International Journal of Analysis and Applications

Volume 18, Number 2 (2020), 172-182

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

DOI: 10.28924/2291-8639-18-2020-172

A NOTE ON GENERALIZED INDEXED PRODUCT SUMMABILITY

A. MISHRA1, B. P. PADHY1,∗, B. K. MAJHI2 AND U. K. MISRA3

1Dept. of Mathematics, KIIT, Deemed to be University, Bhubaneswar, Odisha, India

2Dept. of Mathematics, Centurion University of Technology and Management, Odisha, India

3Dept. of Mathematics, National Institute of Science and Technology, Berhampur, Odisha, India

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

Abstract. In the past, many researchers like Szasz, Rajgopal, Parameswaran, Ramanujan, Das, Sulaiman,

have established results on products of two summability methods. In the present article, we have established

a result on generalized indexed product summability which not only generalizes the result of Misra et al [2]

and Paikray et al [3] but also the result of Sulaiman [7].

1. Introduction

If we look back to the history, it is found that, in 1952, Szasz [8] published some results on products of

summability methods. Subsequently, Rajgopal [5] in 1954, Parameswaran [4] in 1957, Ramanujan [6] in 1958

etc. published some more results on products of summability methods. Later Das [1] in 1969 proved a result

on absolute product summability. In 2008, Sulaiman [7] published a result on indexed product summability

of an infinite series. The result of Sulaiman was then extended by Paikray et al.[3] in 2010 and Misra et al

[2] in 2011.

Let
∑
an be an infinite series with the sum of partial sums {sn}. Let {pn} be a sequence of positive real

Received 2019-11-02; accepted 2019-12-02; published 2020-03-02.

2010 Mathematics Subject Classification. 40D25.

Key words and phrases. |(R,qn)(R,pn)|k-product summability; |(N,qn)(N,pn)|k-product summability;

|(N,qn)(N,pn),αn|k-product summability; |(N,qn)(N,pn),αn; δ|k-product summability; |(N,qn)(N,pn),αn,δ,µ|k-product

summability.

c©2020 Authors retain the copyrights

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

172

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-172


Int. J. Anal. Appl. 18 (2) (2020) 173

constants such that

Pn = p0 + p1 + p2 + ... + pn →∞ as n →∞ (P−i = p−i = 0). (1.1)

The sequence-to-sequence transformation

tn =
1

n

n∑
ν=0

pνsν (1.2)

defines the (R,pn) transform of {sn} generated by {pn}.

The series
∑
an is said to be summable |R,pn|k,k ≥ 1, if

∞∑
n=1

nk−1|tn − tn−1|k < ∞. (1.3)

Similarly, the sequence-to-sequence transformation

Tn =
1

n

n∑
ν=0

pn−νsν (1.4)

defines the (N,pn) transform of {sn} generated by {pn}.

Let {τn} be the sequence of (N,qn) transform of the (N,pn) transform of {sn}, generated by the sequence

{qn} and {pn} respectively.That is

τn =
1

Qn

n∑
r=0

qn−r
1

Pr

r∑
ν=0

pr−νsν

Then the series
∑
an is said to be summable |(N,qn)(N,pn)|k,k ≥ 1, if

∞∑
n=1

nk−1|τn − τn−1|k < ∞, (1.5)

and the series
∑
an is said to be summable |(N,qn)(N,pn),δ|k,k ≥ 1, 1 ≥ δk ≥ 0 if

∞∑
n=1

nδk+k−1|τn − τn−1|k < ∞. (1.6)

Similarly, if {αn} is a sequence of positive numbers, then the series
∑
an is said to be summable

|(N,qn)(N,pn),αn|k,k ≥ 1, if
∞∑
n=1

αn
k−1|τn − τn−1|k < ∞, (1.7)

and the series
∑
an is summable |(N,qn)(N,pn),αn; δ|k,k ≥ 1, 1 ≥ δk ≥ 0, if

∞∑
n=1

αn
δk+k−1|τn − τn−1|k < ∞. (1.8)

For, µ a real number, the series
∑
an is summable |(N,qn)(N,pn),αn,δ,µ|k,k ≥ 1, 1 ≥ δk ≥ 0, if
∞∑
n=1

αn
µ(δk+k−1)|τn − τn−1|k < ∞. (1.9)



Int. J. Anal. Appl. 18 (2) (2020) 174

We assume through out this paper that Qn = q0 +q1 +...+qn →∞ as n →∞ and Pn = p0 +p1 +...+pn →∞

as n →∞.

2. Known Theorems

In 2008, Sulaiman [7] has proved the following theorem.

Theorem 2.1. Let k ≥ 1 and {λn} be a sequence of constants. Let us define

fν =

n∑
r=ν

qr
pr
, Fν =

n∑
r=ν

prfr (2.1)

Let pnQn = O(Pn) such that

∞∑
n=ν+1

nk−1qn
k

Qn
kQn−1

= O

(
(νqν)

k−1

Qk−1ν

)
. (2.2)

Then the sufficient condition for the implication
∑
an is summable |R,rn|k ⇒

∑
anλn is summable

|(R,qn)(R,pn)|k are

|λν|Fν = O (Qν) , (2.3)

|λν| = O (Qν) , (2.4)

pνRν|λν| = O (Qν) , (2.5)

pνqνRν|λν| = O (QνQν−1rν) , (2.6)

pnqnRn|λn| = O (PnQnrn) , (2.7)

Rν−1|∆λν|Fν−1 = O (Qνrν) , (2.8)

and

Rν−1|∆λν| = O (Qνrν) , (2.9)

where Rn = r1 + r2 + ... + rn.

Subsequently Paikray et al [3] generalized the above theorem by replacing the (R,pn) summability by A

summability. He proved:

Theorem 2.2. Let k ≥ 1 and {λn} be a sequence of constants. Let us define

fν =

n∑
r=ν

qrarν, Fν =

n∑
r=ν

fr (2.10)



Int. J. Anal. Appl. 18 (2) (2020) 175

Then the sufficient condition for the implication
∑
an is summable |R,rn|k ⇒

∑
anλn is summable

|(R,qn)(A)|k are

m+1∑
n=ν+1

nk−1qn
k

Qn
kQn−1

= O

(
1

λν
k

)
, (2.11)

(
n∑
r=ν

qr
k

k−1

)
= O(qν), (2.12)

(
n∑
r=ν

akr,ν

)
= O

(
νk−1

)
, (2.13)

Rν = O (rν) , (2.14)

qn
Qn

= O (1) , (2.15)

qnλnan,n
Qn−1

= O (1) , (2.16)

(∆λν)
k

qνk−1
= O

(
νk−1

)
, (2.17)

∆λν
λν

= O (1) , (2.18)

and

λν
k

qνk−1
= O

(
νk−1

)
, (2.19)

where Rn = r1 + r2 + ... + rn.

In 2011, Misra et al [2], generalize the above theorems and proved the following theorem.

Theorem 2.3. For the sequences of real constants {pn} and {qn} and the sequence of positive numbers

{αn}, we define

fν =

n∑
i=ν

qn−ipi−ν
Pi

and Fν =

n∑
i=ν

fi (2.20)

Let

Qn = O (qnPn) (2.21)

and

m+1∑
n=ν+1

{f(αn)}k(αn)k−1qnk

Qn
kQn−1

= O

(
(νqν)

k−1

Qkν

)
as m →∞. (2.22)



Int. J. Anal. Appl. 18 (2) (2020) 176

Then for any sequence {rn} and {λn}, the sufficient conditions for the implication
∑
an is summable

|R,rn|k ⇒
∑
anλn is summable |(N,qn)(N,pn),αn; f|k,k ≥ 1, are

|λν|Fν = O (Qν) , (2.23)

|λn| = O (Qn) , (2.24)

RνFν|λν| = O (Qνrν) , (2.25)

qnRnFn|λn| = O (QnQn−1rn) , (2.26)

Rν−1Fν+1|∆λν| = O (Qνrν) , (2.27)

Rν−1|∆λν| = O (Qνrν) , (2.28)

qnRn|λn| = O (QnQn−1rn) , (2.29)
∞∑
n=1

nk−1|tn|k = O(1), (2.30)

and

∞∑
n=2

{f(αn)}k(αn)k−1|tn|k = O(1), (2.31)

where Rn = r1 + r2 + ... + rn.

In what follows, we established a theorem on generalized product summability of the infinite series
∑
anλn

in the following form:

3. Main Theorem

Theorem 3.1. For ’µ’ a real number, the sequences of real constants {pn} and {qn} and the sequence of

positive numbers {αn}, we define

fν =

n∑
i=ν

qn−ipi−ν
Pi

and Fν =

n∑
i=ν

fi (3.1)

Let

Qn = O (qnPn) (3.2)

and

∞∑
n=ν+1

αn
µ(kδ+k−1)qn

k

Qn
kQn−1

= O

(
(νqν)

k−1

Qkν

)
as m →∞. (3.3)



Int. J. Anal. Appl. 18 (2) (2020) 177

Then for any sequence {rn} and {λn}, the sufficient conditions for the implication
∑
an is summable

|R,rn|k ⇒
∑
anλn is summable |(N,qn)(N,pn),αn,δ,µ|k,k ≥ 1, are

|λν|Fν = O (Qν) , (3.4)

|λn| = O (Qn) , (3.5)

RνFν|λν| = O (Qνrν) , (3.6)

qnRnFn|λn|αnµδ = O (QnQn−1rn) , (3.7)

Rν−1Fν+1|∆λν| = O (Qνrν) , (3.8)

Rν−1|∆λν| = O (Qνrν) , (3.9)

qnRn|λn|αnµδ = O (QnQn−1rn) , (3.10)
∞∑
n=1

nk−1|tn|k = O(1), (3.11)

and

∞∑
n=2

(αn)
µ(k−1)|tn|k = O(1), (3.12)

where Rn = r1 + r2 + ... + rn.

4. Proof of Theorem 3.1

Let {tn
′
} be the (R,rn) transform of the series

∑
an. Then

tn
′

=
1

R

n∑
ν=0

rνsν

tn = tn
′
− t
′

n−1 =
rn

RnRn−1

n∑
ν=1

Rν−1aν

Let {sn} be the sequence of partial sums of the series
∑
anλn and {τn} be the sequence of (N,qn)(N,pn)-

transform of the series
∑
anλn. Then

τn =
1

Qn

n∑
r=0

qn−r
1

Pr

r∑
ν=0

pr−νsν

=
1

Qn

n∑
ν=0

sν

n∑
r=ν

qn−νpr−ν
Pr

=
1

Qn

n∑
ν=0

fνsν (4.1)



Int. J. Anal. Appl. 18 (2) (2020) 178

Hence

Tn = τn − τn−1

=
1

Qn

n∑
ν=0

fνsν −
1

Qn−1

n−1∑
ν=0

fνsν

= −
qn

QnQn−1

n∑
ν=0

fνsν +
fnsn
Qn−1

= −
qn

QnQn−1

n∑
r=0

fr

r∑
ν=0

aνλν +
fn

Qn−1

n∑
ν=0

aνλν

= −
qn

QnQn−1

n∑
r=0

arλr

r∑
ν=0

fν +
fn

Qn−1

n∑
ν=0

aνλν (4.2)

= −
qn

QnQn−1

n∑
ν=1

Rν−1aν

(
λν
Rν−1

n∑
r=ν

fr

)
+

q0p0
PnQn−1

n∑
ν=1

Rν−1aν

(
λν
Rν−1

)

= −
qn

QnQn−1

[
n−1∑
ν=1

(
ν∑
r=1

Rr−1ar

)
∆

(
λν
Rν−1

n∑
r=ν

fr

)
+

(
n∑
ν=1

Rν−1aν

)
λn
Rn−1

fn

]

+
q0p0

PnQn−1

[
n−1∑
ν=1

(
ν∑
r=1

Rr−1ar

)
∆

(
λν
Rν−1

)
+

(
n∑
ν=1

Rν−1aν

)
λn
Rn−1

]

= −
qn

QnQn−1

[
n−1∑
ν=1

{
λνFνtν +

Rν−1
rν

fνλνtν +
Rν−1
rν

(∆λν) Fν+1tν

}
+
Rn
rn
λnFntn

]

+
q0p0

PnQn−1

[
n−1∑
ν=1

{
λνtν +

Rν−1
rν

(∆λν) tν

}
+
Rn
rn
λntn

]

=

7∑
i=1

Tn,i,say. (4.3)

In order to prove this theorem, using (4.3) and Minokowski’s inequality, it is sufficient to show that

∞∑
n=1

αn
µ(δk+k−1)|Tn,i|k < ∞ for i = 1, 2, 3, 4, 5, 6, 7.

On applying Holder’s inequality, we have

m+1∑
n=2

αn
µ(δk+k−1)|Tn,1|k

=

m+1∑
n=2

αn
µ(δk+k−1)|

qn
QnQn−1

n−1∑
ν=1

λνFνtν|k



Int. J. Anal. Appl. 18 (2) (2020) 179

≤
m+1∑
n=2

αn
µ(δk+k−1) qn

k

Qn
kQn−1

n−1∑
ν=1

|λν|kFνk|tν|k

qνk−1

(
1

Qn−1

n−1∑
ν=1

qν

)k−1

= O(1)

m∑
ν=1

1

qνk−1
|λν|kFνk|tν|k

m+1∑
n=ν+1

αn
µ(δk+k−1)qn

k

Qn
kQn−1

= O(1)

m∑
ν=1

1

qνk−1
|λν|kFνk|tν|k

(νqν)
k−1

Qν
k

, using (3.2)

= O(1)

m∑
ν=1

νk−1|tν|k
(
|λν|Fν
Qν

)k

= O(1)

m∑
ν=1

νk−1|tν|k using (3.4)

= O(1) as m →∞.

Next

m+1∑
n=2

αn
µ(δk+k−1)|Tn,2|k

=

m+1∑
n=2

αn
µ(δk+k−1)|

qn
QnQn−1

n−1∑
ν=1

Rν−1
rν

fνλνtν|k

≤
m+1∑
n=2

αn
µ(δk+k−1) qn

k

Qn
kQn−1

n−1∑
ν=1

Rν
kFν

k|λν|k|tν|k

qνk−1rνk

(
1

Qn−1

n−1∑
ν=1

qν

)k−1

= O(1)

m∑
ν=1

Rν
kFν

k|λν|k|tν|k

qνk−1rνk

m+1∑
n=ν+1

αn
µ(δk+k−1)qn

k

Qn
kQn−1

= O(1)

m∑
ν=1

νk−1|tν|k
(
RνFν|λν|
rνQν

)k

= O(1)

m∑
ν=1

νk−1|tν|k using (3.6)

= O(1) as m →∞.

Further

m+1∑
n=2

αn
µ(δk+k−1)|Tn,3|k

=

m+1∑
n=2

αn
µ(δk+k−1)|

qn
QnQn−1

n−1∑
ν=1

Rν−1
rν

Fν+1(∆λν)tν|k

≤
m+1∑
n=2

αn
µ(δk+k−1) qn

k

Qn
kQn−1

n−1∑
ν=1

(Rν−1)
k
(Fν+1)

k|∆λν|k|tν|k

qνk−1rνk

(
1

Qn−1

n−1∑
ν=1

qν

)k−1



Int. J. Anal. Appl. 18 (2) (2020) 180

= O(1)

m∑
ν=1

(Rν−1)
k
(Fν+1)

k|∆λν|k|tν|k

qνk−1rνk

m+1∑
n=ν+1

αn
µ(δk+k−1)qn

k

Qn
kQn−1

using (3.3)

= O(1)

m∑
ν=1

νk−1|tν|k
(
Rν−1Fν+1|∆λν|

rνQν

)k

= O(1)

m∑
ν=1

νk−1|tν|k using (3.7)

= O(1) as m →∞.

Again,

m+1∑
n=2

αn
µ(δk+k−1)|Tn,4|k

=

m+1∑
n=2

αn
µ(δk+k−1)|

qn
QnQn−1

Rnλnfntn
rn

|k

≤
m+1∑
n=2

αn
µ(δk+k−1)|tn|k

(
qnRnFn|λn|
QnQn−1rn

)k

=

m+1∑
n=2

αn
µ(k−1)|tn|k

(
qnRnFn|λn|αnµδ

QnQn−1rn

)k

= O(1)

m+1∑
n=2

αn
µ(k−1)|tn|k, using (3.7)

= O(1) as m →∞.

Next,

m+1∑
n=2

αn
µ(δk+k−1)|Tn,5|k

=

m+1∑
n=2

αn
µ(δk+k−1)|

p0q0
PnQn−1

n−1∑
ν=1

λνtν|k

≤ O(1)
m+1∑
n=2

αn
µ(δk+k−1) 1

Pn
kQn−1

n−1∑
ν=1

|λν|k|tν|k

qνk−1

(
1

Qn−1

n−1∑
ν=1

qν

)k−1

= O(1)

m∑
ν=1

|λν|k|tν|k

qνk−1

m+1∑
n=ν+1

αn
µ(δk+k−1)

Pn
kQn−1

= O(1)

m∑
ν=1

|λν|k|tν|k

qνk−1

m+1∑
n=ν+1

αn
µ(δk+k−1)qn

k

Qn
kQn−1

using (3.2)

= O(1)

m∑
ν=1

νk|tν|k
(
|λν|
Qν

)k

= O(1)

m∑
ν=1

νk|tν|k using (3.6)

= O(1) as m →∞.



Int. J. Anal. Appl. 18 (2) (2020) 181

Again,

m+1∑
n=2

αn
µ(δk+k−1)|Tn,6|k

=

m+1∑
n=2

αn
µ(δk+k−1)|

p0q0
PnQn−1

n−1∑
ν=1

Rν−1
rν

(∆λν)tν|k

≤ O(1)
m+1∑
n=2

αn
µ(δk+k−1) 1

Pn
kQn−1

n−1∑
ν=1

(Rν−1)
k|∆λν|k|tν|k

rνkqνk−1

(
1

Qn−1

n−1∑
ν=1

qν

)k−1

= O(1)

m∑
ν=1

(Rν−1)
k|∆λν|k|tν|k

rνkqνk−1

m+1∑
n=ν+1

αn
µ(δk+k−1)

Pn
kQn−1

= O(1)

m∑
ν=1

νk−1|tν|k
(
Rν−1|∆λν|
rνQν

)k

= O(1)

m∑
ν=1

νk−1|tν|k using (3.9)

= O(1) as m →∞.

Finally,

m+1∑
n=2

αn
µ(δk+k−1)|Tn,7|k

=

m+1∑
n=2

αn
µ(δk+k−1)|

p0q0
PnQn−1

Rn
rn
λntn|k

= O(1)

m+1∑
n=2

αn
µ(δk+k−1)|tn|k

(
Rn|λn|

PnQn−1rn

)k

= O(1)

m+1∑
n=2

αn
µ(δk+k−1)|tn|k

(
qnRn|λn|
QnQn−1rn

)k

= O(1)

m+1∑
n=2

αn
µ(k−1)|tn|k

(
qnRn|λn|αnµδ

QnQn−1rn

)k

= O(1)

m+1∑
n=2

αn
µ(k−1)|tn|k, using (3.10)

= O(1) as m →∞.

This completes the proof of the theorem.



Int. J. Anal. Appl. 18 (2) (2020) 182

5. Conclusion

For µ = 1, the summability method |(N,qn)(N,pn),αn,δ,µ|k reduces to the summability method

|(N,qn)(N,pn),αn,δ|k. For, f(αn) = (αn)δ and δ ≥ 0, |(N,qn)(N,pn),αn,δ; f|k - summability re-

duces to |(N,qn)(N,pn),αn,δ|k - summability. Again, for δ = 0, |(N,qn)(N,pn),αn,δ|k - summabil-

ity reduces to |(N,qn)(N,pn),αn|k- summability and for αn = n, |(N,qn)(N,pn),αn|k- summability re-

duces to |(N,qn)(N,pn)|k-summability. When pn = 1 = qn, |(N,qn)(N,pn)|k-summability is same as

|(R,qn)(R,pn)|k-summability. Also, |(R,qn)(R,pn)|k-summability reduces to |(R,qn)(A)|k-summability

when (R,pn)-summability is replaced by A- summability. From the above results and discussions, we are in

a conclusion that our results are more generalized and in particular generalizes the results of Sulaiman [7],

Paikray et al [3] and Misra et al [2].

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

References

[1] Das, G., Tauberian theorems for absolute Norlund summability, Proc. Lond. Math. Soc. 19 (2) (1969), 357-384.

[2] Misra, M., Padhy, B.P., Buxi, S.K. and Misra, U.K., On indexed product summability of an infinite series, J. Appl. Math.

Bioinform. 1 (2) (2011), 147-157.

[3] Paikray, S.K., Misra, U.K. and Sahoo, N.C., Product Summability of an Infinite Series, Int. J. Computer Math. Sci. 1 (7)

(2010), 853-863.

[4] Parameswaran, M.R., Some product theorems in summability, Math. Z. 68 (1957), 19-26.

[5] Rajgopal, C.T., Theorems on product of two summability methods with applications, J. Indian Math. Soc. 18 (1) (1954),

88-105.

[6] Ramanujan, M.S., On products of summability methods, Math. Z. 69 (1) (1958), 423-428.

[7] Sulaiman, W.T., A Note on product summability of an infinite series, Int. J. Math. Sci. 2008 (2008), Article ID 372604.

[8] Szasz, O., On products of summability methods, Proc. Amer. Math. Soc. 3 (2) (1952), 257-263.


	1. Introduction
	2. Known Theorems
	3. Main Theorem
	4. Proof of Theorem 3.1
	5. Conclusion
	References