General Mathematics Vol. 14, No. 3 (2006), 23–26
On the means of sequences
1
Ioan Ţincu
Abstract
In this paper we investigate the invariancy of a class of real sequences with respect to the transformation A : a → A(a).
2000 Mathematics Subject Classification: 40G05
1
We consider the set of real sequences K, the set Km of all sequences which
are convex of order m (m ∈ N) and the operator ∆r : K → R, r ∈ R,
defined by
∞
X
(−r)k
r
[r]
∆ an = (−1)
(1)
an+k ,
k!
k=0
with the convention ∆0 an = an for every n ∈ N, where:
(x)l = x(x + 1)...(x + l − 1), l ∈ N, (x)0 = 1
[r]-represent integer part of the real number r.
Definition 1.1. We say that a real sequence (an )∞
n=1 is of Mr class if and
only if
(2)
∆r an ≥ 0 for every n ∈ N (see [5]).
1
Received 20 July, 2006
Accepted for publication (in revised form) 17 August, 2006
23
24
Ţincu Ioan
µ
r
If r ∈ N then Mr = Kr ∆ an =
r
P
(−1)r−k kr
¡¢
k=0
an+k
¶
Property. For real numbers r, r1 , r2 we have:
i) ∆r+1 an = ∆r an+1 − ∆r an , for every n ∈ N
ii) ∆r1 +r2 an = ∆r1 (∆r2 an ) = ∆r2 (∆r1 an ) (see[5])
Let A(a) = (An (a))∞
n=1 , be the sequence of the means, that is
n
P
1
An (a) = n +
1 k=0 ak , n = 0, 1, 2, ...
If S denotes a certain class of real sequences, then an interesting problem
is to investigate if this class is invariant with respect to the transformation
A : a → A(a); in other words if A(S) ⊆ S. For instance, it is wellknown that A(S0 ) ⊆ S0 , S0 being the class of all real sequences which are
convergent.
In [1]-[4] it is shown that from the n-th order convexity of a = (an )
follows the convexity, of the same order, of the sequence A(a) = (An (a)),
i.e. A(Km ) ⊆ Km .
We shall find a representation of ∆r An (a) as a positive linear combinations of ∆r a0 , ∆r a1 , ..., ∆r an .
Theorem 1.1. For r ≥ 0 and n = 0, 1, 2, ... the equality
∆r An (a) =
(3)
n
X
ck (n, r), ∆r ak with
k=0



n!
,
k=0
(r + 1)n+1
ck (n, r) =
(r + 2)k−1
n!

·
, k = 1, 2, ...n.

k!
(r + 2)n
(4)
is verified.
Proof. For k = 0, 1, 2, ... we have:
ak = (k + 1)Ak (a) − kAk−1 (a)
r
[r]
∆ ak = (−1)
∞
X
(−r)i
i=0
[r]
= (−1)
∞
X
(−r)i
i=0
= (−1)[r]
"
∞
X
(−r)i
i=0
i!
i!
i!
ak+i =
[(k + i + 1)Ak+i (a) − (k + i)Ak+i−1 (a)] =
(k + i + 1)Ak+1 (a) −
∞
X
(−r)i
i=0
i!
#
(k + 1)Ak+1−1 (a) =
On the means of sequences
= (−1)[r]
−
∞
X
i=0
"
25
∞
X
(−r)i
i!
i=0
(k + i + 1)Ak+i (a)−
#
(−r)i+1
(k + i + 1)Ak+i (a) − kAk−1 (a) =
(i + 1)!
(∞
X
)
¸
(−r)
(−r)
i
i+1
= (−1)[r]
(k + i + 1)Ak+1 (a)
−
− kAk−1 (a) =
i!
(i + 1)!
i=0
#
"∞
X (−r)i
1+r
[r]
(k + i + 1)Ak+i (a)
− kAk−1 (a) =
= (−1)
i!
i+1
i=0
"
#
∞
∞
X
X
(−r)
(−r)
1
+
r
i
i
= (−1)[r] (1 + r)
Ak+1 (a) + k
·
Ak+i (a) − kAk−1 (a) =
i!
i!
i
+
1
i=0
i=0
#
"∞
X (−r − 1)i+1
Ak+i (a) − Ak−1 (a) =
= (1 + r)∆r Ak (a) − k(−1)[r]
(i
+
1)!
i=0
"∞
#
X (−r − 1)i
= (1 + r)∆r Ak (a) − k(−1)[r]
Ak+i−1 (a) − Ak−1 (a) =
i!
i=0
·
= (1 + r)∆r Ak (a) − k(−1)[r]
∞
X
(−r − 1)i
i=0
r
i!
Ak−1+i =
= (1 + r)∆ Ak (a) + k∆r+1 Ak−1 (a).
From property ii), ∆r+1 Ak−1 (a) = ∆r Ak (a) − ∆r Ak−1 (a).
We obtain
(5)
∆r ak = (1 + r + k)∆r Ak (a) − k∆r Ak−1 (a),
(6)
(r + 2)k r
(r + 2)k−1 r
(r + 2)k−1 r
∆ ak =
∆ Ak (a) −
∆ Ak−1 (a).
k!
k!
(k − 1)!
By summing these equalities we obtain
n
X
(r + 2)k−1
k=1
k!
∆r a k =
(r + 2)n r
∆ An (a) − ∆r A0 (a).
n!
1 ∆r a .
In virtue of (5), for k = 0, ∆r A0 (a) = r +
0
1
26
Ţincu Ioan
We obtain
n
X
(r + 2)k−1 r
n!
n!
1
r
·
∆ a0 +
∆ ak ,
∆ An (a) =
r + 1 (r + 2)n
(r + 2)n k=1
k!
r
n
X
n!
n!
(r + 2)k−1 r
r
∆ An (a) =
∆ a0 +
∆ ak .
(r + 1)n+1
(r + 2)n k=1
k!
r
Theorem 1.2. Let a = (an ), A(a) = (A(an )); then:
i) An (Mr ) ⊆ Mr
ii) if there exists C ∈ R, such that
|∆r (an )| < C, n = 0, 1, 2, ...
then for n = 1, 2, ...
|∆r An (a)| <
C
r+1
Proof. The assertions i), ii) are consequences of the equalities (3) and (4).
References
[1] I. B. Lacković, S. K. Simić, On weighted arithmetic means which are
invariant with respect to k-th order convexity, Univ. Beograd Publ. Electrotehn. Fak. Ser. Mat. Fiz. No. 461 - No. 497 (1974), 159-166.
[2] A. Lupaş, On convexity preserving matrix transformations, Univ.
Beograd, Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 634 - No. 677 (1979),
189-191
[3] A. Lupaş, On the means of convex sequences, Gazeta Matematică, Seria
(A), Anul IV, Nr. 1-2 (1983), 90-93.
[4] N. Ozeki, On the convex sequences (IV), J. College Arts Sci. Chibo Univ.
B 4 (1971), 1-4.
[5] I. Ţincu, Linear transformations and summability methods of real sequences, Ph. D. Thesis, Cluj-Napoca 2005.
”Lucian Blaga” University of Sibiu
Department of Mathematics
Str. Dr. I. Raţiu, no. 5–7
550012 Sibiu - Romania
E-mail address: tincu ioan@yahoo.com