c Journal of Applied Mathematics & Decision Sciences, 1(2), 151{157 (1997)
Reprints Available directly from the Editor. Printed in New Zealand.
Classroom Note
An Inductive Derivation of Stirling Numbers of
the Second Kind and their Applications in
Statistics
ANWAR H. JOARDERy
Department of Mathematical Sciences, King Fahd University of Petroleum and
Minerals, Dhahran 31261, Saudi Arabia.
MUNIR MAHMOOD
Department of Econometrics, Monash University, Clayton, Victoria, Australia 3168.
Abstract. An inductive method has been presented for nding Stirling numbers of the second
kind. Applications to some discrete probability distributions for nding higher order moments
have been discussed.
Keywords: Stirling numbers of the second kind, raw moments, factorial moments.
1. Introduction
Stirling numbers of the second kind are used to express
where is a nonnegative integer, as sums of powers of factorial polynomials not higher than the
-th. For every non-negative integer the function can be expressed as
n
x
n
n
x
n
n
x
= ( 1)
S n
x(1)
+ ( 2)
S n
x(2)
+
:::
n
+ (
)
S n n x(n)
(1.1)
where ( ) = ( ; 1) ( ; + 1) is the factorial polynomial of order ( =
12
) and ( 1) ( 2)
( ) are called Stirling numbers of the second kind. It may be mentioned that ( ) is the number of ways to distribute
distinguishable balls into indistinguishable urns with no
urn empty. These numbers are called Stirling numbers of the second kind after the
British mathematician James Stirling (1692 - 1770).
x r
:::
n
x x
S n
::: x
S n
r
:::
r r
S n n
S n r
n
r
In this note we present an inductive proof to derive Stirling numbers of the
second kind. For a combinatorial proof the reader is referred to Roberts (1984, pp.
182-183). We also demonstrate the application of Stirling numbers in calculating
moments of some discrete distributions. The usual method of deriving raw moments of higher order of an integer-valued random variable is to derive the moment
y
The work was done in part while the rst author was at the University of Sydney, Australia.
A. H. JOARDER AND M. MAHMOOD
152
generating function and then dierentiate as many times as the order of the moment
required. In this paper we take an advantage of factorial moments which are easily
derived for integer-valued random variables. These moments can be combined with
the help of Stirling numbers of the second kind for deriving raw moments and hence
central moments of integer-valued random variables. The advantage of the method
presented here for calculating raw moments or central moments of a distribution is
that it avoids using derivatives of higher order.
2. The Description of the Method
Tables showing Stirling numbers up to = 8 is well-known (see e.g. Beyer
(1981, p.450)). These tables can be prepared by simple algebraic manipulations.
Suppose that we want to calculate Stirling numbers in (1.1) for = 3 so that
3
(2.1)
= (3 1) (1) + (3 2) (2) + (3 3) (3)
= (3 1) + (3 2) ( ; 1) + (3 3) ( ; 1)( ; 2)
Equating the coecients of 2 and 3 we have the following triangular
system of equations:
(3 1) ; (3 2) + 2 (3 3) = 0
(3 2) ; 3 (3 3) = 0
(3 3) = 1
Solving the equations we have (3 1) = 1 (3 2) = 3 and (3 3) = 1 If the
above method of calculating Stirling numbers is repeated for = 2 3 etc., then
it is easy to prepare a table showing Stirling numbers for dierent and The
following well-known recurrence relation (see e.g. Beyer, 1981, p. 450) is similar to
the relation among binomial coecients in Pascal triangle except a multiple to the
second term
( ) = ( ; 1 ; 1) + ( ; 1 ) = 1 2
(2.2)
with ( 1) = ( ) = 1 and ( ) = 0
For a simple algebraic proof see
Berman and Fryer (1972, pp. 217-218).
n
n
x
S
x
S
x
S
x
S
x
S
x x
S
x x
x x
S
S
S
S
x
:
x
S
S
:
S
S
S
:
n
n
S n r
S n
r
S n
S n n
rS n
r
S n r
r
:::
r:
n
r > n:
In this section we describe that the well-known formula for a Stirling number
of the second kind given by
X
( ; )
( ) = 1! (;1)
=0
X
= 1! (;1) ;
(2.3)
=0
r
S n r
r
i
r
i
r
r
r
i
r
i
i
r
i
i
n
i
n
AN INDUCTIVE DERIVATION OF STIRLING NUMBERS OF THE SECOND KIND
153
(Roberts, 1984, pp.182-183) where 0 and is any non-negative integer
can be derived from (2.2) by induction. The last line in the identity (2.3) follows
by changing the index of summation to where = ;
r
i
n
n
j
j
r
i:
By the repeated use of (2.2), it is easy to check that the Stirling numbers of
the second kind ( ) = 2 3 4 can be written as
S n r
( 2) =
S n
;2
X
n
r
2 ( ; 1 ; 1) = 2 ;1 ; 1
i
S n
n
i
2
n
i=0
( 3) =
S n
;3
X
n
3 ( ; 1 ; 2)
i
i
S n
3
n
i=0
=
;3
X
3 (2 ;2; ; 1) = 16 3 ; 3(2 ) + 3]
n
i
i=0
( 4) =
S n
;4
X
n
n
i
n
4 ( ; 1 ; 3)
i
S n
i
n
4
n
i=0
=
;4 1
X
4 6 f3 ;1; ; 3(2 ;1; ) + 3g
n
i
n
i
n
i
i=0
1 4 ; 4(3 ) + 6(2 ) ; 4]
= 24
4
X
= 4!1
(;1) 4 (4 ; )
=0
n
n
n
i
where
r
i
i
n
i
i
is the combination of things taking at a time.
r
i
We assume that the identity (2.2) is true and try to prove the identity in
(2.3). The identity (2.3) holds for ( 0) = 0 and ( 1) = 1 It may be proved
by induction that for all real numbers
S n
X
n
(;1)
n
i
i
i=0
S n
( ; ) = !
r
i
n
n
(Ruiz, 1996) and consequently (
and so 1
we have
) = 1 Applying induction hypothesis for ; 1
S n n
:
n
< r < n
1
( ; 1 ; 1) = ( ; 1)!
S n
r
:
r
r
= ( ;1 1)!
r
;1
X
r
(;1) ;1;i
r
r
r
i=0
(;1)(;1) ; (1 ;
r
i
i=0
;1
X
;1
i
i=r
)
n
i
;1
r
i
i
;1
n
A. H. JOARDER AND M. MAHMOOD
154
;
X
(;1) ;
= ( ;;11)!
1
r
r
r
+ ( ; 1)!
r
( ;1 ) =
rS n
r
;1
X
r
(;1) ;i
r
i=0
1 X (;1) ;
! =0
r
r
r
r
i
1
= ( ; 1)!
r
;1
X
r
r
i
r
r
i
i
i
(;1) ;i
i
;1
n
;1
i
i=0
n
r
;1
n
i
i
r
i
i=0
1
and
r
i
n
i
;1 +
r
n
;1
( ; 1)!
r
so that
(
S n r
) = ( ; 1 ; 1) + ( ; 1 )
;1
;1
X
;1 +
= ( ;1 1)!
(;1) ;
( ; 1)!
=0
S n
r
r S n
r
r
r
i
i
r
i
r
r
i
i
n
r
n
r
The summand in the second term in the above expression is exactly the same
as the rst term if = and consequently the two terms add to the expression in
(2.3). The proof is thus complete.
The above proof has pedagogic values especially to readers who want to
avoid combinatorial proof.
i
r
3. Applications
Stirling numbers connect the factorial moments to raw moments and vice
versa. Taking expectations in the both sides of (1.1) with and replaced by
and respectively, we have the general relationship between raw moments and
factorial moments as
n
r
r
i
0
r
=
X
r
( )
S r i
(3.1)
0
(i)
i=1
where = ( ) is the -th ( = 1 2
) raw moment and ( ) = ( ( ) )
is the -th ( = 1 2
) factorial moment. Harris (1966, p. 110) rst used this
formula clearly in the context of statistics though it has not been popular.
0
r
r
E X
r
r
r
:::
r
:::
n
0
E X r
r
n
Stuart and Ord (1994, p.84) presented the formula (3.1) for 4 and
also mentioned that Frisch gave general formulae showing factorial moments about
one point in terms of ordinary moments about another, and vice versa. However,
r
AN INDUCTIVE DERIVATION OF STIRLING NUMBERS OF THE SECOND KIND
155
formulae given by Frisch involve Bernoulli polynomials and are not easy to use.
The central moments are connected to the raw moments by the following relation:
r
=
X
r
(;1)
r
i
i=0
0
r
i
;i (1 ) :
0
(3.2)
i
The factorial moments of integer valued distributions e.g. binomial and negative
binomial distributions can be easily derived. Then these moments can be combined
by (3.1) and (3.2) to derive raw and central moments without having to resort to any
dierentiation. Some examples are presented below to illustrate the application.
Example 1. Let us calculate the fourth raw moment of the binomial distribution
(
B n p
) by the use of (3.1). Here we have
0
4
= (4 1)
S
0
(1)
+ (4 2)
S
0
(2)
+ (4 3)
S
0
(3)
+ (4 4)
S
0
(4) :
It follows from (2.3) that (4 1) = 1 (4 2) = 7 (4 3) = 6 and
(4 4) = 1 The -th factorial moment of the binomial distribution is easily shown
to be ( ) = ( )
=1 2
Therefore the fourth raw moment of the binomial
distribution is given by
+ 7 (2) 2 + 6 (3) 3 + (4) 4
4 =
S
S
:
0
i
S
S
i
n i p
i
i
:::
n:
n
p
0
np
n
p
n
p :
Example 2. Consider deriving the moments of the geometric distribution. Let
X
be the number of trials prior to the rst success with probability density function
(p.d.f.)
( = )=
0
1 + =1 =0 1
It can be easily veried that ( ) = !( ) = 1 2
By the use of the
formula (3.1) we have
1 =
(1) =
(2 1) (1) + (2 2) (2) = ( ) + 2( )2
2 =
) + 6( )2 + 6( )3
3 = (
) + 14( )2 + 36( )3 + 24( )4
4 = (
Now by the use of the moment relationship in (3.2) followed by some algebraic
manipulations, we nd that the second, third and fourth order central moments of
the geometric distribution are
;2
2 =
2 ;3
)
3 = ( +
2
=
(
+
7
+ 3 ) ;4
4
P X
x
q p
x
< p <
q
p
0
0
i
q=p
::::
i
i
0
q=p
S
0
i
x
0
0
S
q=p
q=p
0
q=p
q=p
q=p
0
q=p
qp
q
q
q=p
q
q=p
p
q
q
p
:
q=p
:
::::
A. H. JOARDER AND M. MAHMOOD
156
Example 3. Let us derive moments of the negative binomial distribution. Let
be the number of trials prior to s-th ( 1) success with p.d.f.
+
;
1
( = )=
+ =1 0
1 =0 1
X
s
P X
s
x
x
x s
q p
x
q
p
< p <
It can be easily veried that ( ) = ( + ; 1)( ) (
By the use of the formula (3.1) we have
0
s
i
i
i
x
)
q=p
i
:::
i
=1 2
::::
= (1) =
(1 + ) ;2
2 =
( ) + 3( + 1)(2) ( )2 + ( + 2)(3) ( )3
3 =
( ) + 7( + 1)(2) ( )2 + 6( + 2)(3) ( )3 + ( + 3)(4) ( )4
4 =
Then by the use of (3.2) we nd that the second, third and fourth order
central moments of the negative binomial distribution are
;2
2 =
( ) + 3 ( ) 2 + 2 ( )3
3 =
= ( + 2 ) ;3
( ) + (7 + 3 )( )2 + 6 (2 + )( )3 + 3 (2 + )( )4
4 =
= + (3 + 4) 2 + 3 )] ;4
(cf. Evans, M. et al., 1993 p. 110).
0
1
0
sq=p
sq
sq
0
p
0
s q=p
s
q=p
s q=p
s
q=p
sqp
s q=p
s
q=p
0
s q
s q=p
q
q=p
s
q=p
:
s q=p
p
s q=p
s q
s
s
s
s
q
q=p
q
s
s
q=p
s
s
q=p
p
Acknowledgments
The authors thank Dr. David Easdown, the University of Sydney and Dr. M. Iqbal,
King Fahd University of Petroleum and Minerals for their constructive suggestions
on an earlier draft of this paper. The authors are also grateful to the editor Dr.
Mahyar Amouzegar and an anonymous referee for their constructive suggestions
that considerably improved the presentation of this paper.
References
1. Berman, G. and Fryer, K.D. Introduction to Combinatorics. Academic Press New York,
1972.
2. Beyer, W.H. Handbook of Mathematical Sciences. CRC Press, Florida, 1981.
3. Evans, M., Hastings, N., and Peacock B. Statistical Distributions. John Wiley, New York,
1993.
4. Harris, B. Theory of Probability. Addison-Wesley Pub. Co., Massachusetts, 1966.
AN INDUCTIVE DERIVATION OF STIRLING NUMBERS OF THE SECOND KIND
157
5. Roberts, F.S. Applied Combinatorics. Prentice Hall, New Jersey, 1984.
6. Ruiz, S.M. An algebraic identity leading to Wilson's theorem. Mathematical Gazette, 80,
579-582, 1996.
7. Stuart, A. and Ord, J.K. Kendal's Advanced Theory of Statistics, Vol. 1. Charles Grin
and Co. Ltd., London, 1994.