; δ – Summability of A Series

A Note on

n k

p

N ,

^α

; δ – Summability of A Series

S.K. PAIKRAY

¹

, R. K. JATI

²

, U. K. MISRA

³

and N. C. SAHOO

⁴

1

Dept. of Mathematics,

Ravenshaw University, Cuttack, Odisha, INDIA.

2

Dept. of Mathematics, DRIEMS, Cuttack. Odisha, INDIA

3

Dept. of Mathematics,

Berhampur University, Odisha, INDIA.

4

S. B Women’s College (Auto), Cuttack, Odisha, INDIA (Received on: May 22, 2012)

ABSTRACT

In this paper a theorem on

n k

p

N ,

^α

; δ

– summability has been proved.

Keywords: Summability, Bounded, partial sum.

AMS Classification No.: 40 G05.

1. INTRODUCTION

Let ∑ a

n

be an infinite series and

{ } s

n

be the sequence of its partial sums.

{ } p

n

be a sequence of non-negative numbers with

0

,

₀

0

≠

= ∑

=

p p P

n n

γ γ

(1.1) Let us define

α

p

n

= ∑

=

−

− n

n

p

A

0 1

γ α γ

γ

(1.2)

Where ^A

^nα

⁼ ( )

^αn+α

^{, α ≥ 1} (1.3)

Let ∑

=

=

ⁿ

n

p

P

γ 0 α γ

α

, with

= 0

=

₋

−α α

i

i

p

p , i

≥

1 (1.4)

Let ∑

=

=

ⁿ

n

n

p s

T P

0

1

γ α γ

α γ α

Then the sequence { } ( ^T

^nα

^{is N , p}

^nα

) ^{, mean}

of the sequence { } s

n

generated by the

sequence of coefficients { } ^p

^nα

^.

321

The series ∑ a

n

is said to be summable 1

,

, p k ≥

N

n k

α

If ∑

^∞

= −

−

∞

<

 −







 





1

1 1

n

k n n k

n

n

T T

p

P

α α

α α

Again, the series ∑ a

n

is said to be summable

n k

p

N ^,

^α

^; δ , k ≥ 1, δ ≥ 0

If ∑

^∞

= −

− +

∞

<

 −







 





1

1 1

n

k n n k k

n

n

T T

p

P

α α

δ α α

If

α

= 0 ,

δ

= 0, k = 1, then

n k

p N ,

^α

, δ summability is same as

p

n

N , – summability.

2. KNOWN THEOREMS Dealing with

n k

p

N , summability Bor

¹

proved the following theorem :

Theorem – A

Let k

≥ 1 and let the sequence

{ } p

n

and { } λ

n

be such that

(i) 



 



= 

∆ X

_n

O n 1

(2.1)

(ii) ∑

^∞

=

−−

1 1

n k

X

n

+

₊

< ∞ n

k n k

n

| | |

| λ λ

₁

(2.2)

(iii) ^∑

^∞

( )

=

∞

<

∆ +

1

|

| 1

n

n k

X

n

λ (2.3)

Where X

n

=

n n

np P .

If {s

n

} is bounded, then the series

∑

^∞

=1 n

n n

n

X

a λ a

n

λ

n

X

n

is summable

n k

p N , .

Subsequently, Misra, sahoo and Paikray

²

prove the following theorem

Theorem B: Let {s

n

} be a bounded sequence and the sequence { λ

n

} and { p

^αn

} satisfy the following conditions :

(i) P

_n^α

= O ( n , p

_n^α

)

(ii) ∑

=

−

−

−

−

−

− =

n k

k

n v n n

v

n

O

P p P

p

0

1

1

1

( 1 )

γ α

α α α

(iii) ∑

^∞

< ∞

=

+ k

n

n k n k

k

p

n ( ) | |

1

2 ^α

λ

δ

and

(iv) ∑

^∞

∆ < ∞

=

+ k

n k n n

k

k

p

n ( ) | |

1

2 ^α

λ

δ

Then the series ,

∑ λ

n

P

n^α

a

n

^is | N , p

_n^α

, δ | k summability

where k ≥ 1 and α > – 1

3. MAIN RESULT :

In this paper we have extended theorem –B for

n k

p

N ,

^α

; δ – summability

of the series ∑ a

n

λ

n

X

n

Theorem:

Let k

≥

1,

α ≥

1,

δ

k < 1 and

α α

n n

n

np

X = P

If the sequence {s

n

} is bounded and the sequences { p

^α_n

} and { λ

n^α

} are such that

)

1

(

α α

n

n

O n p

P

₋

= (3.1) )

1 ( ) (

|

| X

_γ ^k

γ

^δk

p

_γ^α

+ P

_γ^α

= O (3.2)

∑

^∞

=

∞

<

1

|

|

n

k n

n

λ (3.3)

∑

^∞

=

∞

∆ <

1

|

|

n n

n

λ (3.4)

 

 



= 

 





 



 ∆

− −

=

∑

k

n k

O

X γ

γ γ

1

1 1

1

(3.5)

 



 



= +

∆

+

1

1

|

|

|

|

1

γ

γ

γ

γ

O

X X

k

k

(3.6)

 

 



= 

 





 





⁻

O n p X

P

k

n k

n

n

1

|

|

δ 1 α α

(3.7) Then the series,

∑

^∞

=1 n

n n

n

X

a λ is summable

n k

p

N ,

^α

, δ .

4. PROF OF THE THEOREM

α

T

n

be the { N , p

_n^α

} mean of the series ∑

^∞

=1 n

n n

n

X

a λ

∑

=

=

ⁿ

n

n

p s

T P

0

1

γ α γ

α γ α

= 1 ( 0 )

0 0

0

=

 

 



 ∑

∑

= =

x X a

P p

z

z z z n

n

υ

γ α

α γ

λ

= 



 



 



 

 +  +

 

 



+  ∑ ∑

=

=

n

z

z z z n

z

z z z n

X a p

X a p

X a

P p

0

1

0 1 0 0 0

0

( ) ...

1

_α ^α

λ

^α

λ

^α

λ

= ¹ [ ^p

⁰

^{( a}

^{0 0}

^X

⁰

^{) p}

¹

^{( a}

^{0 0}

^X

⁰

^a

^{1 1}

^X

¹

⁾

P

_nα ^α

λ +

^α

λ + λ

+ ^... + P

n^α

⁽ a

0

λ

0

X

0

+ a

1

λ

1

X

1

+ ^... + a

n

λ

n

X

n

⁾ ]

323

= ¹ [ ^{( p}

⁰

^p

¹

^{... p ) a}

^{0 0}

^X

⁰

P

_nα ^α

+

^α

+

^nα

λ

+ ( p

₁^α

+ ... + p

_n^α

) a

₁

λ

₁

X

₁

⁺ + p

_n^α

. a

_n

λ

_n

X

_n

]

= ¹ [ ^{P ( a}

^{0 0}

^X

⁰

^{) ( P P}

⁰

^{) a}

^{1 1}

^X

¹

P

_nα ^nα

λ +

^nα

−

^α

λ

+… + ( P

_n^α

− P

_n^α₋₁

) a

_n

λ

_n

X

_n

]

= 1 ( ) ( ) , 0

1 0

1

 =





 

 −

₋

= −

∑

^α

γ α γ γ γ

γ α

α

^{P P a} λ ^{X where P} P

n n n

For n

≥

1,

 

 



 −

=

− ∑

= −

−

n n n

n

n

P P a X

P T

T

0

1

1

1 ( )

γ α γ γ γ

γ α α

α

α

λ

 

 



 −

− ∑

⁻

= − −

− 1

0

1 1

1

) 1

ⁿ

(

n n

X a p

p

γ

p

α γ γ γ

γ α

α

λ

= ∑ ∑

= = − −

−

−

− −

n

−

n

n n

n n

X a P P P

X a P

P

1

P

1

1 1

1

1

1 ( )

) 1 (

γ γ α γ γ γ

α γ γ α

γ α γ

α γ

α

λ λ

= ∑ ∑

= −

=

−

ⁿ

n n

n n

X a p P

X a

P P

1

1 1

1 1

γ α γ γ γ

α γ

γ α γ γ γ

α

λ λ

∑ ∑

= −

= −

−

−

+

−

^{n n}

n n

n

X a P P

X a

p p

1

1

1 1

1 1

1 1

γ α γ γ γ

γ α γ γ γ α γ

α

λ λ

= ∑

= −

−

 





 



 − +

ⁿ r

n n

X a P P

P

1

1 1

1 1

γ α γ γ γ

α

α

λ

= ∑

= −

− n

n n

n

P a X

P P

p

1 1

1 γ α γ γ γ

α γ α

α

λ

= 



 





⁻

∆

₋

+

₋

− =

∑ ⁽

1

^{) (}

1

⁾

1

1 1

n n n n n

n n

n

s P X s P X

P P

p

α

λ

γ γ ^α

λ

γ γ γ α

α α

by Abel’s lemma

= 



 ∑

⁻

− + ∆

− =

) ( {

1

1 1

γ γ α γ α γ γ

γ γ γ

α α

α

^s λ ^{X p p X} λ

P P

p

ⁿ

n n

n

^{+ P}

^γα

^λ

^γ+1

^{( ∆ X}

^γ

^{)} + s}

ⁿ

^{( P}

^nα−1

^λ

ⁿ

^X

ⁿ

^)] ]

= ∑ ∑

⁻

=

−

= −

−

∆ +

−

¹

1 1

1 1

1

) (

n n

n n

n n

n

n

P X s

P P X p p P s

P p

γ α γ γ γ

γ α α

α γ

γ α γ α γ

α

α

λ λ

∑

⁻

= −

− +

−

+

∆

+

¹

1

1 1

1 1

) (

n

n n n n n n

n n

n

n

s X P

P P s p X P P

P p

γ

α α

α α γ

γ α γ

α γ α

α

λ λ

= T

_n,1

+ T

_n,2

+ T

_n,3

+ T

_n,4

The theorem will be proved if we can prove that

∑

^∞

=

− +

=

∞

 <







 





1

, 1

4 , 3 , 2 , 1 ,

n

k r n k k

n

n

T r

p P

^δ

α α

(A) ∑

^∞

=

− +

 





 





1

1 , n

1

P

n

k n n

p T

k δk

α α

=

k

n

n

n n

n k

k

n

n

p X s

P P

p p

∑

^∞

P ∑

=

−

− =

−

+

−

 





 





1

1

1 1 1

γ α γ γ γ

α γ α δ α α

α

λ

Let us consider

n k

n n

n m

n

k k

n

n

p X s

p p

p p

p ∑

∑

⁻

− = +

=

−

+

−

 





 





¹

1 1 1

2

1

γ α γ γ γ

α γ α δ α

α

α

λ

=

1

1 1

1

2

1

1

1

⁻

+

− +

=

−

 





 





 





 



∑ 

k

n n

m

n

k

n n

p p

p p

α α

δ

α

α _n ^k

k k

k

p

p X

∑

⁻

s

= 1 − 1

1 1

) ( ) (

γ

α γ α γ γ γ γ

λ

=

 





 





 





 





 





 





 





 





^{− −}

− =

−

− = +

=

−

∑

∑

∑

1 1

1 1 1

1 1 1

2

1

1

1

^{n k}

n n

k k k

n m

n

k

n

n

p

p p X p s

p p

γ γα γ α

γα γ γ α γ

δ α

α

λ

= 0 (1) ∑ ∑

⁻

= +

= −

−

 





 





¹

1 1

2 1

1

1

^h k k k

m

n n

k

n

n

s X p

p p

p

γ

α γ γ γ α γ

δ α

α

λ

=

_α

γ

δ α α

γ

α γ γ γ

γ

λ

1 1

1

1

1

) 1 1 ( 0

− +

+

=

−

=

∑

∑ _ ^





 





n m

h

k

n n

m k k k

p p

p p

X

s

325

= ∑

=

m k k k

p X s

γ 1

α γ γ γ

γ

λ ( )

∑

⁺

( )

+

= − −

1 −

1 1

1

1

1

m

n n

k n

k n

p p p

γ α δ α

α δ

≤

( )

∑ ∑ ( )

=

+ +

= −

−

m m −

n

k n

k k n

k k

p p p

X s

1

1

1

2 1

1

γ γ α δ

α δ α

γ γ γ

γ

λ ^{(1 –}

δ

k>0)

≤

( )

∑ ( )

∑

⁺

+

= −

−

−

=

1

1

2 1

2

1

m

n

k n

k n

m k k k

p p p

X s

γ α δ

α δ

γ

α γ γ γ

γ

λ

≤

∑

=

m k k k

p X s

γ 1

α γ γ γ

γ

λ ∑

⁺

+

=

−

 

 





1

1

1

2 m

n

k

γ

n

δ

(Using 3.1) ≤ ∑

=

m k k k

p X s

γ 1

α γ γ γ

γ

λ ^O ( ) ^γ

^δk−1

≤ ∑

= m

k k k

P X

γ 1

α δ γ γ

γ

γ

γ λ

≤ ∑

= m

k O

1

) 1 (

γ γ

γ

λ using 3.2

< ∞ as m → ∞ using 3.3

(B) ∑

^∞

=

− +

 





 





1

2 , 1

n

k n k k

n

n

T

p P

^δ

α α

=

n k

n n

n n

k k

n

n

P X X s

P P

p p

p ∑

∑

⁻

− =

∞

=

− +

 ∆







 





¹

1 1 1

1

) (

γ α γ γ γ

α γ α δ α α α

Now,

_γ

γ γα

α

∑

⁻

∆ λ

− = 1

1 1

1

ⁿ

n

P P

∑

⁻

− =

−

∆

≤

¹

` 1 1 1

1

ⁿ

n n

P P

_γ ^γ

α

α

λ

= O(1)

Let us consider

n k

n n

n m

n

k k

n

n

P X X s

P P

p p

P ∑

∑

⁻

− = +

=

− +

 ∆







 





¹

1 1 1

2

1

) (

γ α γ γ γ

α γ α δ α α α

( )

n k

k n m

n

k

n

n

X P s

p P

P ∑

∑

⁻

− = +

=

−

 ∆







 



≤ 

¹

1 1 1

2

1

) )(

1 (

γ α γ γ

γ α γ

δ

α

α

λ

^{k n}

( ) (

^k

)

^{kk k}

n n m

n

k

n

n

X s P P

P P p

P 



 



 ∆ ∆

 





 





 





 



= 

^{− −}

=

−

−

− +

=

−

∑

∑

1 1 1

1 1

1 1 1

2

1

) ( 1 (

1

γ α γ γ α γ γ γ γ α

α δ α

α

λ λ

1 1

1 1 1

1 1

2 1

1

) 1 (

)) ( (

|

|

|

1 |

^{− −}

− =

−

= +

= −

−

 





 



 ∆

 ∆







 



≤ ∑  ∑ ∑

n k

n n

k k m

n n

k

n

n

P

P P s p X

p p

γ α γ

α γ

γ α γ

γ γ α γ

δ α

α

λ λ

( )

^{1 1}

1 1

1 1 1

2

1

|

| ) (

|

|

|

1 |

^{− −}

=

−

− = +

=

−

 





 



 ∆

 ∆







 



≤ ∑  ∑ ∑

n k n

k k n

m

n

k

n

n

X s P

P p

P

γ γ

γ α γ

γ γ α γ

δ α

α

λ λ

∑ ∑

⁺

+

= −

−

=

 





 



∆ 

≤

¹

1 1

1

1

| 1 ) (

|

|

|

|

| ) 1 (

m

n n

k

n n m

k

P p

P p X O

γ α

δ α α γ γ

α γ

γ

λ

| | | | | | (

¹

)

1

−

=

∆

≤ ∑

^m

^X

^k

^P

γ

^O

^δk γ

α γ

γ

λ γ

=

^{k k}

m

P X

_{γ γ}^{α δ}

γ

γ

γ

γ

λ | | | | |

|

1

∑

=

∆

= | | ( 1 )

1

O

m

∑

=

∆

γ γ

γ

λ using 3.2

< ∞ as m → ∞ using 3.4

(C)

_n ^k

n

k k

n

p T

^,3

1

1

P

n

∑

^∞

=

− +

 





 





^δ

α α

= ∑

^∞

=

− +

 





 





1

1

P

n n

k k

p

n δ α

α n ^k

n n

n

P X s

P P

p ∑

⁻

= +

− 1

∆

1

1 1

) (

γ α γ γ γ

α γ α

α

λ

Let us consider,

∑

⁺

∑

=

−

= +

−

− +

 ∆







 





1

2

1

1

1 1

1

) (

m

n

n k

n n

n k k

n

n

P X s

P P

p p

P

γ α γ γ γ

α γ α δ α

α

α

λ

327

=

k k n k

k k

n m

n

k

n

n

s P X P X

P p

P

^{1 1}

1

1

1 1

1

2

1

)) ( ( ) (

1

⁻

(

⁻

= +

− +

=

−

∆

 ∆







 





 





 



 ∑

∑

α γ

γ α γ γ

γ γ α γ

δ

α

α

λ

=

1 1

1 1 1

1 1 1

1 1

2

) 1 (

|

1 |

^{− −}

− =

−

= +

− + −

=

 





 



 ∆

 





 



 ∆

 





 



 ∑ ∑

∑

n k

n n

k n

m k

n n

n

P X

X P P P

p P

γ α γ

α γ

γ α γ

γ α γ

δ

α

α

λ

=

1 1

1 1

1 1 1

2 1

1

) (

|

1 |

^{− −}

=

−

= +

+

= −

−

 





 



 ∆

 





 



 ∆

 





 



 ∑ ∑

∑

n k n

k m

n n

k

n

n

P X X

P p

P

γ γ

γ α γ

γ α γ

δ

α

α

λ

= 



 



∆ 

 





 



 ∑

∑

⁻

= +

+

= −

−

k n

k m

n n

k

n

n

P X O

P p

P

λ

α γ

γ

γ γ γ α

δ

α

α

1

) (

| 1

¹

|

1 1 1

2 1

1

using 3.5

= ∑ ∑

⁺

+

= −

−

= +

 





 





 

 



∆ 

¹

1 1

1

1 1

1

| 1

|

|

|

m

n n

k

n n k

m

k

P p

O P X P

γ α

δ α α γ γ

α γ

γ

γ

λ

^∑ ( )

=

+



−





 

∆ 

≤

^{m k}

P X O

_k

O

^k

1

1 1

| 1

|

|

|

γ

γ δ γα

γ

γ

λ γ

=

^k

m

k

k

P X

^δ

γ α γ

γ

γ

γ

λ γ

∑

= + +

∆

1

1

1

1

| |

|

|

=

_k

m

k k

X X

|

|

| 1 |

|

|

1 1 1

γ γ

γ

λ

γ

γ ^∆

∑

= + +

using 3.2

= 





 



∑ +

= +

1

| 1

|

1

1

γ

λ

γ γ

m

k

O using 3.6 <

∞

as m

→∞

using 3.4

(D)

_n ^k

n

k k

n

n

T

p P

4 , 1

1

∑

^∞

=

− +

 





 





^δ

α α

=

k

n n n n n n

k k

n

n

s X

P p p

P

_α^α

λ

δ α α

∑

^∞

=

− +

 





 





1

1

Let us consider

k

n n n n

n m

n

k k

n

n

s X

P p p

P

_α^α

λ

δ α α

∑

=

− +

 





 





1

1