.2. Let qnnG0 be a hyperbolic persistent con¨olution sequence. Then either:

nn ⅷ Ž. Ž . qxn scxr2n!with Ž. ᎏdelta operator f Dbbs D rc, and ᎏroots rn s 0 for all n G 0, or else nn12 ⅷ Ž.Ły Ž . qxnisc s02xyi r with Ž.Ž Ž . . ᎏdelta operator f Db s cosh 'rD2 b y1rcr2 , or more explicitly,

22 fDŽ .Ž. px px''r px r 2pxŽ. 2cr , b sž/ž/ž/ž/ž/q''22q y y 2 Ž.2 ᎏand roots rn s n y 1 r2 for n G 1. The hyperbolic forward difference operator is thus given by '' ⌬bpxŽ.spxŽ.Ž.q2xq1qpxy2xq1y2pxŽ..

Ž.Žr2 . Proof. By Proposition 2.1, fDbbsE y1rcr2. Now, by Proposi- tion 5.1,

22 fDŽ .Ž. px px''r px r 2pxŽ. 2cr . b sž/ž/ž/ž/ž/q''22q y y 2 Ž . Ž. Ž. Ž . Ž. We now show that fDbn q xsqx ny1 by verifying that fDbn q r has the correct roots.

22 fDŽ.Ž q i 1 .2r qi Ž11 .r qi Ž11 .r bnŽ.y 2sžnž/ž/ž/ž/yq''2 qn yy 2

2qiŽ.1r 2cr ynŽ.y 22/r

222 sž/qirnŽ.2qqinŽ.Ž.Ž.y2r2y2qin Ž.y1r22r2cr s0q0y0. 5.3. Sheffer Sequences The result for Sheffer sequences is surprisingly similar. Other than those mentioned above, up to scalingŽ. vertical and horizontal the only persistent Sheffer sequence in the divided difference umbral calculus is Ž.Žx y 1 x y 2 9.Žxy25 .Žx y 49 .иии Žx y Ž2n y 1 . .Žr 2n .Ž! see Table II . .

ROPOSITION Ž. P 5.3. Let snnG0 be a hyperbolic persistent Sheffer sequence. Then either:

k ⅷ Ž. Ž . r1s0 in which case the polynomials sn x s cxxyr2 = Ž.ŽŽ.2.Ž. Ž. xy4r22иии x y n y 1 r rk 2n ! with roots rns n y 1 r2 form in fact a con¨olution sequence as in Proposition 5.2, or else 52 DI BUCCHIANICO AND LOEB

k ⅷ Ž. Ž .Ž . r1/0in which case, the polynomials sn x s cxyrx11y9r= Ž.ŽŽ.2.Ž. xy25r11иии x y 2n y 1 r rk 2n ! are Sheffer kŽ.Ž.ŽŽ.2.Ž. ᎏrelati¨etocx xy4rx11y16 x иии x y 2n y 2 r1rk 2n !, ᎏwith delta operator

coshž/ 2'rD1by1 fDŽ.bs 2cr1 Ž.Ž. Ž''22.Ž . gi¨en by f Db p x s wpxwxq2''r12qpxwxy2r y Ž. 2pxxr2cr1, Ž. Ž . ᎏin¨ertible operator a Db s cosh 'rD1 b gi¨en by Proposition 5.1, and Ž.2 ᎏroots rn s 2n y 1 r1.

Proof. By Proposition 2.2, either qxnŽ.is a convolution sequence in Ž.r1 Ž . which case we refer to Proposition 5.2, or else aDbbsE scosh 'rD1b and fDŽ.b is equal to either

r2 kDb⌽ЈŽ.rD1bb E Qor Q k 1 crŽ.r 01ss r1y 2y1 c⌽Ž.rD1bbž/E depending on whether or not r12s r . Without loss of generality, we will take r1 s c s k s 1.

Case r12s r . Application of the hyperbolic transfer formulawx 13 allows us to calculate the first few polynomials sxnŽ.,

sx0Ž.s1

sx1Ž.s Žxy1 .r2 2 sx2Ž.s Žxy1 .r24 SEQUENCES WITH PERSISTENT ROOTS 53

2 sx3Ž.s Žxy1 .Žxq7 .r720

2 2 sx4Ž.s Žxy13 .Žxq90 x q 163.r40320.

We note that y7 is a root of s34, but not of s . This contradicts the assertion that sn has persistent roots.

Case r12/ r . Again by the hyperbolic transfer formula,

sx0Ž.s1

sx1Ž.s Žxy1 .r2

sx22Ž.s Žxy1 .Žxyr .r24

sx32Ž.s Žxy1 .Žxyrx .Žy4r2q11.r720

sx42Ž.s Žxy1 .Žxyrgx .Ž.Žr120960 . , Ž. 2 Ž.2 where gxs3xq129 y 39rx222q108r y 641r q 696. Since 4r 2y 11 is a root of s3, it must also be a root of gxŽ.. Thus,

0 s g Ž.4r2y 11

s40Ž.r2y 9.

3 Hence, we must have r2 s 9, and using the sum of cubes identity ␣ q 3 22 ␤sŽ.Ž␣q␤␣y␣␤ q ␤ ., we have

coshŽ. 3't ftŽ.sy12 ž/coshŽ.'t

3''t 3t eqey 1 sy''tt 2Ž.eqe2

2''t 2t eqey sy1 2

scoshŽ. 2't y 1.

As in the proof of Proposition 5.3,

22 pxž/ž/''q2qpxy2y2pxŽ. fDŽ.Ž.b pxs , 2 Ž.Ž.Ž.Ž . Ž Ž .2. so that fDb xy1 xy9 xy25 иии x y 2n q 1 has roots 2 1, 9, 25, . . . ,Ž. 2n y 1. 54 DI BUCCHIANICO AND LOEB

Note that this proof distinguishes itself from the analysis of persistent Sheffer sequences in Propositions 3.2 and 4.2 in that the necessary condi- tions of Proposition 2.2 are not sufficient, so a certain amount of case analysis is required.

6. COMPLETENESS

In this section, we will show that as enumerated in Table I, the three umbral calculi treated above give all examples of persistent convolution sequences.

HEOREM Ž. T 6.1. Let qnnG0 be a persistent con¨olution sequence in some umbral calculus b. Then without loss of generality, one of the following holds: Ž. Di¨ided Difference bn s 1 for all n, Ž. Classical bn s n! for all n, Ž. Ž. Hyperbolic bn s 2n ! for all n, or else Ž.Ž. Ž.n Tri¨ial qnn x s cx rb . Note, for example, that there are no nontrivial persistent convolution Ž. sequences in the ‘‘ultrabolic’’ umbral calculus bn s 3n !. We must first explain what we mean by ‘‘without loss of generality.’’ The first three generalized factorials b01, b , and b 2do not affect the umbral calculus. In particular, the first three generalized factorials can be arbitrar- ily changed while keeping exactly the same sequences of binomial type.

LEMMA 6.1. Let b be an umbral calculus. Then for any nonzero constants X X X ␤, ␥, there exists another umbral calculus bЈ where b01s 1, b s ␤, b 2s ␥ Ž. such that any , qnnG0 is a con¨olution sequence in b if ŽŽ..Ј Ž ␤ . and only if ann q x nG00 is in b where a s 1 and ans b1r for n G 1.

Proof. We will consider successively b01, b , and b 2. Suppose qxnŽ.is a Ž.Ž. Ž. Ž.␦ convolution sequence, so fDbn q xsqx ny1 and qnn0 s 0. Ž0. Ž0. Ž0. Let bnns b rb0for all n, then bnnrb y1s bnnrb y1, and hence the Ž.ŽŽ.Ž.. generalized derivative is unchanged Dbb0s D . Thus, fD bn0 q xs yy Ž . Ž. Ž.Ž Ž0. fDbn q xsqx ny1 . Note that Ebbs E rb0. It is conventional to take Ž0. 0 . b0 s1 as here and in the examples above so that Eb is the identity. Ž1. n Ž0. Ž. Let bnns cb . In particular, c s ␥ b21r␤ b . Obviously, the general- Ž1. Ž . Ž. Ž. ized derivative is rescaled Dbbs cD . Thus, fD bnnrcq x sqxy1. ycy Ž Ž. . Note that Ebb1s E rb0. XŽ1. XŽ. Finally, let bnns kb for k G 1 and b0s 1. In particular, k s ␤rc. n n DxbЈ is equal to DbŽ1. x unless n s 1 in which case the former is k times Ž.Ž. Ž. Ž. Ž. Ž .Ž. the latter. Thus, if rxsDpxbb1 , then DpxЈsrxqky1r0. SEQUENCES WITH PERSISTENT ROOTS 55

kk Ž. Ž. Ž. Ž More generally, if r s Dpbb1 , then by induction, DpxЈsrxqky .Ž. Ž.Ž1. Ž.Ž. 1r0 . Thus, since qny1s fDbnrcq , we also have fD bЈrcqn x s Ž. Ž . Ž. Ž. Ž. qxny1 qky1qny1 0sqxny1 for n ) 1 since qny1 0 s 0. Ž. The root r1 is automatically zero, since qn 0 s 0 for n G 1. By Proposi- tion 2.1, r2 cannot be zero unless qn is trivial, so we may without loss of Ž. generality set r22s 1 replacing x with xrr otherwise . Similarly, the generalized factorials b012, b , b have just been shown to be arbitrary. To complete the proof of Theorem 6.1, we will show how bn for n G 3is determined by r3. Conversely, we will show how bnndetermines r for nG3. At this point, the only variable left to determine is r3. Ž. Recall that by Proposition 2.1, the delta operator Q s fDb is deter- mined by b. Thus, by the general transfer formulawx 13 , qn is determined by b. Requiring that q53has the correct root imposes conditions on r . These conditions have six solutions of which three can be eliminated by consider- ing q6. The other three lead to the classical, divided difference, and hyperbolic umbral calculi, respectively.

EMMA Ž. L 6.2. Let qnnG0 be a persistent con¨olution sequence with roots rn in the b umbral calculus. Without loss of generality, b012s b s b s 1, Ž. Ž. Ž . qx12sx,and q x s xxy1.For n G 3, each generalized factorial bn is determined by the root r3. In particular,

r3 q 1 b3 s 2 2 Ž.r333q 1 Ž.r q r q 1 b4s r3q5

2 32 Ž.r3q1Ž.Ž.r 33qrq1r 333qrqrq1 b5s 2 4r33q6rq7

2 32 432 Ž.r3q1Ž.Ž.Ž.r 33qrq1r 333qrqrq1r 3333qrqrqrq1 b6s 43 2 . 3r33q8rq14r 3q 14r 3q 21 Note that the numerators are the Gaussian factorials. Proof. As in Proposition 2.1, we use the generating function Ýϱ Ž.n ⌽Ž Žy1.Ž.. ns0 qxtn s xf t evaluating now at x s r3 2 Žy1. 1 q rt333qrrŽ.y1ts⌽Ž.rf3 Ž.t . Now, substitute t s fuŽ.s⌽ Ž.uy1. 2 ⌽Žru3 .sAqB⌽ Ž.u qC⌽ Ž.u ,5Ž. Ž.2 Ž. Ž. where A s r3333y 1,Bsr3y2r, and C s rr3y1. 56 DI BUCCHIANICO AND LOEB

␾ ⌽Ž. Ýϱ ␾ nn Let nns 1rb so that u s ns0uu . Identify coefficients of u in Eq.Ž. 5 for n G 3

n ␾ n ␾ ␾␾ nr3 s B ninq C Ý yi is0 ny1 ␾␾␾ sŽ.Bq2CniqCÝnyi is1 Cny1 ␾nisÝ␾␾ni. rnB2Cy 3yyis1 By hypothesis,

qx0Ž.s1

qx1Ž.sx

qx2Ž.sxx Žy1 .

qx33Ž.sxx Žy1 .Žxyr .rb3.

Using the general transfer formulawx 13 and Lemma 6.2, we compute

2 qx43Ž.sxx Žy1 .Žxyrx .Ž.yŽ.5r3q1rŽ.r3q5rb4 32r7r3 2 q33q qx53Ž.sxx Žy1 .Žxyrx .Ž.yrx3q2b 5 ž/2r33q3rq7

2 qx63Ž.sxx Žy1 .Žxyrx .Ž.yr3

43 2 2 Ž.3r33q8rq14r 3q 14r 3q 21 x 65432 = yŽ.7r333333q14r q 35r q 24r q 11r q 15r q 14 x 3b6. 7543 qŽ.21r33q 7r q 14r 3q 14r 33q 3r q 1

Ž2.Ž . Ž. Now, the root r43s 5r q 1 r r 3q 5 must persist in qx5. Thus, either 2 Ž 3.Ž 2 . r43sr or r 4sy3q2r3q7r 3r2r 3q3r 3q7 . Solving for r3yields Ž.' the following solutions: r3 s 1, 2, 4, y1r3, y1 " i 3 r2 which we shall consider individually.

r3 s 1 leads to the divided difference umbral calculus.

r3 s 2 leads to the classical umbral calculus.

r3 s 4 leads to the hyperbolic umbral calculus. SEQUENCES WITH PERSISTENT ROOTS 57

Ž.Ž.Ž.Ž. r344s1r3 leads to r sy1r3, qxsxxy1 xq1r3 xy1r3rb4, Ž.Ž.Ž.Ž.Ž. and qx55sxxy1 xq1r3 xy1r3 xy1r9rb so that r5s 1r9. Ž.Ž.Ž.Ž.Ž2 . Then qx6 sxxy1 xq1r3 xy1r9 12852 x y 7084 x y 280 r 12852b64. Note that r sy1r3 is not a root of q6, a contradiction. Ž.'' Ž . Ž.Ž. r345sy1"i3r2 leads to r sy1.i3r2 and qxsxxy1= Ž.Ž.2 Ž..' xqxq1xy2 which implies r56s 2. However, q 2 s 91 y 7i 3 r 3/0. We now have completely classified all persistent convolution sequences Ž.Table I . Nevertheless, consider the following result communicated to us by H. Niederhausen: Ž. Let qnb be the con¨olution sequence for f D in some umbral calculus b. Then qŽ. x x is a Sheffer sequence in the umbral nX q1 r Ј . calculus b where bnns b q1 Ž. Ž . Ž Ž .2. Since qxn sxxy1иии x y n y 1 is a convolution sequence in the hyperbolic umbral calculusŽ. see Subsection 5.3 , it follows that the persis- Ž. Ž .Ž . Ž 2. tent sequence sxn sxy1xy4иии x y n is Sheffer. Note however that sn does not appear in Table II. We hope to continue working on the classification of persistent Sheffer sequences.

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