Sums of Powers of Integers, Bernoulli numbers, Eulerian numbers and Stirling numbers Amrik Singh Nimbran Abstract: This paper explains methods for deriving a general formula for ∑ one of the three: (1)
, (2)
The author employs the concept of ℎ ℎ
, (3)
. These formulae involve
in the second and that of
ℎ
.
in the third
method. He also gives some results of his own relating to these numbers. AMS Subject Classification: 05A19 Key words and phrases: Bell numbers, Bernoulli numbers, Eulerian numbers, Higher Order sums, Power sums, product sums, Stirling numbers of the second kind.
The sums of like powers of consecutive natural numbers have fascinated mathematicians since antiquity. In the beginning, the need to compute areas and volumes motivated them to look for such formulae. It is not known when, where and by whom the formula, first
(
)
, for the sum of
positive integers was discovered. However, Archimedes (287-212 BC) gave, in his On Spirals
(c. 225 B.C.), the formula for the sum of squares in order to compute the area inside a spiral. The neo-Pythagorean Nicomachus of Gerasa (60-120 AD), is credited with the theorem that the cubic number is the sum of
th
consecutive odd numbers, beginning with ( − 1) + 1. The formula
for the sums of cubes is implicit in this theorem. Āryabhaṭa, the ancient Indian mathematician, stated the formula for the sum of the squares and of the cubes in verse 22 of his Āryabhaṭīya (499 A.D.).[5] Abū ‘Alī al-Ḥasan ibn al-Haytham (965–1039), an Arab mathematical physicist, derived the formula for summing the biquadrates to compute the volume of a general paraboloid of revolution. A German mathematician named Johann Faulhaber (1580-1635) extended these particular formulae for powers up to the 17th in his monograph Academia Algebrae, darinnen die miraculosische Inventiones, zu den höchsten Cossen weiters continuirt und profitiert werden (1631). His notable contribution is the use, in his formulae for odd powers, of polynomials not in as was being done formerly, but in N=
(
)
with alternating signs. E.g., ∑
(
=
)
. [1][9]
Pascal (1623-1662) employed Newton’s binomial theorem in his Traite du Triangle Arithmetique to formulate the idea of
for deriving a particular formula, using formulae for the
lower exponents.
1
1.
Jakob Bernoulli (1654-1705), the Swiss mathematician from Basel, made real breakthrough in this field. After closely examining the particular formulae, he discerned a pattern and introduced the numbers
(named after him) in Ars Conjectandi (published posthumously in 1713). His
numbers yield a general formula for the sums of ∑
∑
=
(− )
To put it succinctly: ∑
=
The numbers ∑
!
,
th
power of integers:
(1)
(
)
,
, . . . are obtained through the generating function:
, where
is replaced by
after expanding the binomial. =
, | |1.
Euler found this identity: ∑
=−(2 + 1)
, >1.
D.H. Lehmer [8] discovered these recurrence relations, as mentioned by Carlitz [3]: ∑ ∑
=2 +1, =
.
I shall now explain two methods for deriving alternative general power sum formulae. The whole scheme logically flows from this simple ( )=∑ ( + )( + ) … ( +
− )=
(1) can be proved straightaway for
(
)(
)…(
: )(
)
,
∈ ℕ.
(2)
=1 by first writing the terms in a row in ascending
order, then writing them in the descending order in the second row exactly below each term of the first row, adding the pairs having the same sum and finally dividing the combined sums by 2. For ≥ 2, we apply induction on
(not on
) for each
2
individually.
2.
Let me introduce the concept of
. Begin with the natural number sequence { }
and compute its partial sums getting triangular numbers as ∑ (
( ). At the next step we get: ∑ obtain the
( )=
th
( ) !
(
=
)(
∴
( )=
Or, ( + 1)
)(
)…(
(
)!
)
=
}: 1, 3, 6, . . .; I denote it by ( ). We go on and
.
+ 1)
( )+
( )=2
( ) and
( )=( +
( )−
+ 1)
(2a)
( )=
( )=
( ) −
( − 1).
( )
)
( ).
( − 1), using definition.
( )+(
Let us now take up
(2aa)
( − 1).
− 1)
(2ab)
of the natural number sequence and use the symbol ∑
for partial sums of terms up to the . .; ∑
) !
:
We deduce from (2): ( ( )=(
(
=: 1, 4, 10, . . . , and denote it by
!
We have by definition, ∑
i.e.,
)
={
th
. Then, ∑
term all having power
:1, 3, 6, . . .; ∑
:1, 5, 14, .
:1, 9, 36, . . .; and so on. Let us now list the numbers of both types of sums: Order sums
Power sums
1
1
3
6
10
15
21...
1
3
6
10
15
21...
2
1
4
10
20
35
56...
1
5
14
30
55
91...
3
1
5
15
35
70
126...
1
9
36
100
225
441...
4
1 6 21 56 126 252... ............................................. A close examination of
( ) and ∑
reveals that they are inextricably linked and can be
equated through suitable coefficients: ∑
= ( )
∑
= ( )+
∑
= ( ) + 4 ( − 1) +
∑
= ( ) + 11 ( − 1) + 11 ( − 2)+
1 17 98 354 979 2275… .. .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .. .. .. .. .. .
( − 1) ( − 2) ( − 3).
....................................................... 3
The coefficients relating the two types of sums may be obtained in one of these ways: (a) By employing the identity connecting two successive formulae along with (2aa)/(2ab): ∑
={ ∑
} − ∑ ∑( − )
={( + ) ∑
} − ∑(∑
(3a) )
(3b)
Let me illustrate with two examples: ∑
={ ∑ } − ∑(∑ − 1), from (3a). ( )} − ∑
={
( − 1), by definition.
={ ( )+2 ( − 1)} −
( − 1), from (2aa) and by definition.
= ( )+ ( − 1). ∑
= ∑
− ∑(∑( − 1) ), from (3a).
= { ( )+ ( − 1)} − ∑{( + 1)(∑ − 1) − ∑(∑ − 1)}, from above and from (3b). ( )+
=
( − 1) − ∑{( + 1) ( − 1) − ∑
( − 1)}, by definition.
={ ( ) + 3 ( − 1)} + { ( − 1) + 3 ( − 2)} − ∑{ ( − 1) + 2 ( − 2) −
( − 1)},
from (2aa), from (2ab) and by definition. = ( ) + 4 ( − 1) + 3 ( − 2) − ∑{2 ( − 2)}. = ( ) + 4 ( − 1) +
( − 2).
Other coefficients can also be found similarly. It would be apt to use the symbol these coefficients, known as
( ) for
, because they were first introduced by Euler in
1736 in his paper, Methodus universalis series summandi ulterius promota. [6] (b) We may also use the following recurrence relation [which actually comes from (3a)/(3b)], combined with two simple facts:(i) ( ) = 1 and (ii) ( )=0 for every > : ( )=
(
− )+(
Thus it follows that
−
+ )
(
− ), for every
(2 + 1) =( +1){ (2 ) +
≥
≥ .
(4)
(2 )}=2( +1){
(2 )}, is even for
∈ ℕ.
(c) These numbers resurfaced later [7] in Euler’s Differential Calculus, Part II, chap. VII (1855), when he sought to compute sums 1 ( )=∑
+2
+3
( )
+ . . . up to ∞ which led him to
. These numbers and polynomials can be
generated by repeated differentiation of the geometrical series 1+ (convergent for | |1). The arithmetical
>1, accelerates after attaining the least value at
=3. We thus have
= . ( ):
(d) Euler gave an explicit general formula for ( )=∑
(− )
( − ) for every
( + 1) 3
+ {∑
(5)
( ) from the recurrence relation (4):
I derived this particular formula for ( )= 3
≥1.
2
−
− 3 }, for
≥3.
(6)
On comparing (6) with Euler’s formula in (5), we get the following identity: 2∑
( + 1) 3
2
−
− 3 ={52(3
)} − {(
+ 1)(2
−
)}, for
≥3.
(7)
We thus derive a general formula for summation of any power: ∑
( )
=∑
( −
+ ).
(8)
I have computed nine rows of
′
( ), for
consisting of
=1 to 9:
1 1 1 1 1 1 1 1 1
502
11 66
302 1191
4293
14608
1
11
57
247
4
26
120
1
1 26
302
2416
1 57
1191
1 120
15619 15619 4293
1
247
1
88234 156190 88234 14608 502
1
We easily discern these simple, already known, properties of the Eulerian numbers: ( )= ( ) for every ≥1. (i) (ii)
∑
( )= !, for every
≥1.
I discovered these divisibility properties of these numbers: Theorem 2.1: – prime ⇒
( )≡1 (mod ) and
( − 1)≡1 (mod ) ∀ . 5
Theorem 2.2: In general, – prime ⇒ (
) ≡1 (mod ) ∀
where ∈ ℕ∪{0}.
≡2 +1 ⇒ ( )≡0 (mod 2), 1< < , and
Theorem 2.3:
∈ ℕ.
These theorems can be proved by using (5), (i), (ii), Fermat’s little theorem [which states that if g.c.d.(a, )=1, then
≡1 (mod )], and Wilson’s theorem [ – prime ⇔ ( −1)!≡–1 (mod
)]. Worpitzky’s identity [11] expresses power as linear combination of the Eulerian numbers and the binomial coefficients: sum formula: ∑
( )
=∑ ( )
=∑
. From this identity, we deduce the power
. This exactly is our formula (8). Thus our method is
validated. 3.
=∑
(− )
( )
ℎ
( ) defined in (2) and Stirling numbers:
Now I give method using the product sum ∑
( ).
(9)
Note that in (8), n varies but m remains constant, it is the reverse in (9). Further, while the sign is always positive in (8), it alternates in (9) in such a way that the last (highest power) term is always positive. I discovered the following recurrence relation between the coefficients ( )=
(
− )+
(
− ), for every
≥
≥ .
(10)
( ), smaller than corresponding
We have an asymmetric triangle of
( ):
( ), for 1< < :
1 1 1
3
1 1 1 1 1 1
31
127 255
6 25
90 301
966 3025
1
7 15
63
1
1 10
65 350
1701 7770
1 15
140
1050
1 21
266
6951 2646
1 28
462
1 36
1
I have discovered certain divisibility properties of these numbers modulo odd primes: 6
Theorem 3.1:
( )≡0 (mod ), 1< < . It follows from (10).
– prime>2 ⇒
This can be proved by using formula (11) given ahead and Fermat’s little theorem, that is, if g.c.d.(a, )=1, then
≡ (mod ).
This beautiful theorem completely identifies all such numbers divisible by odd primes: Theorem 3.2:
–
> 3 ⇒
+ 1, + 2, . . . , − 1; ℎ
ℎ
( +
≥ 2,
=
= 1, =
+ 1, + 2, . . . , .
ℎ
(5) for =2 to 4,
For example, 5 divides
), = 1, 2, 3, . . . , − 1;
− 1) ≡ 0 (
. ℎ
(6) for =3 to 5,
(
)(
(7) for =4 to 5, and
)
.
(8)
for =5. Nine in all. If we inspect these coefficients meticulously, we recognize their true identity – they are the famous Stirling numbers [after James Stirling (1692-1770)] of the second kind denoted by S(n, k) or by
. Actually,
is the number of ways of partitioning an –element set into
disjoint subsets, called blocks. A
nonempty
of a set is defined to be a mutually disjoint class of its
non-empty sub-sets of whose union is the whole set itself. These coefficients can be computed somewhat like the binomial coefficients as −1
. Further,
−1
=
(
)
(
)
!
> ;
1
=
=1; and
=
−1 + −1
.
We have this explicit formula for ( )=∑
=0, if
( ):
.
(11)
It is known that: ∑
( )= ( ),
where
(12)
( ) stands for
th
the Bell number, after Eric Temple Bell (1883-1960), who first
thoroughly studied them in print during mid 1930s almost 25-30 years after Ramanujan had derived several of their properties in his Notebooks.[2] Combinatorially, ( ) is the number of ways of portioning an –element set. It is defined by the exponential generating function:
= ∑
( ) !
and
( )=∑
(
)!
.
Stirling numbers of the second kind are also known to be related to the Bernoulli numbers: =∑
(−1)
!
( ).
(13) 7
We deduce these first seven particular formulae from (11): ( )=
=1.
!
( )=
.
!
( )=
.
!
.
( )=
.
! .
( )=
.
.
!
( )=
.
.
.
.
.
.
( )=
.
.
!
The above formulae give have
.
!
discovered
seven
corresponding
( ) consists of the coefficients ( ) = 1,
( )=
= (
) (
)
(
)
(
( )=
,
= =
(
)(
ℎ .I
ℎ
.
+ ), =1, 2, 3, . . . These formulae are: (14)
=1, 2, 3, . . ., i.e., (
)( (
( )= ( )=
(
=1, 2, 3, . . .
( )= ( )=
)
)(
) (
)
,
)(
)
,
=2, 3, 4, . . .
(15)
=1, 2, 3, . . . , i.e.,
=3, 4, 5, . . .
(16)
=1, 2, 3, . . ., i.e.,
=4, 5, 6, . . .
(17)
, =1, 2, 3, . . . , i.e.,
=5, 6, 7, . . .
(18)
, =1, 2, 3, . . ., i.e.,
=6, 7, 8, . . .
(19)
)
, =1, 2, 3, . . ., i.e.,
=7, 8, 9, . . .
(20)
I discovered these formulae through an unorthodox method. References: 1. 2. 3. 4. 5.
Beardon, A. F., Sums of Powers of Integers, Amer. Math. Month., 103 (March 1996), pp. 201–213. http://mathdl.maa.org/images/upload_library/22/2975368.pdf.bannered.pdf Berndt, B.C., Ramanujan’s Notebooks: Part I, Springer-Verlag, New York, 1985, pp. 11, 44, 85. Carlitz, L., Bernoulli Numbers, The Fibonacci Quarterly, volume 6, number 3, June 1968, pp. 71-85. http://www.fq.math.ca/Scanned/6-3/carlitz.pdf Carlitz, L., Some Remarks on the Bell Numbers, The Fibonacci Quarterly, volume 18, number 1, February 1980, pp. 66-73. http://www.fq.math.ca/Scanned/18-1/carlitz2.pdf Clark, W.E, The Āryabhaṭīya of Āryabhaṭa: An Ancient Indian Work on Mathematics and Astronomy, University of Chicago Press, Illinois, 1930, p. 37.
8
6.
Euler, L., Methodus universalis series summandi ulterius promota, in Commentarii academiae scientiarum Petropolitanae 8, 1741, pp. 147-158, §13, p.155. http://www.math.dartmouth.edu/~euler/pages/E055.html 7. Euler, L., Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum, Part II, chapter VII, Para 173, pp. 389-390. http://www.math.dartmouth.edu/~euler/pages/E212.html 8. Lehmer D. H., Lacunary Recurrences for the Bernoulli Numbers, Annals of Mathematics (2), Vol 36 (1935), pp. 637-649. 9. Knuth, D.E., Johann Faulhaber and Sums of Powers, Mathematics of Computation, vol. 61, no. 203 (July 1993), pp. 277-294. http://arxiv.org/pdf/math/9207222v1.pdf 10. Sitgreaves, R. Some Properties of Stirling Numbers of the Second Kind, Fibonacci Quarterly, Vol. 8, No.2 (March 1970), pp. 172-181. http://www.fq.math.ca/Scanned/8-2/sitgreaves.pdf 11. Worpitzky, J., Studien über die Bernoullischen und Eulerischen Zahlen, Journal für die reine und angewandte Mathematik 94(1883), pp. 203-232.
Amrik Singh Nimbran, IPS 6, Polo Road, Patna 800002 Email address:
[email protected]
9
