,:;i
o 7 • , 1 152
o6 5
7 1 0
.72
, • 7
1 , •
I 2 t
5 , J
2 I ,
, 5 ,
, 0 1
~ ,
, 1 7 • , 2 , 350
5 0 Z 7 • , 3 _ , 1
, , 1 I 7 • 9 502
• , 1 3 .
, t • 7 1 Z 3
702
,
, S , • 2 J _
1 0 , t J , 5 ' 1 S 5
o7 • ,
J 2 10_ 5 ,
•. , 7 •
7 ~ 3 2 560 I • , 7
• 1 5 , , • 1
, • 7 , 0 1 2
5 , 0 1 7 • ,
0 1 2 3 • , 7
23' 5 , 7 •
COMBINATORIAL
MATHEMATICS
'EAR
,~
J'" - .,.. 1"D
ON '.fHE FOUNDATIONS OF COt-lliINATORIAL THEORY III.
Theory of Hinomial Enumeration
by
Ronald !1ullin
University of Waterloo
Waterloo,
Ontario
Gian-Carlo Rota
,
Massachusetts Institute •
of Technology
.
Cambridge, ~mssachusetts
Department of Statistics
University of North Carolina at Chapel Hill '
Institute of Statistice ~ameo Series No. 600.24
April 1970
This is the final version of lecture notes for a series of lectures given by
tJU3 secortd author at thB Swrrner Study Institute on Coroinatol'ial Matherratics
and Its Applioations in the iVatu:raZ SoUmOGS, held on June 2-27, 1969, in
Chapel BiZZ, and sponsored by the u.s. Air Force Office of Scientific Research
. under Grant No. AFOSR-68-i406 and the U.S. Army ResearCh Office, Durham, under
G~t No. DA-ARO-D-31-124-G910.
ON THE FOUNDATIONS OF COMBINATORIAL THmRY
III.
Theory ot Binomial Jmumeratlon
by
Ronald Mullin
and
Glan-Carlo Rota
1.
2.
3•
4.
5.
6.
7.
8.
9.
10.
11.
Introduction
Reluctant Functions and Trees
Fundamentals
ExPansions
Closed Forms
The Automorphism Theorem
umbral Notation
The Exponential polynomials
Laguerre polynomials
A Glimpse ot Canbinatories
Bibliography
••
. . . ~;, " .
1.
1-1
1.
Introduction.
The present work is born trom the interplay of
. .
.;.
'\.,...,
.. .
two seemingly dj:'sparate branches 'of combiriator1al: theory.•
~,'
.",
."
The first is the ,cl£ss1cal calculus of ·:fin1 te .41fferences l
Which has been in the paatmore otten related to numerical
analysis than to problems of em.nneration.
In the calculus
of finite differences l there occur several sequences of
polynomials which are used ininterpolat1onl numerical
othe~
ClJ.ua.dra.ture, and several
such se,uences ot
connections.
polynomials~are thelower
Typical of
,factorials
and the upper factorials
p
;
,~
e.
(2)
'f
Less well known, but·
e~ualiy. s1gn1~1cant
are the Abel POlyn·bmiais". . s;tud1eci
J~
:
:
!'
by
polynomial sequences
Abell HurWitz and others:
r
(3)
and the exponential polynomials, studied by Touchard and others,
...
.I
2.
1-2
where
S(n,k)
second kind.
••
denote the tamiliar Stirling numbers of the
Another significant
sequence is the Laguerre
polynomials
In ( x )
(5)
n! (n-l)(
k
- ~ ltr k-l -x).
-
which have an extensive literature.
polynom1als~
These
.
se~uences
of
as well as a large number of other sequences
that have arisen in classical analysis and comb1natorics.
share a common property: that ot being of binomial tyPe.
We say that a sequence
ot polynomials Pn(x}. Where Pn(x)
is ot exactly of degree of n.
is of binomial type When
it satisfies the sequences of identities
(6)
Pn(X+Y) - k~ (~)Pk(X)Pn-k(Y). n - O,1.2~ .••
-
It Will be shown in the course of' this study (and it is
verified without difticulty using the results below) that
each one ot the seCluences ot polynomials mentioned above
1s ot binomial type.
•
...
'.
3.
1-3
This work 1s a study of certain analYtic or (more
suggestively) algebraic-combinatorial properties of
sequences of polynanials of binomial type.
The main
problem we aim at is the following: given two selJ.uences
Pn(X)
and
~(X)I
exist coefficients
both of binomial typel there clearly
cnk'
Which express one sequence
other.
the so-called connection constants,
o~
polynomials in terms or the
OUr problem is to determine as efficiently as
possible the coefficients
cnk in terms of m1n1mal
on the polynan1als
and
Pn(X)
fIn(x).
data
A few classical
instances of this problem are given below.
In trying to solve this problem we were led to develop.
a systematic theory ot polynan1alsequences of binom1nal
type.
The main novelties we introduce in this theory are,
first, a systematic use of operator methods as against
less efficient generating function methods l which were
used almost exclus1vely in the pastl and secondly a solution
of the cmlnection problem stated above l Which eluded past
workers in the field, and which we belleve to be remarkably
simple.
4.
1-4
Patches and bits ot the theory developed in this work
.
can be f'ound 1n the 11terature ot the last 50 years.. starting
l'1 th the work of' Fincherle and AJnaldi in 1900, tollowing
through the Dan1sh and ItaJ.1an schools ot calculus ot
f'inite ditterences..
cl'l]m~nat1ng
with the work ot the great
Danish actuar.1allst Steftensen.
The statement (thOUgh not,
uas.. the proot) or Theorem 4 below is due to him.
other results, such a8 the EXpansion
~eorem.
least intuited by Pincher1e and hi8 8chool.
that o\u" not1on ot umbral operator
A tew.
where at
But. webelloV8.
(a term introduced by
Sylvester and extensively used by the .invariant theorists
and by E. T. Bell. though never correctly
defined)~ together
With our solution of' the connection constants problem that
it yields, gives a new direction to the calculus of' finite
dif'ferences, even tor workers interested 1n pure1¥. analytic
matters.
It turns out
that there 1s a second and entirely
different point of view f'ram which the theory ot polynanius
of' binomial types can be looked at.
Each ot the polynanial
sefluences listed above can be interpreted 0.8 counting the
number of' ways of' placing
varioua restrictions.
~allsa
into aboxes-, SUbject to
'1'h1s ties in with the cl&lJs1cal
theory ot distributions and occupancJ'. which can be
5.
1-5
al texnatively considered &s making words out
ot an alphabet"
subject to various restrictions on the successions of
letters.
More precisely, we are given & set
elements and
&
set X with
functions from the set
various restr1ctions.
S
x
S
With n
elements" and we cons1der
to the set
subject to
X
'!'he restrict10ns are such that they
do not I1m1 t the range of the tunct10ns but only the dana.1n.
Thus, for example, the lower factorial powors (1) count the
number of one-to-one functions fram a set
to a set of
x
elements.
ot n elements
S1m1larly, the upper factorials
x(n) _ x(x+l)(x+2) ••• (x+n-1)
(8)
count the different ways of placing the balls
boxes
x
S into the
When a linear ordering is to be chosen ot the
1:&lls w1thin each box.
In the same vein, the Abel polynan1ala
x ( x-an )n-1"n - 0,1" ••• , an<x,
( 9)
can be cons1dered in combinatorial terms.
a circle of c1rcumference
of
~ensth
&
x,
Indeed, consider
and a set ot n
arcs
each
and each having the same rad1us ot curvature
•
t.~
"•.
6.
1-6
as the circle.
If we drop the arcs randomly on the
circumference ot the circle then the probability that no
two arcs overlap is easily seen to be
(10)
Thus the Abel polynanials "count" the ways (i.e., the
measure, since this case 1s continuous)
~
which the arcs
may be placed W1thout overlapping.
Whenever we count a set ot functions trom
to a set
letting
X,
set of
S
sUbject to restrictions on the domain, then,
Pn(x)
be the number ot such functions, we see
.1mmediately that
sequence
8.
Pn(x)
Pn(x)
is a
po~vnom1al
and that the
must be ot binanial type.
ot polynomials o-r
binomial
Thus, se,uences
type arise nAturally as the
unifying concept in the theory of distribution and occupancy.
Accordingly, the present study will be divided into
two parts.
In the first (the present) part we concentrate
on the analytic properties of polynomial seiluences ot binomial
type; the relationship to problems ot distributions and
occu~y
is discussed only in Sections 2 and 10, and is
meant only' as an introduction to the second pa.rt.
It turns
out that every sequence ot binomial type With positive
1-7
integral
coefficients can be associated to a counting
problem ot a certain class ot "reluctant ll functions, a.
In the second part ot this
defined in the next Section.
work we shall intrepret the analytic results derived here
in purely combinatorial, that is, set-theoretic ter.m8.
Perhaps the most satisfying
res~ts
ot this
~vest~gat1on
are.. tirst.. the unexpected relations of sequences of binomial
type with problems ot enumeration ot rooted labeled trees,
(Section 2).. and secondlyj the solution ot the problem ot
the connection constants, which has deep combinatorial
implications.
In several special cases, classical analysis has already
answered the problem ot the connection constants.
For
example, we have
~.
(ll)
where
t
-
k>O .
S(k,n)(x)k
(12)
(X)n.
t
s(k,n)~
k>O.
(13)
x(n) _
t 18(k,n)fxk
k.>Ol
s(k,n)
.
-
-
and
S(k,n)
first and second kind.
(14)
"
are the Stirling number ot the
Another example i8
X(n).
t n! (n-l)(x)
100 iT k-l
It
-
8.
1-8
Ln(x) •
(15)
ldlere .In (x)
~
-
(_l)k
*
(~:i);c
are the Laguerre polynam1a18.
We hope that thi8 introuction has given an 1dea ot the
scope ot the pre8ent inve8tigation.
In the next Section
we briet17 outUne 80me comb1na.tori&l connection8J thereatter to diSlll18s th_ in tavor ot the anaJ..ytietheo!7,
until Section 10.
We thank IC. SIt.lot
and S. Bmollar tor help in the
preparation ot thi8 manuscript, and the Statistics
Department ot the Un1ver8ity ot Horth carolina, a& well
&s the U.S.
A:rmy
Research Center at the ml!ver&ity ot
Wisconsin, and Florida Atlantic l1n1ver81ty tor g1v1ng
U8
. the opportunity to present these result8 to an &ct1ve17
contributing audience •
•
2.
Reluctant FUnctiona.
t; s.-x, Where trca now on 8
01ven a funct10n
w1ll be a t1n1te set with n
elements and X w1ll be a
t1n1te .et ot x el_enta. we can assoc1ate w1th 1t
the range d t 6 namely, the
"tunctor1allya two ob"ect.:
sUb-set ot elements ot X Wh1ch are image. ot some element.
ot S \U1der the tunct10n tJ
and the c01mage
~ .~
which
cons1st. ot the partit10n ot the set :I det1ned by the
tolloWing e,u1valence relat1on:
an el_ent a
ot X 1•
...u1valent to an element b ot 8 1t and only it
tea) - t(b).
set
ibua. the co1m&ge of t
1. a part1tion ot the
s.
We are now going to rather drastically generalize the.e
concept••
We detine a reluctant tunct10n tna 8
tollows.
It 1. a tunct10n t
union SUx.
the element t(t(s»
1t &ncl onJ.7 1t t(.) E 8,
1t and onl.7 1t
t(s),
&
to the disjoint
.ubject to the tollow1.Da re.triction.
ever,v element SE8,
that onl.y
traa 8
to X as
t(t(s»
.1JD1larly"
E 8,
etc.
our
For
1. detined
t(t(t(.») 1. detined
re.u1~ent.
t1n1te number ot tems ot the .84luence a,'
t(t(.», t(t(t(s»), ••• be w.l1-det1ned. A more
suggestive. it 1e•• preci••, _,. ot .tat1D& the ....
•
1.
10.
2-2
cond1t1on 18 the tollow1ng:
the ·orb1t"
tor evert el_ent a E S.
.,t(a). t(t(a». t(f(t(s»). •••
under 1terat1on ot the tunct1an t
in
x..
ot s
-eventu&ll7· enda up
Thus, 011. ID1ght 11&'1 that t
.ere 1t atop..
-reJ.uctant!T' _pa 8 into X.
Dlerange ot a reluctant funct10n
ot tho.e el_ents ot
ot S.
X
Wb1ch
co'Mse ot
aD ord~MZ7
Will cOll.1st
are Dlase. ot seae el_ent
just 11ke 1n the caae ot
on the other hand, we neeel
t
84
ord1na17 funct1on.
to smeNUze the not1on ot
tuncUan, ... defined a'b0Ye, to the
n-17 .1ntroduced conceptd a reluctant tunct1on.1lIherea.
the co1Mse ot an ord1n&17 tunct10D 1a .1ap17 a part1t101l
ot the.et 8"
the co1mase a reluctant funct1on. 1. so1nc
to be .ore thaD
&
partit10n ot 8.
In tact. tor eYe17
eJ.8Ient x ot X Wb1ch 1. in the range ot. the reluctant
tunct10n t,
the inver.e
~e
ot the el_ent x ·1.
det1Ded .a the .et ot all element. •
the ••,uence ot 1t. succe.aor.
enda up in X.
Dl. innr.e
ot 8 auch that
tea), t(t(a»" ...
eventual17
1mas.. ot diatinct el_ent. ot
X are diajoint aubaeta ot S.
1hua, to eve17 reluctant
tunction there 1a aaaoc1&ted a part!t1an ot the .et 8•
.1uat
llJe.e in
W1~
•
the cue ot
aD ord1n&17
tunction.
However,
_ch block ot suCh a par1iit1an there 1a • natural
11.
2-3
structure ot a _to_re
.......8..
t
o~
·h1lJtor~""
rooted trees describing the
ot the elCIDent. ot that block betore they end up in X. 1bus,
.e are led to detine the co1mage ot a reluctant tunction to
be a part1t10n ot tha .et 8,
together w1th a atructure ot
a rooted torest (1.e., set ot rooted tree.) detined on each
block ot the part1tion.
Bach rooted' toreat covering one
block ot the co1.m&Se 1. the ·inver.e 1JDqe- - in the
generalized sen.e juat described - ot an el_ent, x ot x•
. Bote 'that each block ot the co1mage can, be turther.
part1t10ned into the comocted caaponenta, that 1a, the
trees, ot the rooted tore.t.
'1'he result1n& part1t1on 1. a
retinement ot the co1maSe and haa the add1tlonal propert7
that each block has the structure ot a rooted tree.
thi. tiner part1tion
"
ot 8,
We call
together w1th the structure
ot rooted tree (S.. Barary or Moon tor det1n1tion.) on each
block ot w,
(recall th& t
the pre-imye ot the reluctant tunctlon
'&.
rooted tree 18 a part1al17
'Jhua, the co1m&g. ot t
ord.~d
t
set).
1s obtan1ed by ·p1.C!n& togGhtez-ll
all tho•• blocU ot the pre-1ege or t Which are -oventuall7
_pped- to the s..e ol_ent
x
in
Clearq, the'pre:'1m&Ie ot
~
...
l"OO
.......t-.e4
...
reluctant tunct10n 1.
}.&belod tor.st on the set 8,
tel'll1nolog~
t
X.
01~
&
tolJ.ow1D8 clAs.1cal,
&n7 rooted labeled torut
L on 8,
12.
.,'
nth
It block., tbat 1., cooa1.UD& ot It. rooted tree••
there are ev1dent1l' ~ reluctaDt tmcUoca WhOae pre-1Ma.
1.
the tore.t L.
B7 W&7 ot u:aaple, let ua cem.ider tbe .et ot all
reluctant tunetiona trc:a 8
-to-
to X (notic. tbat, our ua.
ot the word. "tra:a- and
i . not .tr1ct~ correct, but 1. , P'.f)"
neverthel••• aUge.tlY• •0 .e .hall keep ua1D8 1t). Let
cDit be the nWlber of rooted labeled tore.t. nth Ie block.
CD the .et
8.
!ben the n_ber 01' reluctaDt tuDct1OD. tI'ca
8 to X 1. mdent17 s1ft1l b7 tb. pollDmaSal
(1)
It 1. e..., to •••, b7 ...1IIpl. ccab1Dator1&1 arsaent
1dd.ch 1II1tat.. th• •taDdard .et-theoret1c proof' 01' til.
bsm.lal tbeoNl, that the ...u_ce Of po1lDaD1&1'
1. 01' b1DollSal tllte.
"n(x)
It 1s le•• obY1oua. and 1t Will
trivia11,. 1'oUow tl'aI 'tile pre.ent theoq (ae. sect10n 10)
, tb&t 'the
(2)
•
po1pal'.'.
"n(x)
ar8 pVeD ,bl'
UM expre••1.OD
.
13.
tbat1., that th87 are a .peclal ca.e of the Abel PO!l"nomial., .cornapond1na to a . -1.
'1'h1. l1.e. uaed1ate17
thecla••ical reau1 t of 0&71e7 count1ng the nuaber of rooted
tr....
.inc. rooted tre.. corre.pond to reluctant tuIlc'Uon.·
"b&v1D& as pr.e-1maIe a Part1t10n w1tho one block, and
the coetficient.
.
.0
are
cnl 1ft (2), which e,ual nn-2.
Ve define a blncD1al cla.. B of reluctant tuDct1CD8
"
as follow..
1'(8.X)
To eve17 .et
8 and .et X we a••iP a .et
of relucteDt tunct1on.a
t~
8
to
X.
1be
....1&ta8llt 1. -functorial- - or. in coab1Dator1&l 1.aD&UAce,
-unlabeled-.
80
. With
aDd X With
of theaeta
:r(8,X)
Xo
With 1'(80'%0).
of' the
denote. the .1.e
Pb(x)
dependa onl7 on the al.. n
'!!lu, 1t the pol1ftcm2ala
••ts P(8.X),
the tunct1cc
the ••t 8
DOW
.
to the crucial condiUon.
tema, the cond1tlon atate. that there 1.
In aet-theoretic
&
natural 1.c:aorph1_
Bere, . ' and E denote d1a~o1nt.,. of aeta,
product of aeta, and •
•
and the ai••..
of ·th. Ht X.
Ve cc:ae
'.
8
induee a natural 1ac.orpb1_
Pn(x)
x
~.
'!b1. . . . . that 1.c.orph1• • of the .eta
•
denote.
atand. tor ft&tUfttl 1·ac.orpblia•
..
'!'"
~
Dle var1able A ranae.over all sub.ets ot the set
We .et (tor lood reason.) . PC .X) • 1
••t.
x.
TaJd.n& the .1ze. ot both
.1d..
s.
tor aU non-_pty
ot· (*) we obta1D the
.,uat1OD .
Wb1ch expre•••• the
~t
that the polu.ca1al.
a ••,uence ot b1nc.1al 'D••
Bo~
Pn(x)
•
.pett'ne. condlt1CQ C*) .tat.. that
an
by "pleclDc
toaebter· tlft) reluctant tunctlcm. 1ft the tu11.7 B. . we aa&1 D
obta1n· a reluctant tunction 1ft the taa117.
It 1. a aenen.u&.d
••t-theoretlc .,..ralon ot the b1Dca1al theon..
TWo 1aportant -7S ot det1n1n& binom.&l cla•••• ·· B .
, ot reluctant functions are the tollo1d.D&.
•
Let
'I be a
r-11.7 ot rooted' tre•• '(It 1. 1I£IIater1&1 Wheth.r they are
labeled or unlabeled).
'!'be r.117 B(T)
will con.l.t ot all- .ra-
reluctaDt t\mctlcna 'Who•• pre-1M6.. are 1&bele4 tore.t. on
8
Mob ot Who••' ccapoa.ent. 1. leCBOrpblc to a tr.. 1ft
tawS17 'I.
C1e&r17 B(~)
tunct1cna.
In the .xa.ple ccmalderecl
~
1•. a _1n<-'al clU. otreluctant
aboY..
the taa1q . T
coul.ted of all rooted tne••
,
.
15.
2-1
. Thus, we see that the enumeration ot labered torests
is closely connected with the theory of polynomia.ls ot
'bir'rJ9.1 type. 'The family
T can be specited in innumerable
ways, which will be considered in the second j)&rt at the
present work.
For the manent, we shall give some illustrations
tbat show that the classical polynaD1a1s listed in Section 1
~
can be interpreted as enumerating binomial classes at
reluctant functions.
We have already seen above that the
. Abel polynomials can be interpreted
.&8
enumerat1ng the
binam1al class ot all reluct&nt tunct1oos, as leaat tor
a --1.
A aaneWhat more elaborate argument would show
that all the other Abel polynam1als, tor a
a negative
1nteger, enumerate other b1nom1al classes ot reluctant' tunct10nr
•
Perhaps the simplest example 1s g1ven by the se,uence
"p.. nns enumerates the binODl1al class B(T), 'where T
cOllsists ot a single tree, With one root.
,
'.""
Another interesting example is the s.,uence ot Laguerre
polynaD1ala In(-X).
Where
T, is the set
'!'hes. enumerate the b1nan1al class
ot all linearly ordered rooted trees.
Ve leave the ...87 veritication' ot thia tact to the reader.
A to1rth example comea tor the inverses. ot the Abel
polynCDiala, considered in Sect10n 10, namely, functional
digrapha, entaerated by the polynomiala
._,,--'......
_-
_ _ _
w
_
.. _
_
_
_
_
••
B(T).
2-9
Which do not appear at first sight to be or binomial type.
We prove that they are. by showing that they enumerate
binomial class B( T) •
Simply take
&
T to be the tamj,ly ot
all rooted trees, allot whose branches have length at most
two!
Given a binomial class
B(T)
ot reluctant functions.
we can consider the subclass ot those functions having the
property. that their cOimage coincides with their pre-1!&S!.
We denote. this subclas8 by Bm(T),
and call it the
monomorphic class associated with B(T); it gcmeraUzes tho
notion ot a one-to-one function.
'!'he monaaorphic class associated with ~
conslsts
precisely ot all one-to-one functions, enumerated by the
lower factorials
(x)n.
'lbe monaDorphic class associated
•
with the Laguerre polynomials tums out to be enumerated by
the upper tactorials
interpretatson ot
x(n)
x{n)
(as tollows fran the combinatorial
g1~en ~bove).
We state without proof (but the proot is 8&s1:) an
important result about monomorphic classes.
It the
seflu~ce
17.
2-9
.'
enumerates the binomial class
~('.r).
then the
e~uenc~
of
polynomials
~ese
examples such suttice to orient the reader to
the combinatorial aspect or the theory we are about to
develop.
The notion or reluctant function
does not exhaust
the interpretation ot setluences ot polyncm1als ot b1nan1&1
. type.
For example it doe8 not interprflt ccmb1nator1ally
those s8f1uences ot polynaa1als ot binomial type which have
••
..
,
18.
2-10
negat1ve or non-lntegraleoeff1cients.
Nevertheless,· we
. shall see in the second part ot this work that all setluenees
of
po~omial
type With non-negative coefficients can be
set-theoreticallY (or probabil1st1eally) interpreted by a
general1zation or the notion ot reluctant function,
whereas those with nega:t1va coett1c1en'ta can be interpl"eted
by sieving methods
(M"obiua inversions, etc.).
There is
also an obvious cormeetion With the theory ot canpound
Poislon·process8s.
Apologizing tor this sketchy introduction, we proceed
to begin the ansJ.ytic theory.
.; f:':.,
..
•
19.
3-1
3.
Fundamentals.
_
rte
'l'hroughout this paper, we shall be concerned with
of'all
the algebra (over .. field of characteristic zero)
-
'polynomials in one variable, to be denoted It •
By a polynomial sequence we shall denote a sequence 01'
polynomials P1(x), i - 0,1,2, •••
ot degree 1, tor all i.
Where P1(x) 18 exactlY
A polyncm1al .seCluence 1s said to be ot bj,nomial
~
it
it satisfies the infinite 8e"uence ot identitie8
All the polyncm1al seCJ.uence8 mentioned above are o:i~inan1al
type.
For seae sequences, such as
>ll,
this i8 • trivial
ob8ervat1on,but tor others, such a8 the Abel and TouCM.rd
pOlync:m1a.18, the verification that they are ot binomial t)'PG
V1ll be
&
consequence (a rather aimple one, to be sure) ot
our theor;y.
OUr stuq W1..ll revolve pr1marily around the study
ot
l1nee.r oparators on P considered as a vec'tor space.
I.
...
Hcncetorth, all operators we conoider Will be t&citly
assumed to be linear.
•
We denote the action ot an operator
T or the poJ.1D.em1al p(x)
by -rp(x);
this notation is not
()
20.
strictly correct; a correct version is
(Tp)(x).
However,
this notational license results in greater readability.
By way
ot orientation, we list scme ot the operators ot
fre4luent occurrence in the theory of binomial enumeration.
The most important are the shift operntora.
if,
operator, written
argument ot
18 an operator Which toran.lates the
& po~an1&l
the field, that 1s,
An operator
T
A shift
by
Where
&,
18 an element
a
ot
Jfp(x) - p(x+a).
which commutes with all shitt operators
.1s called a shift-invariant operator, 1.e."
rnf".Jf'T.
1he tollowing are 1mportant eXA'llples ot shift-invariant
opera.tors:
_.
~
(1)
IdentitY' operator Iz ,[1 ... ,[1.
(11)
Differentiat10n operator D: ,[l ... ~-l.
(111)
Ddtferenee operator
Where we wr1 te
A·
~I:
(X)n'" n(X)n_l'
r,
B in place ot
Where
1s the ident1t)" ot the t1eld.
(1v)
1he Abel operator
IJJf1.
n: .x(x_na)n-l ..
nx(x_(n_l)a)n-l.
(v)
Demou1l1 operator J: p(x) ..
rX+1p(t)dt.
J_
x
1
21.
3-3
(vi)
Backward difference operator A _ 1_£-1: x(n).nx(n-l
Laguerre operator L: p(x) - -r-e-tp'(x+t)dt.
'0
t2
(viii) Hermite operator H: p(x) - ,oTT __e- / 2p(x+t)dt.
(vii)
,g
(1x)
Central difference operator
6 -
(x)
r"
r 112-E- V2 : p(x) - p(x+ 1/2)-p(x- 1/2). .
Euler (mean) operator f\f - (1/2) (1+E) : p(x) ..
(~2)(p(x)+
p(x+l».
We define a delta operator" usually denoted by the letter
Q,
as
a shift-invariant operator for Which
~
18 a non-
zero constant.
Tm denvative" difference" baeltward ditterence" central
difference" Laguerre" and Abel operators are delta operators.
Delta operators possessmany of the properties of the
derivative operator" as we proceed to show.
Lemma
11
It Q 18 a delta operator. then Qa. 0
tor every constant
Proof:
6.
Since Q is shift invariant" then
Q,J/J'x - F!'Q,x.
By
the llneo.r1 ty ot
•
~
22.
3-4
QJf'x •
Q(X+&) • Qx+Qa • e+Qa,
since· Qx 1s equal to some non-zero constant
c
bY' detinition.
But &180
If'Q,x • Itc •
and
80
c+Q&. c.
Lemma 2:
Q
c
Q.E.D.
Hence Qa - 0,
It p(x)
18
1s a delta operator, then
&
~d
polynomial ot degree n
~(x)
18 a polynom1.al ot
degree n-l •
....
pr...o.....o_
.....
t:
It 18 suff1cient to prove the conclusion tor
the special case p(x).~,
polynan1al
that 1s, to show that the
rex). Q;l1 is or degree n-l (exactly).
the b1nan1al theorem and the l1near!tY' ot Q we have
Also by the shitt 1nvanance ot Q
t
so that
FraIl
~,
23.
r(x+a).
Putting x.
O~
we have
r
E (~)a~-k.
k>O .
.
-
eXpressed as a
polynamia~
in
al
The coeff1c1ent of an 18
by
Lemma .1.
Hence
r
J'Urther~ the coefficient of a n - l
.1s of degree
1s
Q.E.D.
n-l~
Let Q be a delta operator.
A p0lJrnan1al seClluence
i8 called the aefluenee ot ,!)as.1..c. E0lynomia18
Pn(x)
tor
~.
1t:
(1)
po(x). 1
(2)
Pn(O). 0 Whenever n>O
(3)
Qpn(x). nPn_l(x)
•
Using Lemma 2 1 it 18 eaaily ahown by induction that
everz
.......
~olta
pporator .!'!!!..! unique sftquenc!!!! !»&s1c Eo!maDials
_a...
a_~o_c_1a
.......t_ed
__
W1_t_~ ll.
For example, the basic pol1n<D1als tor
the der1at1ve operator are
x!'.
2".
We shall now see that severval properties of the
polynomial sequence
se~uor.ce
...
ot
b~sic
polynomials.
r'
noticed about
Jll can be generalized to an arbi tra~
The first property we
was that it was of binomial type.
turns out to be true for every seCjuence
Tb..l,.s
or basic poJ.)lnCDia.la.
and is one of our basic results.
Theorem 1.
(a)
It Pn(x)
operator Q.
is a basic se'luence tor some delta
thm it is a setluenee of polynomials ot
binomial type.
(b)
It Pn(x)
is a se«luence ot. polynomials ot
binomial type" then it is a basic sequence tor sane delta
operator.
Proot:
(a)
Iterating property (3) ot basic polynan1als. we
see that
and h$'lce th3t tor k . n,
.
-
3-7
e.
While
_...
Pn(x)
'ibus1 we may express
in the folloWing torm:
.'
.
Since any polynomial
basic polynan1ala
Pn(x)1
P(X)I
all polynanials
is a linear combination ot the
p(x)
this express10n a180 holds tor
1.e ••
Pk(X)
p(x) - k>0·
t
k'
...
NOlf suppose p(x)
L
(~~(x)J_-n.
,.,.....,
is the polynomial
Pn(ltty) ,
t
k>O
-
Pk(X)
it
•
Pn(x+Y).
k.
lQ'--Pn(X+Y) J_n.
.TtPIV
But
[Q.~n(x+Y)]x-o - [~krPn(x)]x-O
•
iben
[r~~n(x»)x.O
• [EY(n)kPn_k(x)]x.O
• (n)JtPn-k(Y)
-
.
26.
3-8
and so
Wh1cb meana that Pn(X)
(b)
18 ot b1nal1al typ.'.
Conv.rsely. 8UppO.. Pn(X) 18 a ....uenc. ot
b1nCll1al tDe.
Putt1ng 7 . 0
1ft the b1naD1a1 ident1t 7.
we have
81nce each Pi (x)
that
poCO) - 1
it tollows
(8l1d bence po(x) - 1) and Pi (0) - 0
tor &ll other 1.
8 . .uence.
1. exact17 ot degree 1.
~us
propert1es (1) and (2) ot bas1c
are sat1.tied.
We now tind .. delta operator tor wh1cb such .. 80tuerice
Pn(X)
18 the s.,uence
ot bas1c p0J.3nom1al..
Let' Q be
the operator denned bY' the propertY' that Qpo(X) - 0 and
Qpn(X) • DPn_l(x)
zero constant.
Q
tor
~l.
C1ear17 Q;x must be .. ncm-
Hence all that remains to be shoun 18 that
i. 8h1tt-1nvar1ant.
~.
,
3-9
.As betore we -7 tnv1a1ly rewnte the lenere.U.ed .
b1nom1al theora in term. ot Q:
and. bJ' ,linear!t7. thi. JD&7 be extended to all
P(~Y)· t
-
bJ' Qp
p
k_
Ii(x) Q-P(7).
P
k>O
How replace
po~ia1.:
and interchange
x
and
'to get
.
.
P (Y) _Jt..L
(Qp)(-7)· t
\I·-~(X).
QO
-Ir
But
(Qp)(-:r) • ~(Qp)(x) - aYQp(x)
and
• Q(p(x+y»
. ~.., QWp(x).
7
on the' r1gbt
28.
3-10 .
~.
1!1ua we have
rQp(x) • cufp(x),
tor aU
PO~OIIl1al8
p(x>., i.e.,
Q 18 shitt-invariant, Q.E.D.
29.
4-1
4.
!xp&nsions.
We shall study next the various ways ot expressing
a sh,1ft-1nvariant operator in terms of a delta operator and
its powers.
1'he
difficulties caused bY' convergence
tuest1~._.
aro JIl1n1m&l, and we shall get around them in the easiest
possible way.
Consider
on
z:
'lh ...1',
a
sequence or shitt-invariant operatora
Wo say that the &8fluence converges to
if for nery polynom1a1 p(x)
polynClll11als 'litP(x)
Tp(x).
1',
1D
written
the aetuence ot
converges pointwise to the polynomial
The convergence ot an inf1nite aeries or operators
18 to be understood accordingly'.
'lbe fol1oW1ns theor_ generalizes the Taylor expansion
theorem to arbitrar.;y delta operators and bas1c polYnomials.
Theorem 2. (first Expansion Theorem.).
sh1tt-inVariant operator, and let
1f1th bas1c .et
Pn(x) •
Then
~
Let l' be
&
be a delta. operator
30.
4-2
Proof':
are of' b1naD1al
Since the polynan1&la Pn(X)
type then, as usual, ve rewr1 te the b1nca1al f'ormula as
Ko"". mq reprd this a. a
and
~PP17
po~CID1al
1n the variable
'7
T to both side. to seta
Aga1n, b7 linearit7, tb1s expression can be extended to all
polJnCll1ala p. Atter do1ng thi. enel .etUna :I equal to
zero we lot
'l'p(x). t
-
k>O
Obv1ouc~,
. When T. I
~lor's
(T11t(:f)]
-.:r
yJ)
~k.P(x)
the beat-known example of' this Theorem 1•
and ~. D;
e.xpana1on.
then Pn (x) .,[1, and we have
A second example is Newton's o~ion" .
wh1ch h&8 three f'oms •. If' Q.. 6,
the coetf'1e1enta are
~.
then Pn(x). (X)k and
[T(x)k]X-O.
It
Q. 9
then
Pn (x) • x(n)and ~. (Tx(k) ]x-a. ~ basic polynomial" tor
~ • 6 • .,}/3.B-l/2 rill be determined later.
31.
--3
1be folloWing remark will be· uaed accaa1onally:·
Lemma:
~
It
18 a delta operator,
and
p(xh
.(~).
any polynom1als, then
[P(Q).(x»)x-O • Ct(Q)p(x)]x.o.
Proof:
By linearity, we need
When ,(x). PJt(x)
and p('l).
baa1c po!1ncm1als of Q.
on.ly
tfl,
consider the caees
Where Pk(x)
are the
But 1t 18 eaa7 to see that the
relat10n holds in this cale.
'l.B.D.
As a turther example of the use ottbe expansion
theorem, we denve the cla8s1cal lIewtan-Cote8 formulas of
We Wiah to find an expans1on, 111
numerical integrat1on.
tems of A,
of the Bemoulll operator J r
det1ned bT:
·.r
Jz»(x) •
p(t)4t.·
x
J
J
r
•
,<I+6f·1 .6.
1J
A.
jI+d-I
•
A
J
•
.1
1Jb1ch reduces the pl"Obl_ to tind1ng an expana10n ot J l
in tcma of 6. Usq the P1rst ~S1OD i'beorwa., thi- 18
. ta1rlT simple,
Where
where we note that the
the seem! kind.
J2
~
are the Bemoulll numbers of
evaluated in this way gives Simpson's
rule:
•
,x+2h
J_ .
x
1 2
1
4
1
•
p(t)4t ~ 2h(1+6+ 0 6 + 18tS A + 18'0" A +••• )p (x).
.
.
A t1n&l example 1. the classical &ller'. transformation
llh1ch tollows tlUl the 1dentlt1e81
. t (-l)ttr •
DO
J..
...+..
• 1/2
~+
ita
A
-1/2 ~ ¥A'D.
-.
-
.
,,;.
33.
4-5
or
course, in this case
,,~
.'
are diaregard1n& convergence
queationa.
We now tum
_.- .•
OU~
attention to the Abel poJ..ynam1al••
n.
The delta operator in this case 1s
1!\u., the Abel
polynomials are basic polynardals and hence· are ot b1naD1al
type.
'!'heretore, b)" Thm. 1 we have proved Abel's 1dept1t:(:
.. : not .eaa1l¥ PJ'OVM by direct methods.
We can
'I..
.~on 1beortaa to set en Abel expa.ns1on ot
tho
eX. . I;ndo$d,
W. do get the tollOlf1ng beautiful expans10n
convergent tor
A<O •.
be tho r1nS ot tol"Jlal power .$r1es 1n the var1able
tho . . .
t1el~.
on:t2 tho·r1n3 t
t,
over
Then there exiats an isomorphism trom p
of 8h1tt-1h~nant operators, Wh1ch carri."
t(t). t
.
. k>O
.~
-
into . t
bO
~ Qt.•
34.
Proof:
~ans1on
ibe mapping 1s already linear end by
1't1eOr8m" 1t 1s onto.
t~:Hl
Therefore, all we have
to verify· 1s that the map preserves products.
Let
T be
the shitt-invariant operator corresponding to the tomal
power series
t( t)
and let S be the -shitt-invariant
operator correspond1ng to
Where Pn(X) are the basic pollnomia1s ot Q.
.OW
• r
But
Pn(O). 0
tor
n>O
and
po(x). 1.
Beee. it tollowa
thAt· the only Den-zero'tams ot- the double sum occur Uhen
n • r-k.
1'hu8
:
~'.
.
Coro1lan 1..
~.B.D.
A shift-invariant operator If 1.
invertible 1t and onl:r It
':11'0-
In the tolloW1nl. we shall wr1 te
P -p('l),. Where ..
1s' a ab1tt-invar1ant operator ancl pet)
1s .. tormo.l power
••ries, to incUc&te tbat the operator P corresponds. to
under the lsClDOrph1.. ot
'the tormal pewer aerles pet)
. 1heorfD 3.
p(O). 0 and pI (O)PO ldlenover P 18 a
Hote that
shift-invariant delta operator.
For such fOl"DAl pewsI'
.eriea, a un1,ue inver.e tor.zal powor .•erie. p-l(t) exist8.
P,c:x:ollary 2.
Let" Q be
poJ.1rKD1&ll Pn(X).
lin
ZXpsn4
are Pn(a).
delta operctor
and let .(D). Q.
mirene romAl power aenea.
lroo.!.z
&
If
Bence
Let
1114 basic'
.-l(t) be the.
111m
in tems ot~.
'lbe coetficlent
36.
4..8
~
..
a formula which can be considered as a generalization or
Taylor's formula, and which specializes (tor
exampl~
for
Q • A it gives Newton's expansion) to several classical
expansions.
Now use the Isomorphism
Theorem~
with D as the
We get
delta operators.
p (a)
n,
~(t)n. eat,
n>O n.
r
-
Whence the
conclusi~
upon setting' u
= q(t)
and a. x, Q.E.D.
As an aside, we remark at this point a possibly useful
connection between basic polynomials and orthogonal polynomials:
PropoSition.
Let Pn(x), n -
or polynaniala ot binomial type.
inner product
polynomials
•..
be a sequence
Then there exists a unlf{ue
-
(p(x), ,(x») on the vector space P ot all
p(x)
J
under Which the sequence
an orthogonal sCfiuence and
inner product we have
so that
O~1,2,
(Pn(X)' Pn(X»
Pn(x)
is
- n!. Under this
....
31.
4-9
Proot;
-mapping Pn(x)
Let
T be the (un1Cluely det1ned)operator
n
to x,
tor all n.
Define the inner
product as tolloW8:
An argument 8imilar to. the proof at the Lemma. preceding
Theorem 2 8hows that this b1linear torm 18 symmetr1c
(8et p(x) • Pn(X)
and ,(x) • Pk(x». and that Pn(x)
i8 orthogonal to Pk(x)
tor· k,'n.
Finally,
whlchshoW8 that the b1linear torm 18 positive definite•.
It 18 triv1ally veritied that
(p(x).Pn (x) )/./ET.
1bU8
[~l1p(x)Jx-O.
the Expansion Theorem# in the tom
38.
_-10
18 the same &s tlB orthogonal expansion ot .(x)
to
the above inner
product~
relat1ve
ell. B.D.
We note that tor the LaGuerre polynomials, discussed
below, the inner product just introduced reduces to the
classlcal inner product making the Laguerre polynomial8
an orthogonal set.·
Note that tor the operators (1). (11). (111). (lv).
(v1). (vll). (!x)
described at the beg1nn1ng ot this
Sectlon the polyncm1ala defined 'there are
&8
~8
will be shoun in the course ot this study.
baaie .eu"
39-
.
5-1
. 5.
Closed FOrma.
We now introduce a clas8 ot linear operators ot
an altogether ditferent kind.
a factor x.
-
x'N"l. n>o.
i.
e..
replacing each occurrence ot
.;.... we obtain
lie may denote zp(x).
mAy
&
bo rogarded
be a polynomia.l
Multiplying each ter= ot p(:t)
x.
in the parameter
p(x)
Let
&
by
x n by
new polynomial in x which
The t1J"8t
x
in this
expre~slon
a linear open.tor since it roprosenta
&S
linear transformation ot polynanial into pol,yncm.ala.
call this the
mult1p~c:l.t1o.!! pl?er~2!: and
panmeter .!. underlined.
that the operator
-x
Thus.
we denote
.!: p(x) ... xp(x).
J..~
'Va
bi the
Note
1. not shitt-invar1ant.
Beton proceeding turthar. 1t should be noted that
Ji'p(x) • p(x+a)
x+a.
18
&
po1)'nomal in the formal parcmeter
Since the mult1plication operator 1s not sh1tt-1nver1mt.
we have the operator ident1ty:
-.
were .!!:!.' p(x) ... (x+&)p(x).
P!'2P081t1on 1.
It 'I 1s a shift-invario..nt operator,
thm
--
'1" • Tx-=t'r
40.
5-2
1s also a shitt-invariant operator.
11'1e proor is a
'1"
8~ra1ghttOrward
Expan~1on
8ect1OD~
aa a epeelal caae ot U.e
Theorem" that any shitt-invariant operator '1' elm
be Gxprosssd ae an expansion 1n the delta operator
~
'1' • kio lIT D Where
'_k
~. [TJr"]x.o.
corresponding to
T 1s
the indicator ot
T.
~
-
k~
if
t
k •
1" ('t)
.
Proot:
1.e.,
pov~r ~f)r1ea
t(t). Wo call t(t)
lf0p,0n1!:1-on 2 •. It T has indicator t(t),
has
.
D~
P\lrther, b7 the
1acmorphism theorem, (1beorem') tho rormal
'1"
We call
thQ -.P..
in
..c...h_e...r.-l...
e !1er1vative or the operator'!'.
We sa. in the previous
_..
ver1fication.
then
as 1ta indicator.
Straighttorward ver1t1cation ot coetficients
b7 Thoorem 3.
Wo note in passing Pineherle's POr.mula:
Hote that b7 the 1eomorphia theor_ ot tho precee41ng
SGction.. we al"o bave
(Ta)' ~ T'8+T8'.
••
41.
5-3
!rop0Sj. tior 3.
Q
i8
&
delta operator it and only it
for sane shift-invariant operator
Q • DP
Pt# Where
p.l
exists.
PrOOt:
__
i
It Q 18 a delta opera.tor. theft 1t can. be
written
Q.
t
ttl
).dt
~;
But
b7 det1n1tlOD ot
& 4~1t&
'tbm the cODcluucm
optJrator.
tollo,,~
at once.
Dlua it "e ••t
...
COftver.e17, .uppo.e Cl. DP where P 1. Ih1tt1nvar1ant and p.l
exist..
S1nce D and P are Ih1tt-
1nvar1ant, the- Q ilu.t be &1.0.
J\lrther" ahitt-1nvar1aDt
Operator. cOIIDute (b7 DleolWl' 3). '0 that
Q,x • DPx·. PJ)x'.. P1
since P1
Ia 0..
Ia:Q tor an invertible .bitt-invariant operator.
'l.a.D•.
Hence Cl 1. a delta opentor.· .
!tutor_' (Closed toms tor basic polync;a1als).
Pn(x)
I~
1. a .etluenee ot basle pollnCllD1ala tor the delta
operator .. Cl • Dr,
then
Pn(X) • Q,p.n-1x"
Pn(x) • p.n,ll _ (p.n)i'-l
Pn(x) • x:R-n7!'-l
' .
(Rodri.;u.s-twe tol'llU1&)
Proof:.
-1
Pn (.) • x(Cl') .Pn-l (& ) •.
Ve ah&1l t1r.t lhow1;b&t (1) and (2) det1ne the
.. (D....DP' >.~.n-l.
"
\
.
••
1'bus. it we can show that
'nCO).
complete "the pl"OOf that 'n(Jt)
_...
. ~oj~.t'1od.31. for
~
aat1a1)' tormulas (1),
ot .quat1ons
0
tor n>O,
we Will
1. the .e,uence ot b=.61c
end 1 tWill tollow that the7 11111
(2~,
and (3).
(1), (2), (3) we •••
Bov,
t~
eqU1valenc~:
the
that
~
•
. 'n(x). xR-n~-l
.
and hencetn(O) • 0 tor
~v.
ft!O.
-.
'.\bus (1), (2), M4 (3)
been proven, an4 'n(x). Pn(x).
To prove (IJ);, we first invert formula (1),
gettinSi
..
DoUce that
'I
18 invertible,
a.
1s ... 1~ venned.
, ZUert1DS tb11 1I1to tbe ngbt .14e ottol'DlU1& (3) we leta.
Pn(X) • xr-n(QI)-l.pnpn_l(x)
• ~('I)-lpn_l(x)
!he
to~loW1n&
toJ'llU1aa, Dumbered (5) and (6), relate .
the basic pol1D<D1ala ot two d1tterent delta operatora 1ft
an analosoua W&7.' 'Dle1r proot 1a 1JIIIed1&te.
corollal%.
Let
operator. With ba.ic
Napect1ve17. Where
• • DB
JK)~la
.-1
rD(x) aDd Pn(X)'
and .p-l exiata •. · tbel"
(5)
Pn(x) • ~.(•• )-,.-n-l~~n(x)
(6"
Pn(x) - x(.-l.)Dx-'-rn(x).
" , l e 1.
!be. Abel PoJ.Jnc-tal. are the bU1c
h.'
. ~•.ot tbe Abel opel'&~r'
"
, . DP. be delta
and
(3)'
..
.
.'
1D4eed h-ca tolll\ll&
.
. p..(x) .·~-1'
( )"-1
. -x
!!!ARle 2.
x-aD
•
'.Lbe 1oWertaetol'1ala (x)n are the
...umce ·ot b&a1c poJ.Jnc-Sala tor
opel'&tor 6 - "1•. 81nce 6' -.,.
(') sift. ,S-ed1atelJ
~
loWer 41ttennee
the
Biodnsue.
toraula
.,
,
.... ......
:,
• :
"
.
.
'"
., '6.
t
..
,.
•
.ote that tile 'bU1c
dUt.~c:. open.'tcr ·6
po~&1'tor 'tile ·C.tal .
c.._e01»t&1De4 ~ (6) aDd •
1aUCb . . 1M ~ PO~"l:·"". •
wneet
.
~
-.'
.....
..
.
~.
flo
.
..
h.- (3).
.'
.
I
; .
.
'
.~
,
.
.
S'
,
.
.-
'
.....
'.
.
~.
tit·
t ..
. i
.,
.
.
"
•
t'l
:.
47.
6-1
..
Let .'1.. (l)
be tbe. alpbra of all l1Doar operaton
...
en the alaebra of all. P01J'ilca1.aJ.. P.. Let .E be the .ub-
",
.'.
aJ.cebra
CUI'
o~
-
1h1tt-1nv&ri&nt . operaton en P. lie now p:rove'
.un re.ult •
. 'l'heon. 5.
liet T be
Ul
.
nece.aar117 ah1tt-invariaDt. Let
QPen.tor 1..4C!>_
P
and Q be
operator. w1th ba.1e po1J'n<D1&la Pn(X)
_el
not
.
delt&
",(x).
rupeet1ve!J'. Aa. . . 'tbat
'.
theD
T-1
(a)
Uiata and
the-.p 8 .. .,..-1 1. an autaDorph1_ ot the
&1&ebra
E.
(b) If ap. eYeJ7 ....ulne. of baale
po~jal. into
. . . . . . . . 'ot . .ie' po1lD«'W2a1..
(c) Let p. p(D) ADd Cl'" t(D).
t(t)
' ••
_ere pCt) &Del
are tolWJ. power ••rie.. Let the delta operator a,
bav. tonaal POWI' ••n .. .expana1on reD)
~
b&G1c po1pom1ala
I'n(x) • t'bt'lD
...
.
•.
•
._- {t
liS.
c,-~
i~;
a se'1uence of ba.sic polynomials for
P -1
where
that is
t,h(~
delta operator
is the invcree :foJ"fllul pO....'.::H· series of'
pep-let»~
= p-l(p(t»
p(t),
= t.
Pro~f:
(a)
We have the string of identiticB:
ml(; '.:;:,.:-, ':;·:r:ry polyno;,1ial is
hli';,- ', . . <>,',,-:'/:':·;''1<'1.15, by
1"01 ,'.: .:, . . ) ,::..:r!t l ;r.1.i.als
t . ·1.'::.:~
;".
J.,
0.
linea.r ccmbination of the
linearity, we infer that
p(x),
that is,
TP = Q,T.
Tpp(x) =
Q.Tp(X)
It is clear
::'jotible, since it maps polynomials of degree
'-'r'l;.r.ls of decree
n,
for all
n.
Hence
I, ,...
'r',j •
6-3
\'lhence
rn
"•
for all
n>O.
Let
S
let the expansion of
s
be
S
~.n'y
;;Ltft-invariant operator and
in teni5 of
P
be
==
Then
(I)
and thu3
TST- 1
in a shift-invariant operator.
S .... TST- l
more the map
is ont;o since any shift-invariant
c:-: . . "r-~+or can be expanded in tenns of
le
';'.~ ~~,1tomorphismJ
Q..
ThUS" the map
an claimed.
vie have also ::;:-o\o':n t:i1at
OT"~':'~
Further-
T m§l.ps delta
: ,-~-; 1 nt.o del ta operators" since for delta operators
-
r·t.
: ...
"
G
-1
n (x) = Trn (x) and let' S == TRT • By
S is a delta operator since R is.
50.
Also
-
'l'RT-.L sn (x)
-
..• )
_.• '''Rr
J.J.
n\f .....
= n'l'r
.,(x)
n-..L
'1'0 complete the proof that
of
S
sn (x)
we need only silow that
5
n
are the basic polynomials
( 0)
:=:
0
for
fPO.
NO\'l
can write
since
aO
r n\(O)
= O.
=0
becauGe
Hence
R
is a delta
operator~ and
hence
we
"
,
j.l...
O-::J
(c)
in
P,
Now
say
Q, ==
~
and
f(?}
ca~
R
and
be written as power series
R = CI(!"l).
In equation (I)
above let
.,)
H
a
- s(P)
=
-
r'
n pn ;
';;T
k>O n.
-
then
S = TRT -'... == S(Q..) "
and therefore
R == g(p) = g(p(D»
and
S = g(o.) =
g(~{D».
Fina.lly we see that
r ( D) -- g ( p ( D) )
beD) = rep
-Ie D»
and
s _ g(Q)
==
r(p-l(q(D»),
Q.E.D.
52.
7-1
7.
Umbral
Not~tion.
In order to simplify the complex notation which
has been appearinG in
man~,.
of the above formulas" we will
make use a.nd for t:"lC f:irst time malt.e rirr,orous the "umbral
calculus" or "sy1i1bolic notation" first devised by Sylvester
(a (x)1
n
is a polynomial sequence then we simply note that there is
a uniGIue linear operator L on 1: such that L(xn ) = an(x).
and later used infoTrr.al1.y by many authors.
lve say that
[an (x)1.
If
is the umbral representation of the se~uence
L
In particular" an operator
T
with the properties
specified in the preceding 'l'heorem vli11 be called an
umbral .QPerator.
If
f(a(x»
L.
f(x)
is a polynoITlial then we use the notation
to denote the image of
It''or example"
denotes
a 2 (x).
!:(x)
denotes
Similarly,
a 2 (x)+(b+c)a1 (x)+bC.
f(x)
al(x)"
i111der the operator
\'lhile
[.!(X)+b] (!:.(x)+c]
(!.(x)]2
denotes
This is in essence the umbral
notation, which we signify by "boldface lettering.
Loosely speaking, umbral notation is a simple
technique using exponents to denote SUbscripts.
the
defini~G
binomial type
For example,
property for a polynomial sequence to be of
7-2
Pn(X+Y) ; k;O (~)Pk(X)Pn-k(Y)
-
can be restated "mnbrally
Note
L,
that~
a~
in view of our definition in tenas of the operator
this identity has a well-defined meaning.
Theorem 6.
If P and
are delta operators With
Pn(X) and qn(X)' ,and expansions p - p(D)
basic seCluences
and 'l. q(D),
Q
then the umbral composition
is the sequence of basic polynomials for the delta 9perator
R
a
Let
Proof:
petteD»~.
T be the umbral operator defined by
Tx? .,
By
«In(X).
the Autanorphism Theorem of the preceeding Section, it
...
follows that
se,uence.
T takes any basic
se~uence
into another basic
Now if
..
"
"~:~
..
54.
7-3
then
Thus
rn{x}
is a
se~uence
of basic polynomials and by the
Automorphism Theorem l it is the basic seGluence for
R = TPT- l • p(q(D»1
Corollary:
If Pn(x)
is a sequence
nomials then there exists a basic sequence
We say that
~(x)
Q.E.D.
is the ;nverse
se~ue?ce
o~
basic poly-
~n(x)
or
such that
Pn(x).
Theorem 7. (:Su.."nII1ation Fonnula).
~(x)
Suppose Pn (x) and
are the basic sequenceR for the delta operators P
and Q respectively.
If 'n{x)
is inverse to Pn(x),
then
55.
7-4
.,k
k
1
PIle=<) = k~O
J: IT r~· 1 J
A. L X x=O·
-
The proof is similar to the preceedine and is left to
the reader.
We are now in a. position to solve the problem stated in
the Introduction:
given basic sequences
with delta operators
the coefficients
linking the
Pn(x)
P
= p(D)
~
= ~(D),
and
4!n(X)'
how are
cnk
to the
constants" to be determined?
simple.
and
Pn(x)
~(x),
the so-called connection
The answer is dismayingly
Consider the polynomials
and consider the umbral operator
T defined by
".,
Then clearly
'"
56 °
so that
r n (x)
are of bino;:.ial t.y"pe and
q(t) = r(p(t»,
their delta operator, we find
ret)
l)eing
or
Theorem 4 then provides explicit
= Cl(p-l(t»o
eX'~ressions
R == r(D)
for the
rn(x).
One couldn't expect a simpler
answer.
As an exa.,,;;pIe, consider the connection constants bet\'1cen
n
qn{x) = x
and Pn(x) = (x)n0 Here' q(t) = t and pet) =
et_l.
ThUS,
the polynomials
polyno~ials
= log (l+t) and, as
ret)
rn(x)
~n(x),
we shall see below,
turn out to be the exponential
discussed below.
As a second ex~~ple, let
An easy computation shows that
Pn(x)
=
(x)n
and
ret)
=
t/(t-l),
~(x) = x(n).
Whose basic
polynomials are the Laguerre polynomials, also discussed below.
AR instructive example the reader may work out for himself - thereby obtaining a number of classical and new
identities, is to take Pn(x) = x(x_na)n-l and
qn(x) =
1
x {x-nb ) n-l for arb.
These examples could be multiplied
a1
infini UlD1, and a great nwnber of combinatorial identities
in the literature can be seen to fall into the simple pattern
we have just outlined.
Remark.
the algebra
operator
the proof.
T,
It can be shown that every automorphism of
E is of the form S ~ TST- I for some umbral
bu~
this fact will not be needed, so we omit
57.
8-1
8.
The Exponentia 1 Pol:rnordals.
The exponential polynomials, ztudied
by
?ouchard
and other authors, are a good testing ground for the theory
developed so far.
vie sha.ll see that their basic properties
and the identities they satisfy are almost trivial consequences
of the theory.
Consider the sequence of lower factorials
(x)n'
which as we have zeen is the basic sequence for the delta
operator
6
= eD-I.
In this case the inverse sequence is
the sequence of basic polynomials for the operator
~
=
log(I+D).
We denote these polynomials by
~n(x);
these are the e?cponential polynomials.
From the Corollary above we have umbrally
n
= x •
Further by the
sur.~ation
formula
=
where
S(n,k)
I:
k>O
k
S ( k,n)x ,
..
denote the Stirling numbers of the second kind.
Since
Q, =
lq:;(I+D)J
i';e
have
=:
Q,I = (I-i-:J) -'-
and ;:cnce
X.o;'\n_l (x)+x:', '.'1_1 (x),
which is the recursion formula for the
exponcn~ial P01~lC~i~ls.
The next property of these exponential polynomials which
we shall prove is expressed wnbrally as
Let
T
so that
be the urr,bral operator that takes
Txn
= ~n(x).
Hence
Tx(x-l) r..- 1
or changing
n
to
n+l
'Ix(x-I) n = xxn = .xmJ. ( ,.••• ) n
(x)n
into
xl1 ,
3-3
can be rewritten as
We can extend this by linearity to all polynomials
p(x)
so that
Replacing p(x)
p(x+l)
by
we have
TxE -'....p(x+l)
Hence
Since
0=
xTp(x+l).
Txp(x) = xTp(x+l).
T~
= ~n(X)
Tp(x) = P(2(x»
then
and it follows
that
~(x)p(~(x»
-
=
TXp(x)
:; x'I'p(x+l)
=
n
If we let p(x) - x
the~
-
xp(?(x)+l).
cO.
which is what we wanted to prove.
In a. similar vein one can prove
.J~.r~e
remarkable DobinskY-
type formula.:
n
'"·J<k
- '
kI'
which we shall leave as an exercise to the reader •
.
6: -i
.
9-1
9.
Laguerre
.
Polynomial~•
As a further exarr.ple of the a:bove theory, "Ie sha.ll
develop
~guerre
So~e
properties of the LaGuerre polynomials.
operator
iz
L
LP(x)
=
c.cfinE~
I'CIO
_I
The
oy
e-Xp'(x+t)dt.
o
It is a delta operator and as such has a sequence of basic
,
polynomia.ls which we shall call
~a.lt;. . . t lation
Ln(x),
By straightforward
, ''Ie find that the expansion of
L in terms of
D has coefficients
= 0,
n-O
and hence we find that
D
Lsn:Y'
Hence from formula (3) of Theorem 4 we pave
.
(*)
"....
62.
9-2
Since for all
polyno~ial
~(x)
we alse have
= (D-I)P(x)
then
e~e-x =
D-I and hence
Therefore we obtain the classical Rodrigues
for.mula~
From for.mula (*) we find by binomial expansion that
where the coefficients
n! (n-l)
k! k-l
are known as the Lah numbers.
nomials
Ln.
polynomials
Our notation for the poly-
corresponds to the notation in Bateman for the
Ln(-l)~
that is,
63.
9-3
We now
co~e
to
th~
Laguerre polynomials.
most irr.portant fact about the
The indicator of
ret)
=:
L is
t
t=r
and hence
t
f (f (t
» -
t=T
-~.;.---- =
1
tr- -
t
t-t+1 .. t
Thus, by the Automorphism Tneorem we infer that the
Laguerre polynomials are a self-inverse set.
as
~immediate conse~uence
Thus, we have
the beautiful identity
other identities concerning LAguerre polynomials stem
from the fact that
D
L=li=I.
Since Ln(X)
.
are the basic polynomials of L we have
, . ., ...
..
............ ~_(X)' = nI:''',_l(X)
n
;';-J.
and
~ence
the
classic~l r~cursion ~oL~ula
In fact, if we expand
D
b-r =
lITe
-D •
into series for.m
I
:-::5
-=
. 2
-D( I+D+D + •.. )
ca..'"l us e this to get other known recu.rsion
for~1lulas.
65.
10-1
10.
A Glimpse of Combinatorics.
Although we intend to leave most of the combinatorial
applications ot the preceeding theory to the second part ot
this work" we shall outline two typical results which we
hope will orient the reader to applications to problems ot
enumeration" typical ot the second part of this work.
Theorem 8.
operator.
Let P be an invertible shift-invariant
Let Pn(x)
be a se,uence
~of
basic polynomials
satisty1ng
,
for all n>o.
'!ben Pn(x)
is the se,uence ot basic poly-
nomials tor the delta operator
Proof:
I.
~.
DP.
Define the operator Q by Ql. 0 and
and extending by linearity.
Note that Q 1s shift-invariant.
In terms ot Q,. the preceding 1dent1ty, can be rewritten in
the torm
.' ..
~
66.
10-2
By linearity, this extends to an identity for all polynomials
p(x) - a.n argument we have often used in this work.
ThuS., recalling that
p(O) - 0,
[x-lp(x)J x_o
=
[Dp(x)]x_O
whenever
we have
[Dp(x)]~o • [p-1Qp(X)]x=o
for all polynomials
p(o)
~
O.
Setting
p(x),
p(x)
a
including those for which
~(x+a)
we
obt~,
using the
shift-invar1ance of P and Q.,
for all constants
Corollary
~.
cnl' n = 1.2, ••• ,
polynomials
Pn(x)
&.
But 1h1s means that
D. p-1Q.,
or
Given any sequence of constants
there exists a unique se,uence of basic
such that
[X-~n(x)]»=o" ·cnl'
that 1s,
67.
10-3
Corollarl 2.
the above.
Proof:
Let
be the indicator o£
g(x)
Then g - t -1 ,
where
r(t) -
From above
D - ~p
-1
2
E
-
k>O
~
in
tk
Ck,l IIT.
Qk
~ Ck,l iT - t(~)
k>O
-
and the result· tollows.
We now
give
some app11cat+ons of the above theory.
Application 1.
Let
tn, k be the number of forests ot
rooted labeled trees (l.e., trees with a dlstinguished
vertex) With n
vertices and
Proof:
tn,l
on n
vertices..
tree on n
on n-l
Since
then
k
components,
then
is the number of rooted trees
tn,l -
~_l(l)
since each such
vertices may be obt&1ned by mapplng a forest
vertices onto a slngle new root vertex., The
resulting root may be labeled in n
ways.. i.e., either by
using a new symbol or by using one of the
...
and replacing it by the new symbol.
be wrltten
n-l old symbols
But this relatlon may
68.
10-4
and hence the delta operator for ~
8.
is
DE- l
by Tneorem
Thus the associated polynomials are the Abel polynomials
x(x+n)n-l.
Corolla.ry; (cayley).
n
vertices is
Proof:
nn-l
The number of labeled trees on
nn-2.
Since the number of rooted labeled trees is
the number of unrooted trees is
tree can be labeled in n
Application 2.
Let
nn-2
since each tree
ways.
Sn be a symmetric group on n
symbols and let
cn , k be the number of group elements which
consist ot precisely k cycles. It Cn(x). ~ Cn,kxk
k>O
then Cn(x) a X(n).
Proof:
We note that in this case
C· 1 - (n-l)!
n"
Which 1s clearly the number of group elements consisting
or just one cycle, and thus by Corollary 2 this is the
required sequence.
F)mctional Digraphs.
permitted) on
~~lv
A digraph,
D"
(With loops
n symbols is a functional digraph if and
if it satisfies the following two postulates,
1)
each component of
D contains precisely one
cons1stently directed circuit; and
69.
10-j
2)
ea.ch
non-~ircuit
circuit
edge 1s directed towards the
contair~ed
An idempotent 15
a.
in
its CClllponent.
functiona.l digraph all or whose
components contain a distinguished vertex which meets every
edge of that componffilt.
Applica.tion 3.
k
x
1s of binomial type.
idempotent on
Then
V,
Toe polynomial
h
k'
-"h,
n
Let
symbols with
= (~)kn-k
.It
may be chosen in
Pn(x) -
E
k>O
-
(~)kn-k
hn,k be the number of
precis~ly
k
components.
since the k distinguished vertices,
n
ways and the remaining n-k
(k)
points may be directed into
V in
kn-l-:
ways.
However..
we may also view each idempotent as a structure generated
by its components.
coefficients
has indicator
It is interesting to note that the
h_
·n,l - n
fC-l](t)
and the associated delta operator
where
ret) = tete
Thus these
polynomials are the inverse sequence of the Abel polynomials.
Several identities for them may be derived in much the same
way as we related the exponential polynomials to the lower
factorials in Sec.tion 6.
Anticipating some developments in the second part of
'
..
this paper.. we may state the folloWing principle.
to enumerate by a sequence
~nl
In order
a class of rooted trees ..
graded by the number of vertices, one forms the associated
'.
10.
10-6
basic set, which will enumerate a class of reluctant functions,
and then proceedsto apply T'neorem 8 or a variant or it..
Which will retlect the "composition rule" or such class ot
trees.
The connection constants between two polynomial
sequences enumerating sequences of reluctant functions have
a combinatorial interpretation in terms ot "piecing togetherone set ot trees
in
terms of another.
Thus our starting
point 1n the second part ot this work will be:
given two
Pl and F2 ot rooted labeled torests, in how
many ways can be a member of F2 be "pieced together"
fran ,members ot . Pl'l The slmPlest case of this 1s cayley's
families
theorem above, where Pl consists ot a
consists ot all labeled rooted forests.
~1ngle
edge and F2
71.
11-1
11.
Bibliography
Rota, G.-C.,
On the Foundations of Combinatorial
Theory~
I"
Theory of Mobius functions, Zeit. tiir. Wahr.,2 (1964)"
340-366.
.
Crapo, H.H. and Rota~ G.-C., On the Foundation of Combinatorial
Theory, II, Combinatorial Geometries, M.l.T. Press
.
(to appear).
BO&s~
R.P. and Buck, R.C., Poynom1a~ Expansions of &lt1re
Functions" Springer, New York, 1961~.
Martin, W. T., on Expansions in Terms of a Certain General
Class ot Functions, American J. ,Math., 58 (1936), 407-420.
Prings heim" A." Zur Gesch1chte des Taylorschen Lehrsatzes,
Bib110theca Math., (3) 1 (1900), 433-479.
The Powero1d, ~~ Extension ot the Mathematical·Notion of Power, Acta 14athemat1ca, 73 (1941),
333-366.
.
Steffensen, J.F.,
TouChard" J.,
Nombres Exponentiels et Nombres de Ber.noulli,
Canad, J. Math.,8 (1956)" 305-320.
Riordan" J. "
Combinatorial Ident1ties, Wiley, New York" 1968 •
Operational Expressions tor the Laguerre
anet other Polynanials" Duke Math J." 31 (196l~) .. 127-142.
Al-salaam.. W.A...
Carl1tz" L. and Riordan" J."
The Divided Central Differences
ot Zero" C&nadian J. Math., 15 (1963), 94-100.
Hurwitz.. A., Ueber Abel's Verallgemeinerung der binan1achen
Forme1" Acta M&th. " 26 ( 1902)" 199-203.
Lab, I." Eine neue Art von Z8.h1en" 1hre E1gneachaften •••
Mittelungsb1. Math. stat •• 7 (1955). 203-216.
Riordan, J.. . Inverse Relations and Canbinatorlal Ident1 tiesH:~ ...
American MAth. Monthly" 71 (1964), 485-498.
11-2
Rota l G.-C., The Nur.mer of Partitions ot a Set l American
Math. Monthly, 71 (1964), 498-504.
Moon, J.W., Counting Labelled Trees, A Survey ot Methods
and Results, A pUbllcation of the Departmen't of
Mathematics of the University of Albertal 1969.
P1ncherle, S. and Juna.ld1, Le Opel"a~ion1 Distributive,
~anichellil
Bologna, 1900.
Pincherlel S.I
Opere Sct-Ite.. 2 vols." Cremonese.. Rome.. 1954.
Sylvester, J.J., Collected Mathematical Papers.. cambridge..
1904-1912" 4 vols.
Bell.. I.T., postualational Bases tor the Umbral calculus,
American J. Math ... 62 (1940).. 717~724.
Harary.. F...
A Sem1nar in Graph Theory, Holt.. New York, 1967.
Hall" Marshall,
1967.
Harary, Frank,
Combinatorial
~eor,y,
Blc1sdell, Waltham
Graph Theory, Addison-Wesley, Reading, 1969.
Alamos Scientific Laboratory
Brookhaven National Laboratory
The Rand Corporation
University of Waterloo
University of North Carolina
University of Colorado
. Florida Atlantic University
The Rdckefeller University ,
Massachusetts Institute of Technology
Lo~
..
...
~;~ ,