Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 4 Issue 1 (2012), Pages 214-225.
ON SUMMABILITY FACTORS FOR |R, pn |k
(COMMUNICATED BY HÜ SEYIN BOR)
W. T. SULAIMAN
Abstract. Necessary and sufficient
∑conditions are obtained for
|T |k −summable, k ≥ 1, whenever
an is (N, p, q)-bounded.
∑
an λn to be
1. Introduction
Let T be a lower triangular matrix, (sn ) a sequence, then
Tn :=
A series
∑
n
∑
tnv sv .
(1.1)
v=0
an is said to be |T |k −summable, k ≥ 1, if
∞
∑
k
nk−1 |∆Tn−1 | < ∞, (∆Tn−1 = Tn−1 − Tn ) .
(1.2)
n=1
c, with
Given any lower triangular matrix T one can associate the matrices T and T
entries defined by
tnv =
n
∑
b
tnv = tnv − tn−1,v
tni , n, i = 0, 1, 2, . . . ,
i=v
respectively. With sn =
n
∑
ai λi ,
i=0
tn =
n
∑
tnv sv =
v=0
Yn := tn − tn−1 =
n
∑
tnv
v=0
n
∑
i=0
tni ai λi −
v
∑
ai λi =
i=0
n−1
∑
n
∑
ai λi
i=0
tn−1,i ai λi =
i=0
n
∑
v=i
n
∑
tnv =
n
∑
t̂ni ai λi
(1.3)
i=0
t̂ni ai λi as tn−1,n = 0. (1.4)
i=0
2010 Mathematics Subject Classification. Primary 40F05; Secondary 40D25.
Key words and phrases. Absolute summability, infinite series, summability factors.
c
⃝2012
Universiteti i Prishtinës, Prishtinë, Kosovë.
S.P.G is supported by CSIR, New Delhi, India for Emeritus Scientistship, under the scheme
number 21(084)/10/EMR-II.
Submitted November 14, 2011. Published December 1, 2011.
214
ON SUMMABILITY FACTORS FOR |R, pn |k
215
We call T a triangle if T is lower triangular and tnn ̸= 0 for all n. A triangle A
is called factorable if its nonzero entries amn can be written in the form bm cn for
each m and n. We assume that (pn ) is a positive sequences of numbers such that
Pn = p0 + p1 + . . . + pn → ∞, as n → ∞,
The series
∑
)
( an is said to be summable |R, pn |k , or N , pn k k ≥ 1, if
∞
∑
k
nk−1 |Tn − Tn−1 | < ∞ ,
n=1
where
Tn =
n
1 ∑
pv sv .
Pn v=0
(1.5)
Let (tn ) be a sequence of the (N, p, q) −mean of the sequence (sn ) is defined by
tn =
n
1 ∑
pn−v qv sv
Rn v=0
(p−1 = q−1 = R−1 = 0).
(1.6)
The sequence of constants (cn ) is formally defined by the identity
( n
)−1
∞
∑
∑
n
pn x
=
cn xn , c−j = 0, j ≥ 1.
(1.7)
v=0
n=0
By M, we denote the set of sequences (pn ) satisfying
pn+1
pn+2
≤
≤ 1, pn > 0, n = 0, 1, . . . .
pn
pn+1
The series
∑
an is said to be (N, p, q) bounded or
n
∑
∑
(1.8)
an = O(1)(N, p, q) if
pn−v qv sv = O(Rn ) as n → ∞.
(1.9)
v=1
The sequence (λn ) is said
∑ to belong to the class [X, Y ], where
∑ X and Y are two
methods of summability if an λn is Y −summable whenever an is X−summable.
Das , in 1966, proved the following result
Theorem
1.1. [3]. Let (pn ) ∈ M, qn ≥ 0. Then if
is N , qn −summable.
∑
an is |N, p, q| −summable it
Recently Singh and Sharma proved the following theorem
Theorem 1.2. [4]. Let (pn ) ∈ M, qn > 0 and let (qn ) be a monotonic
∑ nondecreasing
an λn
sequence for n ≥ 0. The necessary and sufficient condition that
is N , qn −summable whenever
∑
an = O(1)(N, p, q),
∞
∑
qn
|λn | < ∞,
Qn
n=0
216
W. T. SULAIMAN
∞
∑
|∆λn | < ∞,
n=0
∞
∑
Qn+1 2 ∆ λn < ∞,
q
n=0 n+1
is that
∞
∑
qn
|sn | |λn | < ∞.
Q
n
n=1
Very recently Rhoades and Savas generalized the above result by proving the
following
Theorem 1.3. [1]. Let (pn ) ∈ M, qn > 0 and let (qn ) be a monotonic nondecreasing
sequence
and
) nqn = O(Qn ). A necessary and sufficient condition that
(
λ ∈ |N, p, q|k , N , qn k whenever
∑
(i)
an = O (1) [N, p, q]k ,
∞ q
∑
n
k
|λn | < ∞,
n=0 Qn
(ii)
(iii)
(
∞
∑
n=0
(iv)
(
∞
∑
n=0
(v)
∞
∑
Qn
qn
(
n=0
(vi)
Qn+1
qn+1
(
n
k
|∆λn | < ∞,
Qn+1 ∆qn
qn qn+1
is that
∞
∑
)k−1
k−1
n=1
)k (
)k (
qn
Qn
Qn
qn
Qn
qn
)k−1
k
|∆λn | < ∞,
)k−1
2 k
∆ λn < ∞,
)k
k
k
|sn | |λn | < ∞.
2. Results
We prove the following result
Theorem 2.1. Let (pn ) ∈ M, qn > 0 , (qn ) be a monotonic non-increasing sequence. Let
n−1
∑ (a)
∆v t̂n,v+1 = O (|tnn |) ,
(b)
(c)
(d)
v=0
m+1
∑
(n |tnn |)
k−1
t̂n,v+1 = O(v |tvv |)k−1 , as m → ∞ .
n=v+1
n−1
∑
|tvv | t̂n,v+1 = O (|tnn |) ,
v=0
m+1
∑
n=v+1
(n |tnn |)
k−1
(
)
∆v t̂n,v+1 = O v k−1 |tvv |k , as m → ∞ .
ON SUMMABILITY FACTORS FOR |R, pn |k
(e)
(f )
217
n−1
∑
∆2v tnv = O (|∆tnn |) ,
v=0
∞
∑
n=v+1
((
)k )
k
nk−1 |∆tnn | ∆2v tnv = O v k−1 |∆tvv |
,
Then
∑ the necessary and sufficient conditions that λ ∈ (|N, p, q|k , |T |k ) whenever
(i) an = O (1) [N, p, q]k
k k
∞
∑
k |λv | Qv
= O(1)
(ii)
v k−1 |tvv |
qvk
v=0
(
)
k
∞
∑
1 k
k
(iii)
v k−1 |tvv | ∆
|λv | Qkv = O(1)
qv v=1
∞
∑
k
|∆λv | k
Qv = O(1)
k
qv+1
v=0
k
∞
∑
k |λv |
(v)
v k−1 |∆tvv |
Qkv−1 = O(1)
qvk
v=1
( )k
∞
∑
1 k k
k−1 (vi)
v
∆ qv |∆λv | Qv = O(1)
v=0
∞
k
∑
1 (vii)
v k−1 k ∆2 λv Qkv = O(1)
qv+1
v=1
is that
∞
∑
k
k
k
(viii)
nk−1 |tnn | |sn | |λn | < ∞.
(iv)
v k−1 |tvv |
k
n=1
3. Lemmas
Lemma 3.1. [2]. Let pn ∈ M.Then
(i) c0 > 0, cn ≤ 0, n = 1, 2, . . . ,
∞
∑
(ii)
cn xn is absolutely convergent for |x| < 1, and
(iii)
n=0
∞
∑
cn > 0, except when
n=0
∞
∑
(iv)
∞
∑
pn = ∞, in which case
n=0
cn = 0.
n=0
Lemma 3.2. [3]. If
tn =
n
1 ∑
pn−v qv sv ,
Rn v=0
sn =
n
1 ∑
cn−v Rv tv .
qn v=0
then
Lemma 3.3. [3]. We have
n
∑
v=0
(1)
where cn = c0 + c1 + . . . + cn .
(1)
cn−v Rv = Qn ,
218
W. T. SULAIMAN
4. Proof Of Theorem 2.1.
Let Tn denote the nth term of A-transform of the series
Yn := Tn − Tn−1
n
∑
=
∑
an λn , then
av t̂nv λv
v=0
=
n−1
∑
(
)
sv ∆v t̂nv λv + tnn λn sn
v=0
=
n−1
∑(
)
∆v t̂nv λv + t̂n,v+1 ∆λv sv + tnn λn sn
v=0
= Yn1 + Yn2 + Yn3 .
In order to prove the theorem, by Minkowski’s inequality, it is sufficient to show
that
∞
∑
k
nk−1 |Ynr | < ∞,
r = 1, 2, 3.
n=1
We have
Yn1 =
n−1
∑
∆v t̂nv λv sv
v=0
=
n−1
∑
∆v t̂nv λv
v=0
=
n−1
∑
Rµ tµ
=
µ=0
n−1
∑
cv−µ ∆v t̂nv
v=µ
µ=0
n−1
∑
v
1 ∑
cv−µ Rµ tµ
qv µ=0
(
Rµ tµ
n−1
∑
v=µ
(1)
cv−µ
λn
(1)
+cn−µ−1 ∆tnn
qn
=
λv
qv
(
( )
)
λv
1
∆λv
2
∆v t̂nv
+ ∆t̂n,v+1 ∆
λv + ∆t̂n,v+1
qv
qv
qv+1
)
Yn11 + Yn12 + Yn13 + Yn14 .
Applying Holder’s inequality
ON SUMMABILITY FACTORS FOR |R, pn |k
∞
∑
k
nk−1 |Yn11 | =
n=1
≤
n−1
n−1
∑
∑
λ
v
(1)
nk−1 cv−µ ∆2v t̂nv R µ tµ
µ=0
qv v=µ
n=1
∞
∑
∞
∑
(
n
k−1
n=1
=
n−1
∑
∞
∑
O(1)
n
∞
∑
O(1)
nk−1
∞
∑
O(1)
n=1
=
∞
∑
O(1)
nk−1
O(1)
=
O(1)
2 |λv |
∆v t̂nv qv
)k
v
∆2v t̂nv |λv | ∑ c(1)
Rµ
qv µ=0 v−µ
v=0
)k
(n−1
)k
∑ 2 |λv |
Qv
∆v t̂nv
qv
v=0
n−1
∑
k
∆2v t̂nv |λv | Qkv
qvk
v=0
(n |∆tnn |)
k−1
(n−1
)k−1
∑ 2 ∆v t̂nv
v=0
k
∆2v t̂nv |λv | Qkv
qvk
v=0
n−1
∑
∞ |λ |k
∞
k
∑
∑
v
k−1 Qkv
(n |∆tnn |)
∆v t̂nv k
v=0 qv
n=v+1
∞
∑
v=0
=
(1)
cv−µ
n−1
∑
n=0
=
n−1
∑
v=µ
(
k−1
n=1
=
Rµ tµ
µ=0
n=1
=
219
O(1).
k
k
v k−1 |tvv |
|λv | k
Qv
qvk
(tµ = O(1))
220
W. T. SULAIMAN
∞
∑
k
nk−1 |Yn12 | =
n=1
≤
( ) k
n−1
n−1
∑
∑
1
(1)
λv nk−1 R µ tµ
cv−µ ∆v t̂n,v+1 ∆
q
v
v=µ
n=1
µ=0
∞
∑
∞
∑
(
nk−1
Rµ tµ
∞
∑
O(1)
(
nk−1
=
O(1)
v=0
(
nk−1
n=1
=
∞
∑
O(1)
n=1
=
∞
∑
O(1)
n=1
=
µ=0
)k
( )k
1
|λv | Qv
∆v t̂n,v+1 ∆
qv ( )k
(n−1
)
k−1
∆v t̂n,v+1 ∆ 1 |λv |k Qkv ∑ ∆v t̂n,v+1 qv v=0
v=0
n−1
∑
k−1
(n |tnn |)
( )k
∆v t̂n,v+1 ∆ 1 |λv |k Qkv
q
v
v=0
n−1
∑
n=v+1
( )k
(
)
∞ ∑
∆ 1 |λv |k Qkv v k−1 |tvv |k
O(1)
qv v=1
=
)k
( )k
v
∑
1
(1)
∆v t̂n,v+1 ∆
|λv |
cv−µ Rµ
qv ( )k
∞ ∞
∑
∆ 1 |λv |k Qkv ∑ (n |tnn |)k−1 ∆v t̂n,v+1 qv
O(1)
v=0
=
n
)k
( )
1
∆v t̂n,v+1 ∆
|λv |
qv n−1
∑
v=0
k−1
(1)
cv−µ
n−1
∑
n=1
∞
∑
n−1
∑
v=µ
µ=0
n=1
=
n−1
∑
O(1).
ON SUMMABILITY FACTORS FOR |R, pn |k
∞
∑
k
nk−1 |Yn13 | =
n=1
≤
k
n−1
n−1
∑
∑
∆λ
v
(1)
nk−1 R µ tµ
cn−µ−1 ∆v t̂n,v+1
q
v+1 v=µ
n=1
µ=0
∞
∑
∞
∑
(
nk−1
n−1
∑
∞
∑
O(1)
nk−1
∞
∑
O(1)
k−1
n
n=1
=
∞
∑
O(1)
n
k−1
n=1
=
O(1)
=
O(1)
∞
∑
n
k−1
k
|Yn14 | =
n=1
∞
∑
∞
∑
n
k−1
∞
∑
O(1)
n=1
=
∞
∑
O(1)
n=1
=
∆v t̂n,v+1 |∆λv |
qv+1
)k
n−1
∑
v
∆v t̂n,v+1 |∆λv | ∑ c(1)
Rµ
qv+1 µ=0 v−µ
v=0
n−1
∑
k
∆v t̂n,v+1 |∆λv | Qkv
k
qv+1
v=0
(n−1
)
k−1
∑ ∆v t̂n,v+1
v=0
k
k
v k−1 |tvv |
|∆λv | k
Qv
k
qv+1
O(1).
k
n−1
λn ∑
(1)
Rµ tµ cn−µ−1 ∆n tnn µ=0
qn n
k−1
)k
(n−1
)k
|∆λv |
∑ Qv
∆v t̂n,v+1
qv+1
v=0
O(1).
n=1
≤
(1)
cv−µ
∞ |∆λ |k
∞
∑
∑
v
k−1 Qkv
(n |tnn |)
∆v t̂n,v+1 k
v=0 qv+1
n=v+1
v=1
=
n−1
∑
v=µ
(
n=1
=
Rµ tµ
µ=0
n=1
=
221
|λn |
|∆n tnn |
qnk
k
k
k
n−1
∑
µ=0
k
nk−1 |∆n tnn |
(
|λn | k
Qn−1
qnk
)k
(1)
Rµ cn−µ−1
222
W. T. SULAIMAN
Yn2 =
n−1
∑
t̂n,v+1 ∆λv sv
v=0
=
n−1
∑
t̂n,v+1 ∆λv
v=0
=
n−1
∑
Rµ tµ
=
(
Rµ tµ
∞
∑
n
n−1
∑
v=µ
µ=0
=
cv−µ t̂n,v+1
v=µ
µ=0
n−1
∑
n−1
∑
v
1 ∑
cv−µ Rµ tµ
qv µ=0
(1)
cv−µ
∆λv
qv
)
( )
)
(
1
∆2 λv
∆λv
∆λn
(1)
+ t̂n,v+2 ∆
∆λv + t̂n,v+2
+ cn−µ−1 tn,n+1
∆v t̂n,v+1
qv
qv
qv+1
qn
Yn21 + Yn22 + Yn23 + Yn24 .
k−1
k
|Yn21 | =
n=1
∞
∑
n
k−1
n
k−1
n=1
≤
∞
∑
k
n−1
n−1
∑ (1)
∆λv ∑
R t
c
∆ t̂
µ=0 µ µ v=µ v−µ v n,v+1 qv (
n=1
=
n−1
∑
∞
∑
O(1)
(
n
k−1
∞
∑
O(1)
k−1
n
n=1
=
∞
∑
O(1)
n=1
=
v=µ
µ=0
n=1
=
Rµ tµ
∞
∑
O(1)
n=1
n
k−1
n−1
∑
(1)
cv−µ
∆v t̂n,v+1 |∆λv |
qv
)k
n−1
∑
v
∆v t̂n,v+1 |∆λv | ∑ c(1)
Rµ
qv µ=0 v−µ
v=0
)k
(n−1
)k
|∆λv |
∑ ∆v t̂n,v+1
Qv
qv
v=0
n−1
∑
k
∆v t̂n,v+1 |∆λv | Qkv
qvk
v=0
k−1
(n |tnn |)
(n−1
)
k−1
∑ ∆v t̂n,v+1
v=0
n−1
∑
k
∆v t̂n,v+1 |∆λv | Qkv
k
qv+1
v=0
=
O(1)
∞ |∆λ |k
∞
∑
∑
v
k−1 Qkv
(n |tnn |)
∆v t̂n,v+1 k
v=1 qv+1
n=v+1
=
O(1)
∞ |∆λ |k
∑
v
k−1
Qkv (v |tvv |)
k
q
v=1
v+1
=
O(1).
ON SUMMABILITY FACTORS FOR |R, pn |k
∞
∑
k
nk−1 |Yn22 | =
n=1
≤
k
( )
n−1
n−1
∑
∑
1
(1)
∆λv nk−1 R µ tµ
cv−µ t̂n,v+2 ∆
q
v
v=µ
n=1
µ=0
∞
∑
∞
∑
(
nk−1
n−1
∑
∞
∑
O(1)
nk−1
∞
∑
O(1)
=
O(1)
=
O(1)
=
O(1)
( )
(n−1
)k
∑ |tvv | 1 t̂n,v+2 ∆
∆ |λv | Qv
qv v=0 |tvv |
n
nk−1
n−1
∑
v=0
(n−1
)
t̂n,v+2 ( 1 )k
k−1
∆
|∆λv |k Qkv ∑ |tvv | t̂n,v+2 k−1 qv v=0
|tvv |
k−1
(n |tnn |)
n−1
∑
v=0
∞
∑
1
v=0
|tvv |
O(1)
µ=0
k−1
n
n=1
=
)k
( )
v
∑
1
(1)
t̂n,v+2 ∆
|∆λv |
cv−µ Rµ
qv ( )
(n−1
)k
∑ 1 |∆λv | Qv
t̂n,v+2 ∆
qv v=0
n=1
∞
∑
)k
( )
1
t̂n,v+2 ∆
|∆λv |
qv k−1
n=1
∞
∑
(1)
cv−µ
n−1
∑
v=0
n=1
∞
∑
n−1
∑
v=µ
(
n=1
=
Rµ tµ
µ=0
n=1
=
223
k−1
t̂n,v+2 ( 1 )k
∆
|∆λv |k Qkv
k−1 q
v
|tvv |
( )k
∞
∆ 1 |∆λv |k Qkv ∑ (n |tnn |)k−1 t̂n,v+2 qv
n=v+1
( )k
1 k k
v k−1 ∆
|∆λv | Qv
q
v
v=0
=
O(1)
=
O(1).
∞
∑
224
W. T. SULAIMAN
∞
∑
k
nk−1 |Yn23 | =
n=1
≤
k
n−1
n−1
∑
∑
1
(1)
nk−1 R µ tµ
cv−µ t̂n,v+2 ∆2 λv q
v
v=µ
n=1
µ=0
∞
∑
∞
∑
(
nk−1
n=1
=
n−1
∑
∞
∑
O(1)
nk−1
∞
∑
O(1)
∞
∑
O(1)
∞
∑
O(1)
t̂n,v+2 nk−1
(n−1
∑ t̂n,v+2 v=0
n
k−1
n−1
∑
v=0
(n |tnn |)
=
v=0
|tvv |
∞
∑
O(1)
v=0
=
∞
∑
k
nk−1 |Yn24 | =
n=1
n−1
∑
v=0
1
1
k−1
v k−1
k
qv+1
1
k
qv+1
1
qv+1
1
qv+1
1
qv+1
)k
2 ∆ λv v
2 ∑
(1)
∆ λv cv−µ Rµ
)k
µ=0
2 ∆ λv Qv
)k
(n−1
)
t̂n,v+2 1 k−1
∑
∆2 λv k Qkv
|tvv | t̂n,v+2
k−1 q k
v=0
|tvv |
v+1
k−1
∞
∑
O(1)
t̂n,v+2 n−1
∑
n=1
=
(1)
cv−µ
v=0
n=1
=
v=µ
(
n=1
=
Rµ tµ
µ=0
n=1
=
n−1
∑
t̂n,v+2 1 ∆2 λv k Qkv
k−1 q k
|tvv |
v+1
∞
2 k k ∑
k−1 ∆ λv Qv
(n |tnn |)
t̂n,v+2 n=v+1
2 k k
∆ λv Qv
O(1).
k
n−1
∑
∆λ
n
(1)
nk−1 Rµ tµ cn−µ−1 tn,n+1
q
n n=1
µ=0
∞
∑
∞
∑
≤
k |∆λn |
O(1)
nk−1 |tn,n+1 |
qnk
n=1
=
O(1)
∞
∑
n=1
k
(1)
Rµ cn−µ−1 µ=0
k n−1
∑
k
nk−1 |tn,n+1 |
k
|∆λn | k
Qn−1
qnk
= O(1).
The proof is complete.
Acknowledgment:The author is so grateful to the referee who corrected many
typing mistakes in the paper.
References
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ON SUMMABILITY FACTORS FOR |R, pn |k
225
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W. T. Sulaiman, Department of Computer Engineering, College of Engineering University of Mosul, Mosul, (IRAQ)
E-mail address: waadsulaiman@hotmail.com