Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 3 Issue 3(2011), Pages 122-128.
A STUDY ON N , p, q k SUMMABILITY FACTORS OF INFINITE
SERIES
(COMMUNICATED BY HÜSEYIN BOR)
W.T.SULAIMAN
Abstract. New result concerning N , p, q k summability of the infinite series
∑
an λn is presented.
1. Introduction
∑
Let
an be a given infinite series with sequence of partial sums (sn ). Let (Tn )
denote the sequence of (N, p, q) means of (sn ). The (N, p, q) transforms of (sn ) is
defined by
n
1 ∑
Tn =
pn−v qv sv ,
(1.1)
Rn v=0
where
Rn =
n
∑
pn−v qv ̸= 0, for any n(p−1 = q−1 = R−1 = 0).
(1.2)
v=0
Necessary and sufficient conditions for the (N, p, q) method to be regular are
(i)
lim pn−v qv /Rn = 0 for each v, and
n→∞
n
∑
(ii)
pn−v qv < K |Rn | , where K is a positive constant independent of
v=0
∑
n. The series
an is said to be summable |R, pn |k , k ≥ 1, if
∞
∑
k
nk−1 |φn − φn−1 | < ∞ ,
(1.3)
n=1
where
φn =
n
1 ∑
pv sv .
Pn v=0
and Pn = p0 + p1 + . . . + pn → ∞, as n → ∞ ,
2000 Mathematics Subject Classification. 40F05, 4OD25.
Key words and phrases. Absolute summability,infinite series summability factor .
c
⃝2011
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted June 23, 2011. Accepted July 9, 2011.
122
(1.4)
A STUDY ON N , p, q SUMMABILITY FACTORS OF INFINITE SERIES
123
k
The series
∑
an is said to be summable |N, pn | , if
∞
∑
|σn − σn−1 | < ∞ ,
(1.5)
n=1
where
σn =
n
1 ∑
pn−v sv ,
Pn v=0
(1.6)
and it is said to be summable |N, p, q|k , k ≥ 1, if
∞
∑
k
nk−1 |Tn − Tn−1 | < ∞ ,
(1.7)
n=1
where Tn as defined by (1.1).
For k = 1, ∑
|N, p, q|k summability reduces to |N, p, q|
∑summability.
The series
an is said to be (N, p, q) bounded or
an = O(1)(N, p, q) if
n
∑
tn =
pn−v qv sv = O(Rn ) as n → ∞ .
(1.8)
v=1
By M, we denote the set of sequences p = (pn ) satisfying
pn+1
pn+2
≤
≤ 1, pn > 0, n = 0, 1, . . . .
pn
pn+1
It is known (Das [1]) that for p ∈ M, (1.5) holds iff
n
∞
∑
1 ∑
pn−v vav < ∞ .
(1.9)
nP
n
n=1
v=1
∑
For p ∈ M, the series
an is said to be |N, pn |k −summable k ≥ 1 (Sulaiman [3]),
if
n
k
∞
∑
1 ∑
pn−v vav < ∞ .
(1.10)
k
nPn n=1
v=1
It is quite reasonable to give
∑ the following definition:
For p ∈ M, the series
an is said to be N , p, q k −summable k ≥ 1 if
n
k
∞
∑
1 ∑
vpn−v qv av < ∞ .
k
nRn n=1
(1.11)
v=1
where Rn = pn q0 + pn−1 q1 + . . . + po qn → ∞, as n → ∞ .
We also assume that (pn ), (qn ) are positive sequences of numbers such that
Pn = p0 + p1 + . . . + pn → ∞, as n → ∞ ,
Qn = q0 + q1 + . . . + qn → ∞, as n → ∞ .
A positive sequence α = (αn ) is said to be a quasi-f-power increasing sequence,
f = (fn ), if there exists a constant K = K(α, f ) such that
Kfn αn ≥ fm αm ,
holds for n ≥ m ≥ 1(see [4]).
Das [1], in 1996, proved the following result
1.1. Let (pn ) ∈ M, qn ≥ 0. Then if
Theorem
N , qn −summable.
∑
an is |N, p, q| −summable it is
124
W.T.SULAIMAN
Recently Singh and Sharma [2] proved the following theorem
Theorem 1.2. Let (pn ) ∈ M, qn > 0 and let (qn ) be a monotonic
∑ non-decreasing
sequence for n ≥ 0. The necessary and sufficient condition that an λn is N , qn −summable
whenever
∑
an = O(1)(N, p, q),
∞
∑
qn
|λn | < ∞,
Qn
n=0
∞
∑
|∆λn | < ∞,
n=0
∞
∑
Qn+1 2 ∆ λn < ∞,
q
n=0 n+1
is that
∞
∑
qn
|sn | |λn | < ∞.
Qn
n=1
2. Lemmas
Lemma 2.1. Let (pn ) be non-increasing, n = O(Pn ), Pn = O(Rn ). Then for r > 0,
k ≥ 1,
∞
∑
pkn−v
1
= O( r+k−1 ).
(2.1)
r
k
n Rn
v
n=v+1
In particular,
∞
∑
pn−v
1
= O( ).
nRn
v
n=v+1
Proof. Since pn is non-increasing, then npn = O(Pn ). Therefore
∞
∑
pkn−v
r k
n=v+1 n Rn
2v
∑
pkn−v
r k
n=v+1 n Pn
= O(1)
∞
∑
2v
∞
∑
∑
pkn−v
pkn−v
pkn−v
=
O(1)
+
O(1)
r k
r k
r k
n=v+1 n Pn
n=v+1 n Pn
n=2v+1 n Pn
= O(1)
1
v r Pvk
2v
∑
n=v+1
pkn−v = O(1)
v
1 ∑
pk
v r Pvk m=1 m
v
1 ∑
1
= O(1) r k
=O
pm = O(1)
r
v Pv m=1
v Pvk−1
∞
∑
pkn−v
r k
n=2v+1 n Pn
= O(1)
= O(1)
Therefore
(
)
1
v r+k−1
.
∞
∑
pkm
pkm
=
r k
r k
m=v+1 m Pm
m=v+1 (m + v) Pm+v
∞
∑
∞
∑
1
m=v+1
mr+k
∞
∑
= O(1)
pkn−v
=O
nr Pnk
m=v+1
∫∞
(
x−r−k dx = O
v
(
1
v r+k−1
)
.
1
v r+k−1
)
.
A STUDY ON N , p, q SUMMABILITY FACTORS OF INFINITE SERIES
125
k
Lemma 2.2. For p ∈ M,
∞
∑
|∆v pn−v | < ∞.
v=0
Proof. Since p ∈ M, then (pn ) is non-increasing and hence
m
∑
v=0
|∆v pn−v | =
∞
∑
(pn−v−1 − pn−v ) = pn − pm−v−1 = O(1).
v=0
Lemma
2.3.
[4]. If (Xn ) is a quasi-f-increasing sequence , where f = (fn ) =
( β
γ)
n (log n) , γ ≥ 0, 0 < β < 1 then under the conditions
Xm |λm | = O(1),
m
∑
m → ∞,
nXn ∆2 λn = O(1), m → ∞,
(2.2)
(2.3)
n=1
we have
nXn |∆λn | = O(1),
∞
∑
(2.4)
Xn |∆λn | < ∞.
n=1
3. Result
Our aim is to present the following new general result
Theorem 3.1.
Let p ∈) M, and let (Xn ) be a quasi-f-increasing sequence , where
(
γ
f = (fn ) = nβ (log n) , γ ≥ 0, 0 < β < 1 and (2.2) and (2.3) and
n
k
∑
qv |sv |
v=1
are all satisfied then the series
vXvk−1
= O(Xn ),
(3.1)
∆qv = O(v −1 qv ),
(3.2)
qv+1 = O(qv ),
(3.3)
v = O(Pv ),
(3.4)
Pn = O(Rn ),
(3.5)
∑
an λn is summable |N, p, q|k , k ≥ 1.
126
W.T.SULAIMAN
Proof. Let (Tn ) be the sequence of (N , p.q) transform of the series
we have
Tn =
n
∑
v=1
=
=
an λn . Then,
vpn−v qv av λv
n−1
∑
( v
∑
v=0
r=0
n−1
∑
∑
)
ar
∆v (vpn−v qv λv ) +
( n
∑
)
av
np0 qn λn
v=0
sv (−pn−v qv λv + (v + 1)∆qv pn−v λv + (v + 1)qv+1 ∆v pn−v λv
v=0
+(v + 1)qv+1 pn−v−1 ∆λv ) + np0 qn sn λn
= Tn1 + Tn2 + Tn3 + Tn4 + Tn5 .
In order to prove the result, it is sufficient, by Minkowski’s inequality, to show that
∞
∑
1
k
|Tnj | < ∞, j = 1, 2, 3, 4, 5.
k
nP
n
n=1
Applying Hölder’s inequality, we have
m
∑
1
k
|Tn1 |
k
n=1 nPn
k
∑
1 n−1
=
pn−v qv sv λv k
n=1 nPn v=0
m
∑
∑
1 n−1
k
k
pn−v qvk |sv | |λv |
≤
k
n=1 nPn v=0
m
∑
= O(1)
= O(1)
m
∑
m
∑
v=0
m
∑
k
qvk |sv | |λv |
k
∞
∑
pn−v
n=v+1 nPn
v −1 qvk |sv | |λv |
k
k
k
qvk |sv |
k−1
|λv | |λv |
Xvk−1
k−1
v=0 vXv
= O(1)
m q k |s |k
∑
v v
|λv |
k−1
v=1 vXv
m−1
∑
v=1
= O(1)
pn−v
v=0
= O(1)
= O(1)
)k−1
m P k−1 n−1
∑
∑
k
k
n
pn−v qvk |sv | |λv |
k
n=1 nPn v=0
v=0
= O(1)
(n−1
∑
m−1
∑
v=1
∆ |λv |
v q k |s |k
m q k |s |k
∑
∑
r
r
v v
+
|λ
|
n
k−1
k−1
rX
vX
r=0
v=0
r
v
|∆λv | Xv + |λn | Xn = O(1).
A STUDY ON N , p, q SUMMABILITY FACTORS OF INFINITE SERIES
127
k
k
∑
1 n−1
=
(v + 1) pn−v ∆qv sv λv k
n=1 nPn v=0
m
∑
m
∑
1
k
|Tn2 |
k
n=1 nPn
∑ k
1 n−1
k
k
k
v pn−v |∆qv | |sv | |λv |
≤
k
nP
n=1
n v=0
m
∑
= O(1)
= O(1)
)k−1
pn−v
v=0
m P k−1 n−1
∑
k
k
k
n−1 ∑ k
v pn−v |∆qv | |sv | |λv |
k
n=1 nPn v=0
m
∑
k
k
v k |∆qv | |sv | |λv |
k
v=0
= O(1)
(n−1
∑
m
∑
k
k
∞
∑
pn−v
n=v+1 nPn
v k−1 |∆qv | |sv | |λv |
k
v=0
= O(1)
m q k |s |k
∑
v v
|λv |
k−1
vX
v=1
v
= O(1), as in the case of Tn1 .
∞
∑
1
k
|Tn3 |
k
n=1 nPn
=
k
∑
1 n−1
(v
+
1)
∆
p
q
s
λ
v n−v v+1 v v k
n=1 nPn v=0
m
∑
∑ k
1 n−1
k
k
k
|sv | |λv |
v ∆v pn−v qv+1
≤
k
n=1 nPn v=0
m
∑
= O(1)
= O(1)
∑ k
1 n−1
k
k
k
v ∆v pn−v qv+1
|sv | |λv |
k
n=1 nPn v=0
m
∑
m
∑
v=0
= O(1)
v=0
m
∑
v=0
= O(1)
(n−1
∑
m
∑
v=0
k
k
v k qv+1
|sv | |λv |
k
∞
∑
∆v pn−v
nPnk
n=v+1
k
v k−1 Pv−k qv+1
|sv | |λv |
k
v −1 qvk |sv | |λv |
k
k
= O(1), as in the case of Tn1 .
k
)k−1
|∆pn−v |
128
W.T.SULAIMAN
m
∑
1
k
|Tn4 |
k
n=1 nPn
=
k
∑
1 n−1
(v + 1) pn−v−1 qv+1 sv ∆λv k
n=1 nPn v=0
≤
∑ k k
1 n−1
k
k
v pn−v−1 qv+1
|sv | |∆λv | Xv1−k
k
nP
n=1
n v=0
m
∑
m
∑
=
O(1)
=
O(1)
=
=
O(1)
k
m vk qk
∑
v+1 |sv |
|∆λ |
m−1
∑
∆ (v |∆λv |)
1
k
|Tn5 |
k
nP
n=1
n
m−1
∑
|∆λv | Xv + O(1)
pn−v−1
nPnk
m−1
∑
v ∆2 λv Xv + O(1)m |∆λm | Xm = O(1).
v=0
m
∑
=
1
k
|np0 qn sn λn |
k
nP
n=1
n
=
O(1)
m
∑
n=1
=
∞
∑
v q k |s |k
m q k |s |k
∑
∑
r
r
v v
+
m
|∆λ
|
m
k−1
k−1
r=0 rXr
v=0 vXv
v=0
∞
∑
Xv |∆λv |
v=0
∑ k k
1 n−1
k
k
v pn−v−1 qv+1
|sv | |∆λv |
k
nP
n=1
n v=0
v=0
=
)k−1
m
∑
v
Xvk−1
v=0
n=v+1
k
m qk
∑
v+1 |sv |
O(1)
v |∆λv |
k−1
v=0 vXv
O(1)
(n−1
∑
O(1)
m
∑
n=1
nk−1 Pn−k qnk |sn | |λn |
k
k
k
k
nk−1 qnk |sn | |λn |
= O(1), as in the case of Tn1 .
This completes the proof of the theorem.
References
[1]
[2]
[3]
[4]
G. Das, On some methods of summability, Quart. J. Oxford Ser., 17 (2),244-256, (1996).
N. Singh and N.Sharma, On (N, p, q) summability factors of infinite series, Proc. Nat. Acad.
Sci. (Math. Sci.),110: 61-68, (2000).
W. T. Sulaiman, Notes on two summability methods, Pure Appl. Math. Sci. 31: 59-68, (1990).
W. T. Sulaiman, Extension on absolute summability factors of infinite series,J. Math. Anal.
Appl. 322(2006),1224-1230.
W.T.Sulaiman
Instituto de Matemática - UFRJ Av. Horácio Macedo, Centro de Tecnologia Cidade
Universitária, Ilha do Fundão, Caixa Postal 68530 21941-972 Rio de Janeiro, RJ, Brasil
E-mail address: waadsulaiman@hotmail.com