Bull. Kyushu Inst, Tech.
(M, & N. S.) No. 24, 1977, pp. 23-27
NOTE ON AN EXPONENTIAL GENERATING FUNCTION OF BELL NUMBERS
By
Ky6ichi YosHiNAGA and Masato MoRi
(Received Oct. 23, 1976)
Let B. denote the number ofpartitions ofa set of n distinct objects. B. are sometimes called exponential numbers or Bell numbers and it is known by E. T. Bell and by several other authors, e.g., [1], [2], [3], [4, pp. 37-38], [5] and [7], that the following formula
2:=o #?/- t" =: exp(exp t- 1)
is true for each complex number t, ltlÅq oo, where we have set Bo =1 by convention. The elegant proof given by G.-C. Rota [5], [4, pp, 37-38] to this identity is seen to be rather too formal and so a certain logical gap is unavoidable.
The purpose of the present note is to give a complete proof to this identity by fi11ing up such logical gap. In doing so, it will be found that the theorem on summation by packets ' relative to a summable series [6, p. 172] plays an essential role.
1. Preliminaries
For any pair (x, n) of a real number x and a positive integer n, let us write
[x]. :x(:)c-1)'''(x-n+l).
S; is the Stirling number (of second kind), that is, the number of partitions of a set of n distinct objects into m classes, where m and n, nl}im, are positive integers. The Bell number B., the number of partitions of a set of n distinct objects, is then given by the identity
(1) . Bn=:SA+Sn2+'''+S:•
Another formula requisite for what follows is the next identity, whose proof presenting few diMculties will be found in [4, p. 36].
(2) X"=S,1 [X]i+S.2 [X]2 -l- '''+S: [X]n•
LEMMA L Put for k= 1, 2,...,
24 Ky6ichi YosHiNAGA and Masato MoRi
Mk (t) = 2 ge== o (i+E 'l') ! tk"'.
Then, for any t, Of{: tÅq1/e, the series Mk(t) is eonvergent with
(et)k 1
Mk(t)gv2-.klTv'7)(IN 1='etrm
PRooF. Owing to the identity (2), it follows that sk+ig-ikk++i)iiii-=(k(k+ti-)ikili!.
By means of Stirling's formula:
n!=v2Ttn""'i2e""ept("), pt(n)=-i-ltln- (OÅqOÅqi),
it is not diMcult to see
msÅí.itk+i Åq (.e..!.)k'l4/.L.
(k+l)! - (k+l)!vf2n vfk+lep(k")
Åq.Åqre=C/l.i-. - (-.g.)-ii!
V2n (k+l)!V-k++7 (et)k
f{g vu2'-ik-!v/k (et)i.
Therefore for each t, Os{; tÅq 1le, one obtains
Mk(t)sv-2-;eil,kvk--2ge-o(et)'-v8itl,kvk i2,t
as desired. This completes the proof.
LEMMA2, Put
h(t, x)= 2,co=,M,(ltl)l[x],l•
Then for any t, ltlÅq11e and for any real number x, IxlÅq oo, one observes that the series h(t, x) is convergent.
PRooF. By means of Lemma 1, it follows that
1 (eltl)k Mk(ltl)1[X]klgv7i(1-eltl) VEk!
l [x]k1
Note on an Exponential Generating Function of Bell Numbers 25
E{II'1Åq7E=fi(1l--..,lt1) (i[i) (e1tl)k
for k= 1, 2,.,. . Because of the absolute convergence of the binomial series 2 i51.o(;k'`') sk (l sl Åq 1),
it is not difficult to get the result as desired. This completes the proof.
2. Formula of Bell and formula of Dobinski
The formula of Bell in the following proposition tells us that the entire function exp (exp t- 1) is a generating function of Bell numbers. We here now give a complete proof of this formula.
PRoposiTioN 1. (E. T. Bell) For any comptex number t, ltlÅq oo, it hotds that 1 + 2ee,. i -B.if/L t" : exp(exp t- 1) .
PRooF. Owing to the convergence of the series h(t, J)c), it follows that the double
series
H(t, x) == 1 + Zge., , 2?., ikll-ieL tk [x] ,
is absolutely and hence unconditionally conyergent, that is to say H(t, x) is summable and so the summation by packets may be allowed [6, pp, 172-173]. Therefore according to the identity (2) one obtains
H(t, x)=1+2ff.,-kt-k-! (S,i [x],+•••+Sft[x],)
(tx)k
= exp(tx) , =1+2,co=i
k!
where ltlÅq1/e and lxlÅq co. On the other hand by virtue of the binomial expansion, one observes that
exp(tx) = (1 + (exp t- 1 ))•"
(exp t- 1)t [.X]i =1+2P-i --l!
for each real number t, lexpt-llÅq1, that is -- oo ÅqtÅqlog2. Thus for eachtand x, ltl Åq 1le=:min(11e, log 2), it follows that
26 Ky6ichi YosHJNAcjA and Masato MoRi
1 + 2 P. i 2 kco=i -Åíl kl/ tk [x] i == fl(t, x) =: exp(tx)
==1+2ge.,, (eXPIt !---1)i [.],,
and therefore for l= 1, 2,..., one obtains the next identity (exp lt!- 1)' = 2 ,co--, ani if tk = M, (t)
for t, ltlÅq11e. Owing to the summability of the series H(t, x), it is not diMcult to see the absolute convergence and hence the summability of the double series
2p=,z,-=,-ill4,.,k.
Thus one obtains for itlÅq1/e that
exp(expt-1)=1+2rJ=, (eXP7t-!=l.iL'
- i + ]Åí ge. , 2il?.., tt--i- tk
=1+2kk, 2ts.,,--kSi tk
-= i+2,-=,-,B-?, tk,
where the last equality is seen to be a consequence ofthe identity (l). Since exp(exp t- 1) is an entire function of t, it is not djMcult to get
exp(exp t- l) =1+2ff.)..i-kB-ti,- tk
for t, ltlÅqoo, as desired. This completes the proof.
We next give a complete proof of the formula of Dobinski.
PRoposiTioN 2. (G. Dobinski) For an.v m = 1, 2,..., it hotds that B•n="hSm(i +2'it?, i +2i/-T, i +im`-3i!?,mL+...).
PRooF. We first note that
" :2ff'L-ofirmkii-:/-= :x)ff'.,. [1]'!L'
Note on an ExponLntial Generating Function of Belt Numbers 27
and so
co [k]n 1 == rr 1 2k=n
e k!U
is obtained for every n== 1, 2,.... Therefore it follows that B. =- S,}, + Sk + ••• + Sh.
=:2:=isx 3 zm==, [21,n
= -rr anlwy Z Z'== o mk-1. 7, (S,`n [k] i + Sn2i [k] 2 + ''' + Sfr, [k] ,n)
== " L' "' zkoo-o-li-ll'
.., L-L (1+.L2till,l+..3..1.i?l+h4rltt, .L+...).
This completes the proof.
References
[ 1 ] H. W. BEcK}tR and J. RioRi)AN, 71ie arithmetic of' Bell ctncl Stirlins. nttmbers, Aiiier. J. Math., 70 (1948) 385-394.
[2] E.T. BELL, Exponentialpolynomialb', Ann. of Math., .3.5(1934) .O.58-277.
[3] , The iterated exponential integers, Ann. of Math., 39 (1938) 539-557.
[4] C.BERcJE, .Principesdecombinatoire, Dunod,Paris,1968.
[5] G.-C. RoTA, TIie number ofpartitions ofa set, Amer. Math, Monthly, 71 (1964) 498-504.
[6] L. ScHwARTz, Topologiegtintirale etanalysefonctionnelle, Hermann, Paris, 1970.
[7] J. ToucHARD, Nombres exponentiets et nombres de Bernoulli, Canad. J. Math., 8(1956) 305- 320.
Department of Computer Science Kyusha Institute of Technology