Review of the stirling numbers, their generalizations ...

Communications in Statistics - Theory and Methods

ISSN: 0361-0926 (Print) 1532-415X (Online) Journal homepage: http://www.tandfonline.com/loi/lsta20

Review of the stirling numbers, their generalizations and Statistical Applications Ch, A. Charalambides & Jagbir Singh To cite this article: Ch, A. Charalambides & Jagbir Singh (1988) Review of the stirling numbers, their generalizations and Statistical Applications, Communications in Statistics - Theory and Methods, 17:8, 2507-2532, DOI: 10.1080/03610928808829760 To link to this article: http://dx.doi.org/10.1080/03610928808829760

Published online: 27 Jun 2007.

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Date: 26 October 2015, At: 03:49

COMMUN. STATIST.-THEORY METH., ? 7 ( 8 ) , 2533-2595 (1988)

Depari!iier;t of Statistics, ,,-;n$

Downloaded by [Nat and Kapodistran University of Athens] at 03:49 26 October 2015

8,)-

:z : : n i 7 c r z i tj;

P h ? !,q3e!~nbi-%,

19122

ABSTRACT

The b a s i ? p r o p e r t i e s o f t h e S t i r l i n g nqmbers and t b e l r g e n e r a l i z a t i o n s a r e reviewed.

S t a t i s t i c a l applications of these

numbers i n e x p r e s s i n g ( i )t h e d i s t r l b u t l o n o t s u c c e s s e s I n P o i s s o n ( g e n e r a l i z e d ~ e r n o u l l i )trials, (ii) t h e occupancy d i s t r i b u t i o n s , ( i i i ) t h e c o n v o l u t i o n s o f t r u n c a t e d power s e r i e s d i s t r i b u t i o n s and r e l a t e d minimun v a r i a n c e u n b i a s e d e s t i m a t o r s and (iv) t h e g e n e r a l i z e d d i s c r e t e d i s t r i b u t i o n s and f a c t o r i a l moments a r e presented.

Copyright @ 1988 by Marcel Dekker, Inc.

CHARALAXBIDES A N D SiNGH

.

-

.

2l"-:-::i:L

, .

'ce

-2ef'fLcien's

z m A

--:- 1

----

!:.:.,:&:.-,dI:

,.

,...,

., ---re

variable

-

01-

,:,:. c:.=

::LLcl~>

I n 'ne

:;ex,

"1,"

a t z e " . , ~ wi:ici, i 'd )

*

" ,

>Lrl

'21..

of 'he 5~c:nrFe: Sti:-linr~

liff+z-::;J

:

'7':;

Gf

rqun - -r -

j

.,%rith=u+,

-

:.7u

.

,

2

,

.

:

ralculus

(1

t:ie

f:7,C!...>.'ir.,l:7

r ;; r l i l.t:

ti;

.

C Z T :- ,- ~ ~ r;:L:lii:,el: ~ ti-;e,-,re~j-za A

St%r!~icg n : ~ ~ k r rci' , t h e i'i rst "i-.d L.Li~u2t2

.

, ; l ~ ~ s > ~x-,2.s::-c~~~ i ? ~ ~ zz ~ :2.gi.,!-re!5 l - e ~ rtL-,;-h

Rcci;rrer,ce :elat:.~pj'

y o ~ e r r f ~ ~ rhe .

T e r n rierived b y La,p,ran-ye (1'1713). T A i ? 35T.e

, ZSCZ?-C.i ~

i':'er-ei~ces its t h e ~(jO;-: &{i:e-

.

:>-!r

(;1733). .-.,

=

: :

-

t

-

Into

d

SchlBrnilch ( 1 a 5 2 ) use&

Bate

.. \

i

callin,.; them f-a c t . o r i a l . c o e r f i c i e n t s ( c n e f f ' i c i e n t s d e s facuLtGs, F a c l d t 5 t e n c o e f f i c i e n t e n ) w i t h nomega.ti ve i n t e g e r irlteger.

( f i r s t k i n d ) or r i o n p o s i t l v e i n t e g e r

k

n

and n o n n e g a t i v e (second kind)

a.nd d e f i n e d t h e s e numbers by

I\io7;e t h a t by v i r i u e oi ( 2 . 4 ) aaii ( 2 . 4 3 ) with imply

c=:

s (n,k)1,

tin= S ( k , n ) .

t

i-eijlacid by

-t

T h i s u n i f i e d n o t a t i o n h a s been

su-bsequently u s e d by s e v e r a l o t h e r a u t h o r s .

The f i r s t s h o r t t a b l e o f t h e S t i r l i n g numbers o f t h e second k i n d w a s p u b l i s h e d by S t i r l i n g ( 1 7 3 0 ) up t o

n=

9.

Extensive t a b l e s

o f t h e S t i r i i n g numbers o f b o t h k i n d s were c o n s t r u c t e d by Gupta (i950), Schafer

(1954), F i s h e r

and Yates ( i 9 5 3 ) , David, K e n d a i i and

?,n-tor ( 1T 9T 6 ~ M ) ~i t r i n o v i c ( 1 9 6 0 ) ; --"--- (1,966):Abr..wmvitz and S + ~ ~ I L M i t r i n o v i c and M i t r i r i o v i c (1960, 1961,

1962,

,.-./ ,, Leigk!t,on; Xewmann and Zuckernan ( i i r f u . 1 -

i96';) a n Goidberg, ~


2546

where

CHARALAMRLDES AND S I N G H

C = 0 . 5 7 7 2 1 i s t n e E u i e r ' s c o n s ~ a n t . Other a y m p i u i i c resuiLs

can b e found i n ilammersley ( 1 9 7 1 ) and Moser and Wyman ( 1 9 5 8 ) . t h e sequence o f S t i r i i n g numbers o r tile second k i n d

..

k = Ci,l,2,

;

,r, n > 3

generating function

h

n

For

~(ii,k),

fixed, Harper (1.967)proved t h a t t h e k . i t ) = S(n,k)t n a s n n o n p o s i t i v e real

1

r o o t s a c d hence t h i s sequence i s s l s o unimodal w i t h a p l a t e a u o f two p o i n t s .

(2.20) d e r i v e d for l a r g e

-

nenk r

or

Jordan (1933,1939), using t h e expression n

t h e asymptotic expression

S(n,k)

kii/k!

2547


REVIEW OF THE STIRLING NUMBERS

on u s i n g ( 2 . i 4 ) may be o b t a i n e d as

Aiso, thc aiunbcrs

cn, n = 0,1,2,

... which

a r e t h e i n t e g r a l s of t h e

a s c e n d i n g f a c t o r i a l s and have g e n e r a t i n g f u n c t i o n

a r e g i v e n by

NBrlund ( 1 9 2 4 ) d i s c u s s e d t h e g e n e r a l i z e d B e r n o u l l i numbers

R"),

n = O,l,2,.

Evidently

,B:~'=

..

f o r any complex

z with generating function

\

Bn

and t h e S t i r l i n g numbers o f b o t h k i n d s may be


2548

CHARALAMBIDES AND SINGH

The . i s c e m u i n ~f a . c t o r i a 1 of with t h e descending f a c t o r i a l of Lah

i t 1 = (-l)"(-t),. - -n I

,

k

,

.

.

,

i

t

of d e g r e e

-t

n, Ltln, is connected

o f degree

n , (-t),,

by

(1955) i n t r o d u c e d and s t u d i e d t h e nmheer --I-, . , , , . w~iichiiicy be defir:sc!by

so t h a t

These numbers which a r e i n t e g e r s and have t h e s i g n of (-1)" a r e c a l l e d Lah numbers ( R i o r d a n ( 1 7 5 8 ) )

. Evidently

n [?In =

1 [l(n,k)j(tjk9 n = O , 1 , 2 , . . .

k=O

(3.3)


2549

a n 6 may be c h i l e a s i g n l e s s o r a b s o k ~ eLa3 ~.l;r..aers. ic can b e


e a s i i y s e e n f r o a (2.l), (2.2) an6 (j.i;, t k a t

with

s

a r e a l ( o r more g e n e r a l l y a complex) number a n a l e t

The c o e f f i c i e n t

C(n,k,s)

of t h e kth f a c t o r i a i of

expansion of t h e n t h f a c t o r i a l of

t

t

with s c a l e parameter

i n tne

s

is

c a l l e d c o e f f i c i e n t o f t h e g e n e r a l i z e d f a c t o r i a l o r simply C-number. According t o t h i s d e f i n i t i o n

Expanding t h e f a c t o r i a l virtue of

(st)n

i n t o a Newton s e r i e s , b y

(3.6), i t f o l l o w s t h a t

which may be Used a s an e q u i v a i e n t d e f i n i t i o n o f the C-a-unber. Xote t h a t

(3.6). by

sllhstituti,ng

t = bu

and p u t t i n g

s = ajb,


2550

CMdR4LAYBIDES AND SINGW

.. . ..

: -.----

g e n e r a l l y i t ~f o l l o w s from ( 3 . 6 ) t h a t

Note a l s o t h a t

where

AD

1s t h e d i f f e r e n c e o p e r a t o r wi-cn i n c r e m e n t

g, i i i a i Is

0

a f ( t )= ~ ( t + ~ ) - f ( t ) . 8

st

-1

lim s As = D and l i m j n = tn, from ( 3 . 1 1 ) w i t h sio s-m h = 1, g = s and ( 3 . 7 ) , by v i r t u e o f ( 2 . 8 ) and ( 2 . 1 0 ) , i t f o l l o w s t h a t

Since

l i m slkC(n,k,s) = s ( n , k ) , l i m ~ - ~ i i ( n , k ,= s )S ( n , k j .

s+o

(3.12)

S"

As a d i r e c t conseqlience of ( 3 . 1 0 ) : t h e f a c t o r i a l moments w i t h increment

( X ) n , h ' may be e x p r e s s e d i n terms o f t h e f a c t o r i a l moments with i n c r e m e n t g j h9

i-I(R;,!)

=

(a;g;



2551

Charalamhides i i Y; '(a 1 . Expanding t h ~ nth f a c t o r i a l o f

s2t

tiiese

"---a++:?..,,,,.,,, '

,zGTlpai.+TL""+s

--:-i w A LII

,,

?

'n-:

u,-%.2LL

-3:

..

-?$-..3-*

4:.

....-,'. -LL.L.

L L . ' L ~ ,

!I-,+ ~,lLLbb

-.,,Ql ?

,Q

--&A-

r-. LL~>

4 ? ! :-

A&."-

-01 1 L C - -

!
&ul~-I;~;i->;-,;i~ wr -- -

'I

and. eq11at5ng t h e c o e f f i c i e n t s

tile foLlojiing " t r i a n g u l a r ! '

r e c i x r e n c e r e l a t i o n o f t h e C-number can be d e r i v e d ,

with in.i.tial c o u d i t l o i i ~

The " v e r t i c a l " r e c u r r e n c e r e l a t i o n

can be o b t a i n e d from

1 ~ ( n + l , k + l , s() t )k = s ( ~ t + s - l ) by ~

t h e r i g h t hand s i d e u s i n g t h e Vandermonde formula and The e x p a n s i o n o f t h e i n v e r s e f a c t o r i a i ( s t )

-n

of i n v e r r - 4 f n r t n r i a 1 . s (st+n)

jt)

-ir

OR

u s i n g +he r i . r u r r c n c e

expanding

(3.6).

i n t o a series ( s t )7:. A

:

=

A

and (3.25 j may be o b t d i r l r d as (Cliaraiambides (i3:Saj:


REVIEW OF THE STiRLING NIJMBERS

I-.

....,. ii

iiavc

r"io:-ip~sitivcI-=GI r o ~ t o . V-7-0

f'n-

r,>

3

the

2,rres~cn"in~

sequences of C-numbers are unimoaai with a peak or. plateau or two poi.nts. For large

n, Charalambides (1977a),using tne expression

(3.20) derived the asymptotic expression

For positive integer does not vanish is

s

the maxin~umvalue

n = sk.

Yor this value,

n

for which

Z(n,k,s)


Lq pauTjap

(s'yCU)if s.zaapinu s u ~ . suoynnnj

3rrry.e.zauaS p n s n Su?puodsa.mos aqi jo uoy1esy:xypour s n o 3 o ~ ~upu ~ y%msuoTysunj 8ury.e;cauaB Aq paugap sJaqumu uspaTrqq pu-e Buy-jlr.~~~

aq? /Cq iano pa!.z.res

-

sn?,$saKpz s

~

q=z$y~jrs3bq pahcr.zd ma.zoaq7

~ ? , r r . ~ ' l r i 2 ? pn r r r p?rn??r;r- ? c r ~ jliar!,!

.rasnc;3-$prrz;s "" - ;:qn s;

SVM

-,;;Oy&-!-.x

; I,S,>-,:-

.!XI;:

131

~

*Qp a iF' 1115 ;

-

--

-..-..I

o.AayUilu

.;^ --- -. S T L!LUUICO/I I;I+u*~UI;ISV/I

-

--

;-, -

V C ~ +V M L

'!;YIJJ

;lqr+

2557


REVIEW OF THE STIRLING YCMBERS

We S t i r l i n g , L a t and C n~m&el-s car deeenernte Stirling

numbers j have been e x t e n d e d , m o d i f i e d and g e n e r a l i z e d i n s e v e r a l directions,

I n t h i s s e c t i o n some o f t h e s e numbers, mainly of

s t a t i s t i c a l i n t e r e s t , a r e b r i e f l y presented.

The a m - c e n t r a l S t i r l i n g n 1 3 ~ h e - wo f t h e f i r s t and s e c o n d k i n d may be d e f i n e d by

E v i d e n t l y , t h e s e numbers a r e t h e c o n n e c t i o n c o n s t a n t s of t h e power and f a c t o r i a l moments o f a random v a r i a b l e a b o u t a n a r b i t r a r y p o i n t . Fiere f i r s t .u. - - aA ~ uL.? rw ;i- -iAu- mu-LJ ~ f.l L/n ? ~ I ' / , . :1

In t i i i s i.espect S ( n , k ; r )

WJ

J

J

t h e form

+ [ n n i t + r j L ' ]t = O 7

sin,kir) =

K.

1-

(4.33

2558

CH.A?,.ALAMBIDES

a n d more s e n e r a l l y t h e n o n - c e n t r a l


numbers

C(n,k,s;r):

c(n,k;s,rj

C-numbers

AND S I X G M

or G o ~ i d - E ~ p .Pprr - ..- ..

may be &Line-

;is

d i _ s c ~ s e hy d G o i i l d and Hopper 11962) and a s c o e f f i c i e n t s i n a g e n e r a l i z a t i o n of t h e L a g u e r r e p o l y n o m i a l s ,

d i s c u s s e d by Chack

(1956);!Ihe n o n - c e n t r a l C-numbers were s t u d i e d by

Charalambides and Koutras ( 1 9 8 3 ) . S e v e r a l p a r t i c u l a r c a s e s o f t h e s e numbers were examined by Toscano (1939). iVote tnar,

where

uiril

(4,4)i m p l i e s

( r ) = ECCX-rl

about an a r b i t r a r y p o i n t

1

i s t h e n t h a s c e n d i n g f a c t o r i a l moment

r

and

Y ( ~ ! = E { ( x ) ~ ]t h e u s u a i

2559



r e s p e c t i v e l y -to which a r e , i n g e n e r a l , a s s o c i a t e d by

Recurrence r e l a t i o n s and o t h e r p r o p e r t i e s o f t h e s e riuinbers were d i s c u s s e d by Riordan

(1-9581 , Comtet

( i ~ ' ( 4

and Charalambides

(1974aj .

Although some o f t h e p r o p e r t i e s o f t h e S t i r l i n g and C numbers a r e - .

. .

p r e s e r v e d , s e v e r a i o t n e r s a r e d e s t r o y e d oy t h i s g e n e r a l i z a t i o n . For

r =2

t h e numbers

d(n,k)= s(n,k,2)

and ~ ( n , k=) S ( n , k , 2 ) ,


2560

CHARALAMBIDES 'AND SINGH

-: =

J

a .a !!

. . .a

.L

%

.

Note t h e f o l l o w i n g s p e c l a l case:;

i A130 i ? ~ ra. = (r, - 1 j i ( q - 1 ) ; i = 1 ? ; 1 , 2 , . -

..

t h ~ s emmber's Xdl~.ce t o t h e

c ~ ~ r e s ~ o n d iq n- Sgt i r l i n g nunbers ex-mined by Gouid (19613;.

-

Briit. j - - - - -( . R ll,Zd

" A ;

-7

.

-,-

ri b i L L i r l r

....mi?,.*r. a liuliVLi

Tne

--,- - ( n , k ) and S . - ( n , k ) s t u d i e d hy B e l l

( 1 9 3 9 ) a r e t h e e l e m e n t s o f t h e r t h power o f tile m a t r i c e s s = ( s ( n , k ) ) and S = ( S ( n , k ) ) r e s p e c t i v e l y .

GouSd

i~

9 6 m ) and C a r l i t z ( i 9 7 6 b )

examined g e n e r a l i z e d S z i r l i n g numbers o f t h e f i r s t and second k i n d d e f i n e d by P ( n , n - k ) = ( " r n ) f ( n ) and ~ ( n , n - k )= ( ; ) f k ( - n + k ) w i t h 1 i-c k f k ( n ) a n a r b i t r a r y polynomial o f d e g r e e k such t h a t P k ( 0 ) = 0 qli

v

k

.

They proved t h a t t h e s e n ~ ~ ~ m h esatis* rr,

.

(1963a) s t u d i e d

..,,,

m-

----r~li

) "08

T, n ,

h

_

n7!mhor~ ---,,,_I C ,+_~' .~

(2.23). Taiuber

T * ~hy P T T~uhlh~r ! 11965)

may be


---


*-c . ---LC

d

G 2.

Tne f a c t o r i a l . momenis o f t i l l s ui5tri"uti on a r e

...

whence a i : ( 1 - p ) /p :jrl& ~ i x ; c ; ~ v . = p , i =l ,,;' , -> n n-K t h i s 2 i s t r i b u t i o n reduces t o t h e binomial. s (n.kl a) = ( k ) a A r l o t i l r r s i J z c i a l c a s e - ~ h i c : ia p p e a r e d -3 s a ~ , r e r a ls r , z t - s t i c a l

Noze t h a t i'or

applications i s t h a t with

p i = 8/(8+i-l), i = l , 2 , . ; . A

case

a. = i / 0 , i = C,l,2

,..., s ( n , k / ? ) = j s ( n , k ) l 0 -n'k

.

In this

and (5.1)

reduces t o

P ( x ~ = ~ )~=s ( n , k ~) ~ / [ t 3 1k =~l ,, 2 ,..., n

(5.3)

m e f a c t o r i a l c9,lcldants of t h i s d i s t r i b u t i o n may e a s i l y be o b t a i n e d i n t h e form:


UIJOJ

ayq u~ p a h p a p sv.9

Jaqurnu ay? j o uor3sun.r b x ~ ~ ? q ~ q o .ax ud SBM

u""'

Z ' T = ? ' ( ~ + u ) / :='d '

'

(&&j

u

x

s a s s a a s n s 30

u e p l o r bq p a ~ u a s a ~ d

--

' a s ~ s~ ~ ? s ~a a d y y~o u y

( q ' ~ L L 6 1j ? % o V s a f i puE ( z L 6 - t ) Y"O.XJoYS

( ~ 9 6 1 )? h a g ' ( ~ 9 6 1 )UO?JR$~pup p p e a u?: pun03 a q u-es s ~ p q a paJom



where

=

(~(II), i u =

.

2

i n the

n

,

8,1,2,.

..

is t h e a - f o l d c o n v o l u t i o n of

Qr,

When b a l l s a r e s e q u e n t i a l l y d i s t r i . b u t e d a t random

urns t h e probability

qm(k,n)

r e q u i r e d u n t i l a p r e d e t e r m i n e d number o f g i v e n by ( C h a r a l ~ n b i d e s(1986b) )

that

k

m

balls are

a r n s a r e occupieci i s

2564

-,,,",; . .

.

CHARALAXBIDCS 4 N D S I N G H

..>..r::-.Jti?? -+- .

2:

. ,. . - . " u.2I L U

G

(Stevens


!L? ' :!

!.

~

L

. .-

~

~

LY t

~

'i-,

9 : . 5 2

.

.

>

IIXhi;

,,;, ',

_

+.

\? ,:

,

,

zLll;~.;-

I L'?L

:.ioi

,."Pi.>
9 (biZonin1) ~ ( ) =0 (1.~8j " - 1 or

s,

€I< 0

(negative binomial).

Using ( 3 . 1 8 ) we deduce from

(7.3)

..

p n [ z ; O ) = ~ ( 1 + 0 ) ~ - i i - ~~n(!z , n . s ) 8 ~ / z !z=k,k+l,. , with

z$ s n

when

s

i s a positive integer.

(7.13)

These d i s t r i b u t i o n s

i 9 7 i b j . liie pi=obiem o f were e s s e n t i a l l y d e r i v e d b y Ahuja (1.9'70, ~ c n s + r : _ ! c + l nmr u e f o r

A

was d i s c u s s e d i n Caco7LLlos 2nd

Charalambides ( 1 9 7 5 ) . The b i n o m i a i ani'l n e g a t i v e b i r i o ~ n i a ld i s t r i b u t i o n s t r ~ t c a t e d


REVIEW OF THE STIRLING NCMBESS

s u b c l a s s o f t h e compound d i s t r i b u t i o n s i s i n c l u d e d i n t h e c l a s s o f generaii zed disiri'uutions nie -PI .--",.I. :" U U ~ . U I I L tY f u i i c t i o n ai,d t h e

where

r n ( k ) ( q O ) i s t h e k t h f a c t o r i a l momnt o f t h e power s e r i e s

d i s t r i b u t i o n with s e r i e s function distribution

F

and

of the distribution

qn(uj

G,

f ( q O ) , pk

i s the p.f.

of t h e

i s t h e pf o f t h e u - f o l d c o n v o l u t i o n


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