International Journal of Analysis and Applications

Volume 17, Number 2 (2019), 226-233

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-17-2019-226

A NOTE ON GENERALIZED INDEXED NORLUND SUMMABILITY FACTOR OF

AN INFINITE SERIES

B. P. PADHY∗, P. TRIPATHY AND B. B. MISHRA

Department of Mathematics, School of Applied Sciences, KIIT, Deemed to be University, Bhubaneswar-24,

Odisha, India

∗Corresponding author: birupakhya.padhyfma@kiit.ac.in

Abstract. In the present article, we have established a result on generalized indexed absolute Norlund

summability factor by generalizing results of Mishra and Srivastava on indexed absolute Cesaro summabilty

factors and Padhy et.al. on the absolute indexed Norlund summability.

1. Introduction

In 1930, J.M.Whittaker [18] was the 1st to establish a result on the absolute summability of Fourier series

and in 1932, M. Fekete [6] established a result on generalized indexed summability. Later on the researchers

like Daniel [4] in 1964, Das [5] in 1966, Siya Ram [15] in 1969, Mazhar [11] in 1971, Mishra and Srivastava [13]

in 1984, Sulaiman [16] in 2011 etc. have established results on indexed summability factors of an infinite

series.

Let
∑
an be a given infinite series with sequence of partial sums {sn} . Let tnα be the nth (C,α) mean

(with order α > −1) of the sequence {sn} and is given by

tαn =
1

Aαn

n∑
k=0

Aα−1n−ksk, n ∈ N,where A
α
n =

Γ(n + α + 1)

Γ(α + 1)Γ(n + 1)
,

Received 2018-11-21; accepted 2018-12-18; published 2019-03-01.

2010 Mathematics Subject Classification. 40D15, 40F05, 40G99.

Key words and phrases. absolute summability; summability factors; infinte series.

c©2019 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

226

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-226


Int. J. Anal. Appl. 17 (2) (2019) 227

then the series
∑
anis said to be summable |C,α|k,k ≥ 1, [7] if

∞∑
n=1

(n)k−1|tαn − t
α
n−1|

k < ∞.

Let tn be the nth (C, 1)- mean of the sequence {sn} and is given by

tn =
1

n + 1

n∑
k=0

sk,

then the series
∑
anis said to be summable |C, 1|k,k ≥ 1, [3] if

∞∑
n=1

(n)k−1|tn − tn−1|k < ∞. (1.1)

Suppose {qn} be a sequence of real numbers with qn > 0, such that

Qn =

n∑
ν=0

qν →∞, as n →∞(Q−i = q−i = 0, i ≥ 1) (1.2)

The sequence to sequence transformation

Tn =
1

Qn

n∑
ν=0

qn−νsν (1.3)

defines the sequence {Tn} of the (N,qn)- means of the sequence {sn} generated by the sequence of coefficients

{qn} .

The series
∑
an is said to be summable |N,qn| if the sequence {Tn} is of bonded variation i.e;

∑
|Tn −Tn−1|

is convergent.

The series
∑
an is said to be summable |N,qn|k,k ≥ 1 ,if (see [8])

∞∑
n=1

(
Qn
qn

)k−1
|Tn −Tn−1|

k
< ∞ (1.4)

Clearly,|N,qn|k-summabiity is same as |C, 1|-summabiity,when qn = 1, for all values of n. Further any

sequence {αn} of positive numbers the series
∑
an is said to be summable |N,qn,αn|k,k ≥ 1 if

∞∑
n=1

(αn)
k−1|Tn −Tn−1|

k
< ∞ (1.5)

and is said to be summable |N,qn,αn; δ|k,k ≥ 1,δ ≥ 0 if
∞∑
n=1

(αn)
δk+k−1|Tn −Tn−1|

k
< ∞ (1.6)

For any sequence {µn},
∑∞
n=1 anµn is an infinite series.

We define

∆µn = µn −µn−1, |∆µn| = |µn −µn−1|

Also, for any sequence {µn}, by µn = O(n), we mean that the sequence {µnn } is bounded.



Int. J. Anal. Appl. 17 (2) (2019) 228

2. Known Theorems

Concerning with |C, 1| and |N,qn| summability Kishore [10] has proved the following theorem:

Theorem 2.1. Let q0 > 0,qn ≥ 0 and (qn) be a non-decreasing sequence. If
∑
an is summable |C, 1| then

the series
∑
anQn(n + 1)

−1 is summable |N,qn|.

Later on Ram [15] has proved the following theorem related to absolute Norlund factors of infinite series.

Theorem 2.2. Let (qn) be a non-increasing sequence with q0 > 0,qn ≥ 0. If
n∑
k=1

1

k
|sk| = O(Yn) as n →∞;

where (Yn) is a positive non-decreasing sequence and (µn) is a sequence such that

∞∑
n=1

n|∆2µn|Yn < ∞;

|µn|Yn = O(1) as n →∞,

then the series
∑
anQn(n + 1)

−1 is summable |N,qn|.

Also verma [17] has proved the following summability factor theorem:

Theorem 2.3. Let (qn) be a non-increasing sequence with q0 > 0,qn ≥ 0. If
∑
an is summable |C, 1|k then

the series
∑
anQn(n + 1)

−1 is summable |N,qn|k,k ≥ 1.

In 1984, Mishra and Srivatava [13] proved the following theorem for |C, 1|k summability.

Theorem 2.4. Let (Yn) be a positive non-decreasing sequence and let there be sequnces {βn} and {µn} such

that

|∆µn| ≤ βn; (2.1)

βn → 0 as n →∞; (2.2)

|µn|Yn = O(1) as n →∞; (2.3)
∞∑
n=1

n|∆βn|Yn < ∞; (2.4)

∞∑
n=1

1

n
|sn|

k
= O(Ym) as m →∞, (2.5)

then the series
∑∞
n=1 anµn is summable |C, 1|k,k ≥ 1.



Int. J. Anal. Appl. 17 (2) (2019) 229

Very recently, Padhy et al. [14] have proved a theorem on |N,qn|k-summability by extending theorem 2.4,

in the following form:

Theorem 2.5. Let for a positive non-decreasing sequence (Yn),there be sequences {βn} and {µn} satisfying

the conditions 2.1 to 2.5 and {qn} be a sequence with {qn}∈ R+ such that

Qn = O(nqn); (2.6)
∞∑
n=1

qn
Qn
|sn|

k
= O(Ym) as m →∞; (2.7)

Qn−r−1
Qn

= O

(
qn−r−1
Qn

Qr
qr

)
; (2.8)

m+1∑
n=r+1

(
Qn
qn

)k−1
qn−r
Qn

= O

(
qr
Qr

)
, (2.9)

then the series
∑∞
n=1 anµn is summable |N,qn|k,k ≥ 1,where 0 ≤ r ≤ n.

It should be noted that if we take qn = 1∀n then condition 2.7 will be reduced to 2.5.

In what follows, we have generalized known theorems 2.4 and 2.5 to |N,qn,αn; δ|k - summability in the form

of the following theorem after studying [1] and [2] :

3. Main Theorem

Theorem 3.1. Let (Yn) be a positive non-decreasing sequence and there be sequences {βn} and {µn} such

that the conditions 2.1 to 2.5 are satisfied.Further let {qn} be a sequence of real numbers with qn > 0, such

that

Qn = O(nqn); (3.1)
∞∑
n=1

qn
Qn
|sn|

k
= O(Ym) as m →∞; (3.2)

Qn−r−1
Qn

= O

(
qn−r−1
Qn

Qr
qr

)
; (3.3)

m+1∑
n=r+1

(αn)
δk+k−1 qn−r

Qn
= O

(
qr
Qr

)
, (3.4)

then the series
∑∞
n=1 anµn is summable |N,qn,αn; δ|k,k ≥ 1,δ ≥ 0.

We require the below mentioned lemma to prove our main theorem:



Int. J. Anal. Appl. 17 (2) (2019) 230

4. Lemma [5]

Let (Yn) be a positive non decreasing sequence and there be sequences {βn} and {µn} such that the

conditions 2.1 to 2.5 are satisfied.Then

βnYn = O(1) as n →∞, (4.1)
∞∑
n=1

βnYn < ∞. (4.2)

5. Proof of the Main theorem

Suppose (τn) refers to the (N,qn)- mean of the series
∑∞
n=1 anµn. Then by definition, we have

τn =
1

Qn

n∑
r=0

qn−r

r∑
s=0

asµs

=
1

Qn

n∑
s=0

asµs

n∑
r=s

qn−r

=
1

Qn

n∑
s=0

asµsQn−s

=
1

Qn

n∑
r=0

arµrQn−r

Thus

τn − τn−1 =
1

Qn

n∑
r=1

Qn−rarµr −
1

Qn−1

n−1∑
r=1

Qn−r−1arµr

=

n∑
r=1

(
Qn−r
Qn

−
Qn−r−1
Qn−1

)
arµr

=
1

QnQn−1

n∑
r=1

(Qn−rQn−1 −Qn−r−1Qn)arµr

=
1

QnQn−1

[
n−1∑
r=1

∆{(Qn−rQn−1 −Qn−r−1Qn) µr}

]
n∑
ν=1

aν, with p0 = 0

=
1

QnQn−1

[
n−1∑
r=1

(qn−rQn−1 −qn−r−1Qn) µrsr +
n−1∑
r=1

(Qn−r−1Qn−1 −Qn−r−2Qn) ∆µrsr

]
(By Abel’s transformation)

= Tn,1 + Tn,2 + Tn,3 + Tn,4 (say)

Now, to show
∑∞
n=1 anµn is summable |N,qn,αn; δ|k,k ≥ 1,δ ≥ 0, by 1.6,we need to show that

∞∑
n=1

(αn)
δk+k−1|τn − τn−1|

k
< ∞.



Int. J. Anal. Appl. 17 (2) (2019) 231

i.e; to show that

∞∑
n=1

(αn)
δk+k−1|Tn,1 + Tn,2 + Tn,3 + Tn,4|

k
< ∞.

It will be enough to show that

∞∑
n=1

(αn)
δk+k−1|Tn,j|

k
< ∞ for j = 1, 2, 3, 4.

to establish the main theorem by using the inequality given by Minkowski.

Now we have

m+1∑
n=2

(αn)
δk+k−1|Tn,1|

k

m+1∑
n=2

(αn)
δk+k−1 |

1

QnQn−1

n−1∑
r=1

qn−rQn−1µrsr|
k

≤
m+1∑
n=2

(αn)
δk+k−1 1

Qn

(
n−1∑
r=1

qn−r|µr|
k|sr|

k

)(
1

Qn

n−1∑
r=1

qn−r

)k−1
(Using Holder′s inequality)

= O(1)

m∑
r=1

|µr|
k|sr|

k
m+1∑
n=r+1

(αn)
δk+k−1

(
qn−r
Qn

)

= O(1)

m∑
r=1

|µr|
k|sr|

k qr
Qr

, by 3.4

= O(1)

m∑
r=1

qr
Qr
|sr|

k|µr||µr|
k−1

= O(1)

m−1∑
r=1

∆|µr|
r∑

w=1

qw
Qw
|sw|

k
+ O(1)|µm|

m∑
r=1

qr
Qr
|sr|

k

= O(1)

m−1∑
r=1

|∆µr|Yr + O(1)|µm|Ym , by 3.2

= O(1), as m →∞

(By the lemma and 2.3)

Next,

m+1∑
n=2

(αn)
δk+k−1 |Tn,2|

k

=

m+1∑
n=1

(αn)
δk+k−1 |

1

QnQn−1

n−1∑
r=1

qn−r−1Qnµrsr|
k

≤
m+1∑
n=2

(αn)
δk+k−1 1

Qn−1

(
n−1∑
r=1

qn−r−1|µr|
k|sr|

k

)(
1

Qn−1

n−1∑
r=1

qn−r−1

)k−1
(5.1)



Int. J. Anal. Appl. 17 (2) (2019) 232

= O(1)

m∑
r=1

|µr|
k|sr|

k
m+1∑
n=r+1

(αn)
δk+k−1

(
qn−r−1
Qn−1

)

= O(1)

m∑
r=1

|µr|
k|sr|

k qr
Qr

= O(1), as m →∞, As in proof of the 1st part.

Further,

m+1∑
n=2

(αn)
δk+k−1 |Tn,3|

k

=

m+1∑
n=1

(αn)
δk+k−1 |

1

QnQn−1

n−1∑
r=1

Qn−r−1Qn−1∆µrsr|
k

≤
m+1∑
n=2

(αn)
δk+k−1 1

Qn

(
n−1∑
r=1

Qn−r−1|∆µr||sr|
k

)(
1

Qn

n−1∑
r=1

Qn−r−1|∆µr|

)k−1

Since,

(
1

Qn

n−1∑
r=1

Qn−r−1|∆µr|

)
≤
n−1∑
r=1

|∆µn| ≤ n|∆µr| ≤ nβn

Therefore,

m+1∑
n=2

(αn)
δk+k−1 |Tn,3|

k

≤ O(1)
m∑
r=1

(rβr)
k−1|∆µr||sr|

k
m+1∑
n=r+1

(αn)
δk+k−1 Qn−r−1

Qn

= O(1)

m∑
r=1

|∆µr||sr|
k qr
Qr

≤ O(1)
m∑
r=1

βr|sr|
k qr
Qr

= O(1)

m−1∑
r=1

∆ (βr)

r∑
w=1

qw
Qw
|sw|

k
+ O(1)(βm)

m∑
r=1

qr
Qr
|sr|

k

= O(1)

m−1∑
r=1

|∆βr|Yr + O(1)(βm)Ym

= O(1) as m →∞

Now,

m+1∑
n=2

(αn)
δk+k−1 |Tn,4|

k

=

m+1∑
n=2

(αn)
δk+k−1|

1

QnQn−1

n−1∑
r=1

Qn−r−2Qn∆µrsr|
k

(5.2)



Int. J. Anal. Appl. 17 (2) (2019) 233

≤
m+1∑
n=2

(αn)
δk+k−1 1

Qn−1

(
n−1∑
r=1

Qn−r−2|∆µr||sr|
k

)
1

Qn−1

n−1∑
r=1

Qn−r−2|∆µr|
k−1

= O(1)

m∑
r=1

(rβr)
k−1|∆µr||sr|

k
m+1∑
n=r+1

(αn)
δk+k−1

(
Qn−r−1
Qn

)
, (as above)

= O(1)

m∑
r=1

|∆µr||sr|
k qr
Qr

= O(1) as m →∞. (as above)

This completes the proof of the theorem.

6. Conclusion

If (Yn) is a positive non-decreasing sequence and there be sequences {βn} and {µn} such that the condi-

tions 2.1 to 2.5 along with the conditions 4.1 and 4.2 are satisfied then the series
∑∞
n=1 anµn is summable

|N,qn,αn; δ|k,k ≥ 1,δ ≥ 0, under the conditions 3.1 to 3.4.Thus, our result generalizes the result of Mishra

and Srivastava [13] and Padhy et. al [14].

References

[1] Bor, H., A note on two summability methods, Proc. Amer. Math. Soc., 98(1986), 81-84.

[2] Bor, H., A note on absolute summability factors, Int. J. Math. And Math. Sci., 17(3), (1994), 479-482.

[3] Bor.H, On absolute summability factors of infinite series, Rocky Mt. J. Math., 23(4)(1993), 1221-1230.

[4] Daniel, E.C., On absolute summability factors of infinite series, Proc. Japan Acad., 40(2)(1964), 65-69..

[5] Das, G, On absolute Norlund summability factors of infinite series, J. Lond. Math. Soc., 41(1966), 685-692.

[6] F. Fekete, M., On absolute summability of infinite series, Proc. Edinburgh Math. Soc., 3(2)(1932), 132-134.

[7] Flett, T.M., On an extension of absolute summability and some theorems of Little wood and Palay, Proc. Lond. Math.

Soc., 7(1957), 113-141.

[8] Hardy, G.H., Divergent series, Oxford University press, (1949).

[9] Hsiang,,F.C., On absolute Norlund summability of a Fourier series, J. Austral. Math. Soc., 7(1967), 252-256.

[10] Kishore, N., On the absolute norlund summability factors, Riv. Mat. Univ. Parma, 6(1965), 129-134.

[11] Mazhar,S.M., On absolute summability factors of infinite series, Tohoku Math. J., 23(3)(2008), 433-451.

[12] Misra, U.K., Misra, M. and Padhy, B.P., On the local property of indexed Norlund summability of a factored Fourier series,

Int. J. Res. Rev. App. Sci., 5(1)(2010), 52-58.

[13] Mishra, K.N and Srivastava, R.S.L., On absolute Cesaro summability factors of infinite series, Portugaliae Math.,

42(1)(1983-84), 53-61.

[14] Padhy, B.P., Majhi, B., Samanta, P., Misra, M. and Misra, U.K., A note on the absolute indexed Norlund summability,

New trends Math. Sci., 6(4), (2018), 54-59.

[15] Ram, S., On the absolute norlund summability factors of infinite series, Indian J. Pure Appl. Math., 2(1971), 275-282.

[16] Sulaiman, W.T., On some absolute summability factors of infinite series, Gen. Math. Notes, 2(2)(2011), 7-13.

[17] Verma, R.S., On the absolute Norlund summability factors, Riv. Math. Univ. Parma, 3(1977), 27-33.

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	1. Introduction
	2. Known Theorems
	3. Main Theorem
	4. Lemma cite5
	5. Proof of the Main theorem
	6. Conclusion
	References