A Note on Matrix Summability of an Infinite Series

Abbr:J.Comp.&Math.Sci.

2014, Vol.5(2): Pg.155-161

An International Open Free Access, Peer Reviewed Research Journal www.compmath-journal.org

A Note on Matrix Summability of an Infinite Series

Chitaranjan Khadanga ¹ , Kiran Mishra ² and Lituranjan Khadanga ³

1 Professor,

Department of Mathematics RCET, Bhilai, C.G., INDIA.

2 Assistant Professor, Department of Mathematics,

G.D. Rungta College of Sc. & Tech. Dept. of Mathematics, Bhilai, C.G., INDIA.

3 Assistant Professor, Department of Mathematics,

CEC Bhilai, C.G., INDIA.

(Received on: March 1, 2014) ABSTRACT

In the present Paper we establish an analogue theorem for

1 , , k ≥ A

k

δ summability.

Keywords: Matrix Transformation , Infinite Series, Summability Theory.

INTRODUCTION

Let ∑ ^u _n be an infinite series with sequence of partial sums { } ^s _n .

Let ^T = ( ^a _n , _k ) be an infinite matrix with real and complex elements, then the transform { } ^t _n of { } ^s _n is given by

∑

=

=

n

k

k k n

n a s

t

0

, .

Let _{T =} ( ^a _nk ) ^and σ = { } ^s _{n . Then}

T σ exists for all bounded sequence, if T σ exists for all sequences convergent to 0. A necessary and sufficient condition for T σ to exists for all sequence of either class is that

∑ ^∞

=1 , n

k

a n converges for all m .

The Matrix T transforms all

bounded sequences into sequences if and

only if it transforms all sequences which converge to 0 into bounded sequences. If

( ^a _{n k} )

T

= , , a necessary and sufficient condition for T to transform all sequences of either class into bounded sequences is that there exists a constant C such that

∑ ^∞

=

≤

1 , k

k

n C

a , for all n.

If for k > n , then the method is called triangular matrix method. If

s t

n n =

∞

lim → , then the sequence { } ^s _n ^{or the}

series ∑ ^u _n , is said to be summable ( ^a _{n , k} )

or simply T-summable to s. A series ∑ u n

is said to be absolutely summable by T- method or simply T -summable, if

∞

<

∑ t n − t n −1

Let _{A =} ( ^a _mn ) be a lower-triangular matrix and ∑ a n be an infinite series with sequence of partial sums { } ^s _n such that

∑

=

=

n

v

v nv

n a s

A

0

.

Then the series ∑ ^a _n is said to be summable A k ≥ 1 ,

k , if

∞

<

− ₋

∞

=

∑ − n n ^k n

k A A

n 1

1

1 .

Associate with A we define two lower triangular matrices A ^and A ˆ as follows:

,...

2 , 1 , 0 ,...

2 , 1 , 0

, = =

= ∑

=

v and n

a a

n

v r

nr nv

and

,...

2 , 1 , 0 ,...

3 , 2 , 1

ˆ = a _, − a − ₁ , n = and v = a nv n v n

For any positive integer k ≥ 1 we define Euler-Totient function ^φ ( ) ^k as the number of positive integers not exceeding k _and relatively prime to k _.

In 2007, E.Savas and B.E.Rhodes proved the following theorem on

A k ^-

summability.

Theorem-1: Let A be a lower triangular matrix with non-negative entries satisfying (i) a ₀ = 1 , n = 0 , 1 ,...

n

(ii) a _{n −} 1 , _v ≥ a _nv for n ≥ v + 1 (iii) n a O ( 1 )

nn =

If the sequence { } ^s _n is bounded and the sequence { } ^λ _n is such that

(iv) ( ) 1

1

O

m

n

n =

∑ ∆

=

λ _,

(v) ( ) 1

1

O

a ^k

n m

n

nn =

∑

=

λ _.

Then the series ∑ a n λ n is summable 1

, k ≥ A

k .

KNOWN RESULT Theorem-2

Let A be a lower-triangular matrix with non-negative entries satisfying

,...

2 , 1 , 0 ,

0 1

, = n =

a

n

1

, ,

1 ≥ ≥ +

− a for n v

a _{n v n v}

)

1

(

O

na nn =

and for

∑ ∑

= = + =

= +

n k

n k

k n

k

n a k

n w a

k n

t

1 1

1 log , 1

1 1

( )

( ₁ ) ^{2 2}

1 + +

= _k

vv k

v v

a t

O _,

( )

( ₁ ) ^{2 2}

1 + +

= _k

vv k

v v

a w O

∑ ⁺

+

=

=

∆

1

1

ˆ , m

v n

vv v

n

v a O a

( ) ^{ } 



 





= +

−

− −

=

∑

vv k n k

v

v n v k

a v

O a

t

n 1

1 1

1

, 1

ˆ 1

∑ ^∞ ∑

=

∞

=

=

=

1 1

) 1 (

; ) 1 (

v v vv

v vv

v O

a w O

a t

( ) ( ) ^{ } 



 





= +

 





 





= +

+

+ 2 2 2

2 1

1

; 1 1 1 1

k k

k n k

n n

O w n n

O t n

Then ∑ ^a _n ^λ _n is summable A , k ≥ 1

k .

In the present paper, we establish an analogue theorem for A , , k ≥ 1

k

δ summability. We prove:

MAIN THEOREM

Let A be a triangular matrix with non-negative entries satisfying for

∑ ∑

= = + =

= +

n

k

n

k k n

k

n a k

n w a

k n t

1 1

1 log , 1

1 1

and

∑ ⁺

+

=

=

∆

1

1

ˆ , m

v n

vv v

n v

k a O a

n ^δ

( ) ( )

∑ ^∞ ∑

=

∞

=

= +

= +

1 1

) 1 ( 1

; ) 1 ( 1

v v vv

k v vv

k v O

a w O

a

t _δ

δ

ν ν

Then ∑ ^a _n ^λ _n is summable A , , k ≥ 1

k

δ _.

We need the following lemmas for the proof of our theorem.

Lemma-1

n n nv n

v

v a a

, 1

0

ˆ =

∑ ⁻ ∆

=

Lemma-2 For

( ) ^{k n O} ( ^{n n} )

n

n k

n 3 log

,

1 = = ₂ ² +

> ∑

≤ φ π

λ

Lemma-3

∑

+

=

=

m

v n

v n

a

1

, 1

ˆ

Proof of Main Theorem:

We have

∑ ∑ ∑

= = =

 

 



= 

= ⁿ

i

n

i

i

v v v ni i

ni

n a s a a

T

0 0 0

λ

∑ ∑

= =

=

n

v

n

v i

ni v

v a a

0

λ

∑ ∑

= =

=

=

n

v

n

v

nv v v nv

v

v a a a a

0 1

λ λ

Then

∑ ∑

=

−

=

−

− = −

−

n

v

n

v

v n v v nv

v r n

n T a a a a

T

1

1

1

, 1

1 λ λ

v n n

v

n

v v v nv

v

v a a a a

, 1

1 1

−

= =

∑ − ∑

= λ λ

( ) ⁿ _v

v

v nv n

v

v v v n

nv a a a a

a ∑

∑

=

=

− =

−

=

1 1

,

1 λ ˆ λ

( )

∑

=

+

≤

n

v

v v

n a v v v

a

1

2

, log

ˆ (using Lemma -2)

∑ + ∑

=

n n

v nv v

nv a v a a v v

a

1 1

2 ˆ log

ˆ

( )( ) ∑ ( )( )

∑

=

=

+

=

n

v

v nv n

v

v

nv v a v a a v

a v

1 1

ˆ log ˆ

( ) ( ) ( ) ^



 

 + 

⋅

∆

= ∑ ∑ ∑

=

−

= =

n

v v n

v

v

r

nn r

nv

v v a r a n a va

1 1

1 1

ˆ ˆ

( ) ( ) ( )

∑ ⁻ ∑ ∑

= = =

+

⋅

∆ +

1

1 1 1

ˆ log ˆ log

n

v

v

v

n

v v nn r

nv a r n a a v

a v v

(By using Abel’s Lemma.)

( ) ( ) ( _nn )( ) _n

n

v

v v

n

v v a ˆ v 1 t n a ˆ n 1 t

1

1

, + ⋅ + +

∆

= ∑ ⁻

=

( ) ( ) ( _nn )( ) _n

n

v

v v

n

v v a ˆ v 1 w n a ˆ n 1 w

1

1

, + + +

∆ + ∑ ⁻

=

( ) ( ) ( )

{ } ( ) ( )

∑ ⁻

=

+ +

+

− +

∆

⋅ +

=

1

1

,

, ˆ ( 1 ) 1 ˆ ( 1 )

1 ˆ

n

v

n nn

v v

n v n

v a a v t n a n t

v

( ) ( ) ( )

{ } ( ) ( )

∑ ⁻

=

+ +

⋅ +

− +

∆

⋅ + +

1

1

0 ,

, ˆ ( 1 ) 1 ˆ ( 1 )

1 ˆ

n

v

n n

v v

n v n

v a a v w n a n w

v

( ) ( ) ( )

∑ ⁻ ∑

=

−

=

+ +

+

−

∆

⋅

⋅ +

=

1

1

1

1 , ,

2 ˆ ˆ ( 1 ) ( ˆ ) ( 1 )

1

n

v

n

v

n nn

v v

n v

n v

v a a v t n a n t

t v

( ) ( ) ( )

∑ ⁻ ∑

=

−

=

+ +

+

−

∆

⋅

⋅ + +

1

1

1

1 , ,

2 ˆ ˆ ( 1 ) ( ˆ ) ( 1 )

1

n

v

n

v

n nn

v v

n v

n v

v a a v w n a n w

w v

6 5 4 3 2

1 T T T T T

T − + + − +

= ^.

In order to establish the theorem it is sufficient to prove that

∑ ^∞

=

−

+ < ∞ =

1

,

1 , 1 , 2 , 3 , 4 , 5 , 6

n

k r n k

k T r

n ^δ _.

Now

n k m

n k

k T

n , 1

1

1

∑ ⁺ 1

=

− δ +

∑ ⁺ ∑ ( ) ( )

=

−

=

−

+ + ⋅ ∆

=

1

1

1

1

, 1 2

ˆ 1

m

n

n k

v

v n v v k

k v t a

n ^δ

( ) ( ) ( )

∑ ⁺ ∑ ∑

=

−

−

=

− −

−

=

− +

















 

 

  ∆

 





 



 



 



 + ⋅ ⋅ ∆

≤ ¹

1

1 1

1

1 1 , 1

1

1

1 , 1 2

ˆ 1 ˆ

m

n

k k k n

v

k k

k k v n v k

n

v

k v k n v v k

k v t a a

n ^δ

( )

1 1

1 1

1

1

1

, 1 2

ˆ 1 ˆ

− −

= +

=

−

=

−

+ 



 



 ∆

 

 



 + ∆

= ∑ ∑ ∑

n k

v

nv v m

n

n

v

v n v k v k k

k v t a a

n ^δ

( ) ( ) , ¹

1

1

2 1

1

1 1 ˆ

1 ⁻

−

= +

=

−

+ + ∆ ⋅

≤ ∑ ∑ v n v nn ^k

n

v

k v m k

n k

k v t a a

n

O ^δ

( ) ∑ ⁺ ∑ ( )

=

−

=

∆ +

=

1

1

, 1

1

2 ˆ

1 1

m

n

v n v k v n

v

k v k t a

n

O ^δ

( ) ( ∑ ) ∑

=

+

+

=

∆ +

=

m

v

m

v n

v n v k k

v

k t n a

v O

1

1

1

,

2 ˆ

1

1 ^δ

( ) ( ^m ) _vv

v

k v

k t a

v

O + ⋅

≤ ∑

=1

1 2

1

( ) ( ) ^{= → ∞}

+

= +

⋅ +

= ∑ ∑

= =

+

m as O v

v v

O

m

v

m

v k

k ( 1 ),

) 1 (

1 )

1 ( 1 1 1

1 2 2 1 2

2 .

Next

2 , 1

1 1

n m

n k

k T

∑ ⁺ n

= + δ +

m k

n

v n

v v n k

k a v t

∑ ⁺ n ∑

=

−

=

−

+ +

=

1

1

1

1 ,

1 ˆ ( 1 )

δ

( ) ( ) ( )

∑ ⁺ ∑

=

−

=

− −

+ + ⋅

=

1

1

1

1

1 , 1

,

1 1 ˆ

m

n

n k

v

k k v n k v v n v k

k v t a t a

n ^δ

( ) ( ) ( )

k

m

n

k k n

v

k k

k k v n v k

n

v

k v k n v k

k v t a t a

∑ ⁺ n ∑ ∑

=

−

−

=

− −

−

=

− +

















 

 





 





 



 



 



 + ⋅

= ¹

1

1 1

1

1 1 , 1

1

1

1 ,

1 1 ˆ ˆ

δ

( ) ( )

1 1

1 , 1

1

, 1

1

1 1 ˆ ˆ

− −

=

−

= +

=

−

+ ∑ ∑

∑ + ⋅

=

n k

v

v n v n

v

v n v m k

n k

k v t a t a

n ^δ

( ) ( )

1 1

1 , 1

1

, 1

1

1 1 ˆ ˆ

− −

=

−

= +

=

− ∑ ∑

∑ + ⋅

=

n k

v

v n v k n

v

v n v m k

n

k v t a n t a

n ^δ

( ) ( )

∑ ⁺ ∑

=

−

−

=

− 





 



⋅ + +

= ¹

1

1 1

1

, 1

) 1 ( ˆ 1

1

m

n vv

k n

v

v n v k k

a r

O a t v n ^δ

( )

∑ ⁺ ∑

=

−

=

− + ⋅

≤

1

1

, 1

1

1 1 ˆ 1

m

n vv

v n v n

v k

a a t v n ^δ

( )

∑ ∑

=

+

+

=

⋅ −

+

=

m

v

m

v n

v n k

vv

v n a

a t v

1

1

1

, 1 ˆ

1 ^δ

( )

∑ ₋ ⋅ ∑ ⁺

⋅ + +

≤

m

v

m

v n

v k n

vv

v a

v a

t v

1

1

1

1 ˆ ,

) 1 (

1 1 _δ

∑

=

⋅ +

=

m

v

k vv

v v

a t

1

1 ) 1

( ^δ (using Lemma-3)

∞

→

= O ( 1 ) as m ) Next

m k

n

n k

k T

∑ ⁺ n

=

− + 1

1

3 , 1 δ

m k

n

n nn

k

k n a n t

∑ ⁺ n

=

−

+ +

=

1

1

1 ˆ ( 1 )

δ

( )

m k

n

n

k n k t

∑ ⁺ n

=

− +

≤

1

1

1 1

( )

m k

n

n k k

k t

n n

n

∑ ⁺ n

=

+

=

1

1

1 1 .

δ .

( )

m k

n

n

k k t

n n

∑ ⁺ n

=

+

≤

1

1

2 1

δ 1

( )

∑ ⁺

= + +

=

1

1 2

1 ) 1

1 (

m

n

k

n

n ^δ

= ∑ ⁺ ( ⁺ ) ^{= → ∞}

=

−

m as O n

m

n k ( 1 )

1

1 1

1 2 δ

∞

→

= O ( 1 ) as m , since 2 − k δ > 1 _. Further

∑ ⁺

=

− + 1

1

4 , 1 m

n

k n k

k T

n ^δ ∑ ⁺ ∑ ( ) ( )

=

−

=

−

+ + ∆

=

1

1

1

1

, 1 2

1 ˆ

m

n

n k

v

v n v v k

k v w a

n

δ

( ) ( ( ) ) ( ( ) )

n k

v

k k v n k v

v n v v m

n k

k v w a a

n ∑

∑ ⁻

= + −

=

−

+ + ∆ ∆

≤

1

1

1 , 1

, 2

1

1

1 1 ˆ ˆ

δ

( )

1 1

1 , ,

1

1

2 1

1

1 1 ˆ ˆ

− −

=

−

= +

=

−

+ ∑ ∑

∑ + ∆ ∆

=

n k

v

v n v v n v k v n

v m k

n k

k v w a a

n ^δ

( ) , , ¹

1

2 1

1

1 1 ˆ ˆ ⁻

= +

=

−

+ + ∆

≤ ∑ ∑ v n v n n ^k

k v n

v m k

n k

k v w a a

n ^δ

( )

∑ ⁺ ∑

=

+

=

∆ +

=

1

1

1

1

,

2 ˆ

1

m

n

m

v

v n v k v k k

a w

v

n ^δ ∑ ( ) ∑

=

+

+

=

∆ +

=

m

v

m

v n

v n v k k

v

k w n a

v

1

1

1

,

2 ˆ

1 ^δ

( )

∑

=

+

=

m

v

vv k v

k w a

v

1

1 2

( )

( )

∑

=

+ +

+

=

m

v k

k

v v

1 2 2

2

1 1 1

( )

∑

=

∞

→

= +

=

m

v

m as O

1 v 2 ( 1 )

1 1

Finally

∑ ⁺

=

− + 1

1

5 , m 1

n

k n k

k T

n ^δ

( )

∑ ⁺ ∑

=

−

=

−

+ +

=

1

1

1

1 ,

1 ˆ 1

m

n

n k

v

v v

n k

k a v w

n ^δ

(proceeding in the lines of

∑ ⁺

=

− + 1

1

2 , m 1

n

k n k

k T

n ^δ ₎

and

∑ ⁺

=

− + 1

1

6 , m 1

n

k n k

k T

n ^δ

m k

n

n n n k

k n a w

∑ ⁺ n

=

−

= + 1

1

,

1 ˆ

δ

( )

∑ ⁺

=

+

≤

1

1

2 1

1 . .

m n

k n k k

k w

n n

n n ^δ

(proceeding in the line of ∑ ⁺

=

− + 1

1

3 , m 1

n

k n k

k T

n ^δ )

( )

( )

∑ ⁺

=

 





 



 + +

= ¹

1 2

1 1 1

m n

k

n O

n ^δ

( )

. 1 2 sin

, )

1 ( 1

1 1

1 2

>

−

∞

→

= +

≤ ∑ ⁺

=

−

k ce

m as O n

m

n k

δ

δ .

This completes the proof of Theorem.

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Reihe − + L

+

+ ²

2 ) 1

1 ( x

m m

mx _,

Journal fur reine and angewants mathematik (crelle) I, 311-339 (1826).

2. Hardy, G.H. Divergent Series, Clarendow Press, Oxford, (1949).

3. Misra, M., Misra, M., Rauto, K.

Absolute Banach Summabilty of Fourier Series, International Journal of Mathematical Sciences, Vol. 1, No.1,39 – 45 June (2006).

4. Paikaray, S.K. Ph.D. thesis submitted to Berhampur University, (2010)

5. Petersen, M. Regular matrix transforma-

tions, McGraw-Hill Publishing Company

Limited (1996).