Download Eulerian numbers, generalized arithmetic

U.P.B. Sci. Bull., Series A, Vol. 71,Iss. 2, 2009
ISSN 1223-7027
EULERIAN NUMBERS AND GENERALIZED ARITHMETICGEOMETRIC SERIES
Mircea I CÎRNU1
Pe baza teoremei Cauchy-Mertens de înmulţire a seriilor, se obţin într-un
mod simplu unele rezultate binecunoscute, cum ar fi reprezentarea numerelor
Euleriene, o identitate combinatorială şi formula sumei seriilor aritmeticogeometrice generalizate. Se dau două aplicaţii.
Based on Cauchy-Mertens theorem of series multiplication, we obtain in a
simple manner some
well-known results, namely
the Eulerian numbers
representation, a combinatorial identity and the formula for the sum of generalized
arithmetic-geometric series. Two applications are given.
Keywords: Eulerian numbers, generalized arithmetic-geometric series.
2000 Mathematics Subject Classification: 05A19, 11B68, 65B10.
1. Eulerian numbers and the sum of generalized arithmetic-geometric series
We present the formula that express the sums of generalized
arithmetic-geometric series by the Eulerian numbers ([1]). It can to be
considered as the definition of these numbers. Are included recurrence
and symmetry relations for Eulerian numbers that give the possibility to
compute these numbers.
In entire paper z is a complex variable.
Theorem 1. Let n  0 be a natural number. The sum of a
generalized arithmetic-geometric series (the generating function of the
sequence k n , k  0,1,2, ) is given by the formula

Gn  z    k n z k 
k 1
n 1
E n, k  z
1  z  
z
n 1
k
,
z 1
(1)
k 0
where En, k  are the Eulerian numbers, given by the relations
En,0  En, n  1  1 ,
1
(2)
Prof., Department of Mathematics III, University POLITEHNICA of Bucharest, Romania
26
Mircea Cîrnu
En, k   k  1 En  1, k   n  k  En  1, k  1 , k  1,2, , n  2 . (3)

z
is uniform
1 z
k 1
convergent. By differentiation and multiplication with z , we obtain



z
z
z
k
2 k
,
,


k
z

k
z

1

z
k3zk 
1  4 z  z 2 and



2
3
4
1  z  k 1
1  z 
1  z 
k 1
k 1
so on. Hence the relations (1), (2), (3) are satisfied for n  0,1,2,3 .
We shall prove the relation (1) by mathematical induction after n .
Assuming the relation (1) is true for fixed n  0 , by differentiation and
multiplication with z , we obtain

n 1
z 1  nz  n1
z2
n 1 k
k


k
z

E
n
,
k
z

k E n, k  z k 1 

n2 
n 1 
1  z  k 0
1  z  k 1
k 1
n 1
z

E n, k k  1  n  k z  z k .
n2 
1  z  k 0
We obtain the relation (1) for n  1 if we denote
Proof. For z  r  1, the geometric series
z
k


n 1
 E n, k k  1  n  k z  z
k 0

n
k
  E n  1, k  z k .
(4)
k 0
By identification, from (4) we obtain for the coefficients of z 0 , the
relation En,0  En  1,0 . Because E1,0  1, it follows that En,0  1 . For the
coefficients of z n , we obtain the relation En, n  1  En  1, n . Because
E1,0  1, it follows that En, n  1  1 . For the coefficients of z k , we obtain the
relation
k  1 En, k   n  k  1 En, k  1  En  1, k  , k  1,2,, n  1 .
Replacing n  1 instead of n , the recurrence relation (3) follows.
Consequence. The Eulerian numbers satisfy the following symmetry
relation
(5)
En, k   En, n  k  1 , k  0,1,, n  1 , n  1,2,
Proof. We prove the relation (5) by induction after n . For n  1 the
relation (5) is obvious. By (2) it results the relation (5) is true for k  0 and
k  n  1.
We suppose (5) satisfied for a fixed natural n  1. Replacing k with n  k
in (5), it is obtain the relation
Eulerian numbers and generalized arithmetic-geometric series
En, n  k   En, k  1 , k  1, , n  2 .
27
(6)
Using the relation (3) and induction hypothesis under the forms (5) and
(6), we have
En  1, n  k   n  k  1 En, n  k   k  1 En, n  k  1 
 n  k  1 En, k  1  k  1 En, k   En  1, k  ,
hence the relation (5) is true for n  1 . By induction axiom, the relation (5) is
accomplished for every natural number n  1.
Remark. From relations (2) and (3), the values of Eulerian numbers
En, k  can be computed by recurrence. The first values of these numbers are
given below:
0
1
2
3
4
5
6
k
--------------------------------------------------------------------n
1
1
2
1
1
3
1
4
1
4
1
11
11
1
5
1
26
66
26
1
6
1
57
302
302
57
1
7
1
120 1191 2416 1191 120
1
2. Closed-form expression for the Eulerian numbers
Theorem 2. Let n  0 be a natural number. The Eulerian numbers are
given by the formula
k
j  n  1
 k  j  1n , k  0,1, , n  1 .
E n, k     1 
j 0
 j 
For k  n , we have the following identity
(7)
28
Mircea Cîrnu
n
 n  1
n  j  1n  0 ,
j


  1 
j 0
j
and, for k  n  1, the identity
n 1
n 1
 1 j   k  j  1n  0 .

j 0
 j 
(8)
(9)
Proof. One consider the relation (1) given in theorem 1, in the form

n 1
j 1
k 0
1  z n1  j n z j 1   En, k  z k ,
z  1.
Using Newton's binomial theorem and changing the index j  k  1 in the
left hand series, the above relation becomes

n 1
 n1
k  n  1 k  
n n
k







1
z
k

1
z


  E n, k z .
 k   
 k 0

   k 0
 k 0
(10)
Denoting

k  n  1
 , k  0,1, , n  1
 1 
n
ak  
, bk  k  1 , 0,1, ,
 k 

0
,
k  n 1

and using the Cauchy-Mertens theorem about the multiplications of series (see
for example [2]), the identity (10) can be rewritten as:


n 1

k 
k
k
a
z
b
z

c
z

E n, k z k ,

 k   k   k
k 0
 k 0
  k 0
 k 0
(11)
where the coefficients ck are given by the formula
k
ck   a j bk  j , k  0,1,
j 0
Identifying the coefficients in relation (11), if k  n  1 , we have
(12)
29
Eulerian numbers and generalized arithmetic-geometric series
k
j  n  1
k  j  1n  En, k  , n  0,1,  ,
ck    1 
j
j 0


n
j  n  1
n  j  1n  0 and (8)
hence (7) is obtained. If k  n , then cn    1 
j
j 0


n 1
j  n  1
k  j  1n  0 and follows
is also verified. If k  n  1, then ck    1 
j
j 0


(9).
Remarks. 1) One observe that the identity (8) follows from (9). Namely,
n 1
j  n  1
 n  j  2n  0 hence, if
from (9) for k  n  1, we have   1 
j 0
 j 
n
j n
n 1
substitute n with n  1 , we obtain   1  n  j  1  0 , therefore
j 0
 j
n
n  1
 n
n  j  1n  n  1  1 j  n  j  1n 1 0 .
j 0
 j 
 j
j
  1 
n
j 0
2) Representing the combinatorial coefficients by factorials, the identities
(8) and (9) may be written as:
 1 j n  j  1n1

n  j !
j!
j 0
n
 1 j k  j  1n  0 ,

j ! n  j  1!
j 0
n 1
 0,
k  n  1.
3. Formula for the sum of generalized arithmetic-geometric series
Replacing in formula (1) the closed-form expression of Eulerian numbers
given by the formula (7), it obtains the following result (see [3] or [1]) :
Theorem 3. Let be n  0 a natural number. The sum of the generalized
arithmetic-geometric series (the generating function of the sequence k n ,
k  0,1,) has the following formula

Gn  z    k n z k 
k 1
k
j  n  1
n k







1
k

j

1

 z , z  1. (13)


 j 
1  z n1 k 0  j 0



z
n 1
30
Mircea Cîrnu
4. Applications
4.1. Arithmetic-geometric series
Let a, r , q be complex numbers, q  1 . Using the formula (13) for n  1,
the sum of arithmetic-geometric series is



a  k  1r  q k 1  a  r  q k 1  r  k q k  a  r  r q 2 

q k 1
1  q q 1  q 
k 1
k 1

a 1  q   rq
1  q 2
.
4.2. Z-transforms
For n  1,2, , from (1) and (5), changing the index j  n  1  k , the Ztransform of the sequence of powers of natural numbers { k n : k  0,1,2, }, is
given by the formula

kn
1
n
Z k   k  Gn   
z
k 0 z
1
n 1
n 1
E n, j 
z
z

E n, j  z n 1 j 
n 1 
j
n 1 
z  1 j 0
 1  j 0 z
1  
 z
n 1
n 1
z
z
k
(14)



E
n
,
n

1

k
z

E n, k  z k , z  1.


n 1
n 1
z  1 k 0
z  1 k 0
z
z 1  z 
z 1  4z  z 2
2
3
In particular, Z k  
, Z k  
, Z k 
.
z  12
z  13
z  14
REFERENCES
 
  

[1]. L. Comtet, Advanced Combinatorics, Reidel, 1974.
[2]. W. Rudin, Principles of Mathematical Analysis, 3rd Ed., Mc Graw-Hill, New York, 1976.
[3]. R. Stalley, A Generalization of the Geometric Series, American Mathematical Monthly, 56
(1949) 5, 325-327.