Journal of Applied Mathematics & Bioinformatics, vol. 1, no.2, 2011, 147-157
ISSN: 1792-6602 (print), 1792-6939 (online)
International Scientific Press, 2011
On indexed product summability
of an infinite series
Mahendra Misra1, B.P. Padhy2, S.K. Buxi3 and U.K. Misra4
Abstract
A theorem on indexed product summability of an infinite series has been
established.
Mathematics Subject Classification : 40D25
Keywords: Product summability
1 Introduction
Let
a
n
be an infinite series with the sequence of partial sums s n  . Let
p n  be a sequence of positive real constants such that
1
Government Science College, Malkangiri, Odisha, India,
e-mail: mahendramisra@2007.gmail.com
2
Department of Mathematics, Roland Institute of Technology, Golanthara-760008,
Odisha, India, e-mail: iraady@gmail.com
3
4
Department of Mathematics, Gunupur Science College, Gunupur, Odisha, India.
Department of Mathematics, Berhampur University, Brahmapur – 760007, Odisha,
India, e-mail: umakanta_misra@yahoo.com
Article Info: Revised : July 19, 2011. Published online : November 30, 2011
148
On indexed product summability of an infinite series
Pn  p0  p1    pn  
as n   , Pi  p1  0 .
The sequence-to-sequence transformation
tn 
defines
R,
The series
1
Pn
n
p s
v 0
n v
p n  transform of s n  generated by p n .
a
n
is said to be summable R, pn

n
k 1
n 1
k
, k  1 , if [2]
tn  tn 1   .
k
Similarly, the sequence-to-sequence transforms
Tn 
1
Pn
n
p
v 0
nv
sv
defines the N , p n  transform of s n  generated by p n .
The series
a
n
is said to be summable  N , q n  N , p n  k , k  1 , if

n
k 1
n 1
 n   n 1   ,
k
where  n  defines the sequence of  N , q n  transform of the N , p n  transform of
s n  , generated by the sequence q n  and p n , respectively.
Let  n  be any sequence of positive numbers. The series
a
n
is said to
be summable  N , q n  N , p n ,  n k , k  1 , if


n 1
k 1
n
 n   n 1   ,
k
where  n  defines the sequence of  N , q n  transform of the N , p n  transform
of s n  , generated by the sequence q n  and p n , respectively.
M. Misra, B.P. Padhy, S.K. Buxi and U.K. Misra
149
Let  n  be any sequence of positive numbers. The series
a
n
is said to
be summable  N , q n   N , p n ,  n ;  k , k  1 , if

 
n 1
k  k 1
n
 n   n 1   ,
k
where  n  defines the sequence of  N , q n  transform of the N , p n  transform of
s n  , generated by the sequence q n  and p n , respectively.
Let f be a function of  n , if

{ f (
n 1
then the series
a
n
)}k ( n ) k 1  n   n 1  ,
k
n
is said to be
 N , qn   N , pn  ,  n ; f
k
, k  1 , summable .
Clearly for f ( n )   n ,   0 ,
 N , qn   N , pn  ,  n ; f
k
  N , qn   N , pn  ,  n ; 
k
and for   0
N , q n  N , p n ,  n ; f
  N , q n   N , p n ,  n k .
k
We may assume throughout this paper that Qn  q 0    q n   as
n   and Pn  p 0    p n   on n   .
2. Known Results
In 2008, Sulaiman [4] has proved the following theorem.
Theorem 2.1 [4] Let k  1 and m  be a sequence of constants.
Define
fv 
n
qr
, Fv 

r  v Pr
Let pnQn  O  Pn  such that
n
p
r v
r
fr
150
On indexed product summability of an infinite series
n k 1qnk
(vqv k 1 )
)
 O(
Qnk Qn 1
Qvk


n  v 1
Then
R, rn
sufficient
k
conditions
for
the
implication
a
is
n
summable
  a n  n is summable R, q n  R, p n  k are
v Fv  O(Qv ) ,
v  O(Qn ) ,
pv Rv v  O(Qv ) ,
pv qv Rv v  O(Qv Qv 1 rv ) ,
pn qn Rn n  O( Pn Qn rn ) ,
Rv 1  v Fv 1  O(Qv rv ) ,
and
Rv 1  v  O(Qv rv ) .
Subsequently Paikray [1] generalize the above theorem by replacing the
R, pn  summability by A-summability. He proved:
Theorem 2.2 [1] Let k 1, n  be a sequence of constants. Let us define
fv 
n
 qr ar ,
r v
n
Fv   f r .
r v
Then the sufficient conditions for the implication
R, rn
k
a
  a n  n summable R, q n  ( A) k are
n k 1qnk
1
 O( k ) ,

k
v
n  v 1 Qn Qn 1
m 1
Rv  O(rv ) ,
n

r v
k
k 1
r
q
 O (qv ) ,
qn
 O(1) ,
Qn
n
a
r v
k
r ,v
 O(v k 1 ) ,
q n n a n ,n
Qn 1
 O(1) ,
n
is summable
M. Misra, B.P. Padhy, S.K. Buxi and U.K. Misra
v
( v ) k
 O(v k 1 ) ,
k 1
qv
In
this
v
paper
 N , qn   N , pn  ,  n ; f
k
 O(1) ,
we
vk
and
proved
151
 O(v k 1 ) .
k 1
v
q
the
following
theorem
, k  1 , summability of the infinite series
on
the
a 
. We
n n
prove:
3 Main Theorem
For the sequences of real constants
p n  and q n and
the sequence of
positive numbers  n , we define
fv 
n

i v
qn i pi v
Pi
Fv 
and
n
f
i v
(3.1)
i
Theorem 3.1 Let
Qn  Oq n Pn 
(3.2)
and
{ f ( n )}k ( n ) k 1 qnk
(v qv ) k 1
) , as m   .
 O(

Qnk Qn 1
Qvk
n  v 1
m 1
Then for any sequences
implication
a
n
rn 
and
is summable R, rn
n  ,
k

(3.3)
the sufficient conditions for the
a 
n
n
is
 N , qn   N , pn  ,  n ; f
k
,
k  1 , summable, are
n Fv  O(Qv ) ,
(3.4)
n  O(Qn ) ,
(3.5)
Rv Fv v  O(Qv rv ) ,
(3.6)
152
On indexed product summability of an infinite series
qn Rn Fn n  O(Qn Qn 1 rn ) ,
(3.7)
Rv 1 Fv 1 v  O(Qv rv ) ,
(3.8)
Rv 1 v  O(Qv rv ) ,
(3.9)
qn Rn n  O(Qn Qn 1 rn ) ,

n
k 1
n 1
(3.10)
tn  O(1) ,
k
(3.11)
and

 { f (
n2
)}k ( n ) k 1 tn  O(1) ,
k
n
(3.12)
where Rn  r1  r2    rn .
Proof. Let t n  be the R, rn  transform of the series  a n . Then
t n' 
1
R
n

v 0
rv s v .
Then
tn  tn  tn 1 
n
rn
 Rv1 av
Rn Rn 1 v 1
Let s n  be the sequence of partial sums of the series
a 
n n
and  n  the
sequence of  N , qn   N , pn  -transform of the series  an n . Then
n 
Hence
1
Qn
n

r 0
qn  r
1
Pr
r
 pr v sv 
v 0
n
qn  pr 
1 n
1

s


Qn  0 r 
Pr
Qn
n

v 0
f v sv .
M. Misra, B.P. Padhy, S.K. Buxi and U.K. Misra
153
Tn   n   n 1

1
Qn

v 0
f v sv 
1
Qn 1
n
n 1

v 0
f v sv  
r
qn
f
f r  av v  n

Qn Qn 1 r 0 v 0
Qn 1

n
qn

R 1av 

QnQn 1  1
R 1


n
n

r v
fr 
qn
Qn Qn 1
n

v 0
f v sv 
f n sn
Qn 1
n
a 
v 0
v v
q0 p0
Pn Qn 1

n
 R
v 1
a
1 v
R 1
n 1
v
n
qn
 n

{ ( Rr 1ar ) ( v  f r )  n f n  Rv 1av }
Qn Qn 1 v 1 r 1
Rv 1 r v
Rn 1 v 1

 n
p0 q0 n 1  v

{   Rr 1 av  ( v )  n  Rv 1av }
PnQn 1 v 1  r 1
Rn 1 v 1
 Rv 1
n 1
qn
R
R
R
{  (v Fv tv  v 1 f v v tv  v 1 (v ) Fv 1 tv )  n n Fntn }

Qn Qn 1 v 1
rv
rv
rn

p0 q0 n 1
R
R
{ (v tv  v 1 (v ) t )  n n tn }
Pn Qn 1 v 1
rv
rn


7
T
i 1
n ,i
In order to prove the theorem, using Minkowski’s inequality, it is
sufficient to show that

 { f (
n 1
n
)}k ( n ) k 1 Tn ,i
for i  1, 2,3, 4,5, 6, 7,
Now, on applying Holder’s inequality, we have
k
,
154
On indexed product summability of an infinite series
m 1
 { f (
n2

n
m 1
 { f (
)} ( n )
k 1
{ f ( n )} ( n )
k 1
n2

k
)}k ( n ) k 1 Tn ,1
m 1

k
n
k
n2
 O(1)
1
m
q
v 1
v 1
m
v
k 1
v 1
 O(1)
k
qnk
Qnk Qn 1
m
v
k 1
v 1
tv
k

v 1
n 1

v
Fv tv
v Fvk tv  1
k
k
v 1
qvk 1

qv 

v 1

n 1
k 1
(v qv ) k 1
, using (3.3)
Qvk
 v Fv 


 Qv 
k
k
n 1

 Qn 1
k
k

1
m 1
{ f ( n )} ( n ) qnk
k
k
Fv tv 
Qnk Qn 1
n  v 1
1
k
v Fvk tv
k 1
qv
m
 O(1) 
 O(1)
v
k 1
v
qn
Qn Qn 1
k
k
tv , using (3.4)
 O(1) , as m   .
Next,
m 1
 { f (
n2

n
m 1
 { f (
n2

k
)}k ( n ) k 1 Tn ,2
n
n2
)} ( n )
k
n
k 1
Rvk Fvk v tv
 O(1)
qvk 1 rvk
v 1
k
m
 O(1)
m
v
k 1
v 1
 O(1)
m
v
v 1
qn n 1 Rv 1
f v v tv

QnQn 1 v 1 rv
k 1
k
k
k
qnk n 1 R Fv v  1


Qnk Qn 1 v 1 qvk 1 rvk  Qn 1
k
m 1
 { f (
)} ( n )
k
k 1
tv
k
tv
k
{ f ( n )}k ( n ) k 1 qnk

Qnk Qn 1
n  v 1
m 1
 Rv Fv v 


 rv Qv 
k
 O(1) , as m   .
,
k
using (3.6)

qv 

v 1

n 1
k 1
M. Misra, B.P. Padhy, S.K. Buxi and U.K. Misra
155
Further,
m 1
 { f (
n2

n
m 1
 { f (
n2

k
)}k ( n ) k 1 Tn ,3
m 1
{ f (
n2
)} ( n )
k
n
)} ( n )
k
n
qn n 1 Rv 1
 (v ) Fv1 tv
Qn Qn 1 v 1 rv
k 1
k 1
qnk
Qnk Qn 1
R k 
 O(1)  v k1 k r1 Fvk1 tv
rv qv
v 1
k
m
m
v
 O(1)
k 1
v 1
 O(1)
m
v
k 1
tv
tv
v 1
k
k
k
n 1

v 1
Rvk1
k
v Fvk1 tv
k
k 1
rv qv




k
using (3.8)
 O(1) , as m   .
Again,
m 1
 { f (
n2

n
)}k ( n ) k 1 Tn ,4
m 1

n2

{ f ( n )} ( n )
k
m 1
 { f (
n2
)} ( n )
k
n
m 1
  { f ( n )} ( n )
k
 O(1)
m 1
{ f (
n2
n
k
qn Rn n f n tn
Qn Qn 1
rn
tn
tn
k
k
 qn Rn Fn n 


 Qn Qn 1 rn 
 qn Rn Fn n 


 QnQn 1rn 
)}k ( n ) k 1 tn
 O(1) , as m   .
Next,
k 1
k 1
n2
k
k 1
k
 1

 Qn 1
{ f ( n )}k ( n ) k 1 qnk
, by (3.3)

Qnk Qn 1
n  v 1
m 1
 Rv 1 Fv 1 v


rv Qv

,
k
k
,
k
k
using (3.7)

qv 

v 1

n 1
k 1
156
On indexed product summability of an infinite series
m 1
{ f (
n2

)}k ( n ) k 1
n
m 1
{ f (
n2
)} ( n )
k
n
m 1
{ f (
 O(1)
n2
v
m

 O(1)

 O(1)
n
)} ( n )
tv
k
k
m
v
k
tv
v 1
m
 O(1)  v k tv
k
k

v 1
1
k
Pn Qn 1
v
tv
n 1
v
v 1
qvk 1

k
tv
k
 1

 Qn 1

qv 

v 1

n 1
k 1
{ f ( n )}k ( n ) k 1

Pnk Qn 1
n  v 1
k
tv
k 1
k
n 1
m 1
{ f ( n )}k ( n ) k 1 q nk
,
Qnk Qn 1
m 1

q vk 1
v 1
p0 q0
Pn Qn 1
k
k
v
m
k
k 1
qvk 1
v 1
 O(1)
Tn ,5
n  v 1
 n k

 Qv

,




using (3.2)
k
using (3.7)
v 1
 O(1) , as m   .
Again,
m 1
{ f (
n2

n
)}k ( n ) k 1 Tn ,6
m 1
 { f (
n2
)} ( n )
k
n
k
k 1
m 1
 O(1)  { f ( n )} ( n )
k
p0 q0 n 1 Rv 1
 (v ) tv
Pn Qn 1 v 1 rv
k 1
n2
m
 O(1) 
v 1
m
Rvk1 v tv
rvk qvk 1
 O(1)  v
k
k 1
v 1
 O(1)
m
v
v 1
tv
k 1
k
tv
k
n 1
Rvk1
1
v

Pnk Qn 1 v 1 rvk qvk 1
 O(1) , as m   .
k
tv
k
 1

 Qn 1
{ f ( n )}k ( n ) k 1
 Pk Q
n  v 1
n
n 1
m 1
 Rv 1 v 


 rv Qv 
k
k
k
, using (3.9)
Rv 1 v  O(Qv rv )

qv 

v 1

n 1
n 1
M. Misra, B.P. Padhy, S.K. Buxi and U.K. Misra
157
Finally,
k
m 1
 { f ( n )}k ( n )k 1 Tn,7
n2

m 1
{ f (
n2
 O(1)
)} ( n )
k
n
m 1
 { f (
n2
 O(1)
n2
p0 q0 Rn n tn
Pn Qn 1
rn
n
)} ( n )
n
)} ( n )
k
m 1
 { f (
k 1
k
m 1
k 1
k 1
k
tn
tn
 O(1)  { f ( n )}k ( n ) k 1 tn
k
k
k
 Rn n 


 Pn Qn 1 rn 
k
 qn Rn n 


 Qn Qn 1 rn 
k
, using (3.10)
n2
 O(1) , as m   .
This completes the proof of the Theorem.

References
[1] S.K. Paikray, Ph.D Thesis, Berhampur University, 2010.
[2] B.E. Rhoades, Inclusion theorems for absolute matrix summability methods,
J. Mathematical Analysis Appl., 238, (1999), 82-90.
[3] S.K. Sahu, Ph.D Thesis, Berhampur University, 2010.
[4] W.T. Sulaiman, Note on product summability of an infinite series, Hindawi
publishing corporation, 2008, Article ID 372604, (2008).