Certain results on entire functions defined by bicomplex Dirichlet series

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Sep 20, 2018 - Throughout this paper, the set of Bicomplex numbers is denoted by 2. C and the sets of complex and real numbers are denoted by 1. C and 0.
Int. Res. Adv. 2018, 5(2), 46-51

. Article .

INTEGRATED

RESEARCH ADVANCES Certain results on entire functions defined by bicomplex Dirichlet series Jogendra Kumar1* Government Degree College Raza Nagar Swar, Rampur, Uttar Pradesh –244924, INDIA Received on: 10-JUNE-2018, Accepted and Published on:20-SEPT-2018

ABSTRACT 

In this work, we have introduced and studied the Bicomplex version of Complex Dirichlet Series

f (s )   a n e  n s

. We

n 1 

have derived condition for which the sum function of the Bicomplex Dirichlet Series f () 

  n en 

represents an

n 1

entire function. The Entireness of sum and Hadamard product of two Entire Bicomplex Dirichlet Series are also discussed..

Keywords: Dirichlet Series, Entire Dirichlet Series, Riemann Zeta Function, Hadamard Product

idempotent elements in C2 , denoted as

INTRODUCTION

Throughout this paper, the set of Bicomplex numbers is denoted by

C2 and

the sets of complex and real numbers are

C1

C

denoted by and 0 , respectively. For details of the theory of bicomplex numbers.1-3 The set of Bicomplex Numbers defined as:

C 2 = {x1 + i1 x 2 + i 2 x 3 + i1i 2 x 4 : x1 , x 2 , x 3 , x 4 ∈ C0 , i1



i 2 and i12 = i 22 = -1, i1 i 2 = i 2 i1}

We shall use the notations following sets:

C ( i1 )

and

C ( i2 )

for the

C ( i1 )  {u  i1v : u, v  C 0 } C ( i 2 )  {  i 2 : ,   C0 } 1.1 IDEMPOTENT ELEMENTS:

Besides 0 and 1, there are exactly two non – trivial

and

e2

and

1  i1i 2 1  i1i 2 and e 2  2 2 e1  e2  1 and e1e2  e2e1  0.

defined as e1 Note that

e1



1.2 CARTESIAN IDEMPOTENT SET:

Cartesian idempotent set X determined by denoted as X=

X 1 e X 2

X1 and X 2

is

and is defined as

{

X 1 × e X 2 =  ∈ C 2 :  = 1  e 1 + 2  e 2 , ( 1  , 2  )∈ X 1 × X 2

C 2 = C(i1 ) ×e C(i1 ) = C(i1 ) e1 + C(i1 )e 2 = { ∈ C 2 :  = 1 e1 + 2  e 2 , (1  , 2  ) ∈ C(i1 ) × C(i1 )} C 2 = C(i 2 ) × e C(i 2 ) = C(i 2 ) e1 + C(i 2 )e 2 = { ∈ C 2 :  = 1 e1 +  2 e 2 , (1 ,  2 ) ∈ C(i 2 ) × C(i 2 )} 1.3 IDEMPOTENT REPRESENTATION OF BICOMPLEX NUMBERS

Dr. Jogendra Kumar, Assistant Professor, Government Degree College Raza Nagar Swar, Rampur, Uttar Pradesh –244924, INDIA Email: [email protected] Cite as: Int. Res. Adv., 2018, 5(2), 46-51.

(I) C(i1 ) -idempotent representation of Bicomplex Number is given by

 = z 1 + i 2 z 2 = ( z 1 - i 1 z 2 ) e1 + ( z 1 + i 1 z 2 ) e 2 = 1  e1 + 2  e 2

©IS Publications ISSN 2456-334X http://pubs.iscience.in/ira Integrated Research Advances

}

Int. Res. Adv., 2018, 5(2), 46-51

46

Kumar (II) C( i 2 ) -idempotent Number is given by

representation

of

Bicomplex



 = ( x 1 + i 2 x 3 ) + i1 (x 2 + i 2 x 4 ) = w 1 + i1 w 2

the Dirichlet series

= ( w 1 - i 2 w 2 )e1 + ( w 1 + i 2 w 2 )e 2 = 1 e1 +  2 e 2

convergent for

Note 1.1: Out of the two idempotent representation, we use

12



  z1  z 2 =

x

2 1

2

 1 2  2  2  12  =   2  





σ 0  Re (s)   of conditional

divergence, the region convergence and the line

Re (s)  σ 0 ).

 is called the abscissa of absolute

The quantity



f (s)   a n e  n s

and the line

n 1

Re( s)   is called the line of absolute convergence.

1/ 2

1.5 ENTIRENESS OF COMPLEX DIRICHLET SERIES:

C 2 becomes a modified Banach algebra, in the sense that

.  2 

is absolutely

Re(s)   , and not absolutely convergent for Re(s)   (this region comprise the region Re (s)  σ 0 of

convergence of the series



 x 22  x 32  x 24

f (s)   a n e  n s n 1

C(i1 ) -idempotent representation. All the results also proved with the help of C(i 2 ) -idempotent representation technique. The norm in C 2 is defined as 2

 such that

To every Dirichlet series, there exists a number

THEOREM 1.1:4 

… (1.1) For the complex Dirichlet Series

 a n e 

ns

n 1

1.4 COMPLEX DIRICHLET SERIES:4-6

A

Dirichlet

series

If

is

a

series

of

the

form



f (s)   a n e  ns

where

{λ n } is a strictly monotonically

n 1

increasing and unbounded sequence of positive real numbers, and s    i t is a complex variable.

{λ n } of exponent is to be emphasized, such a series is called a complex Dirichlet series of type λ n .4 When the sequence

A Dirichlet series of the type n is a power series in 

given by

f (s)   a n e

ns

____ log a n n   , Then σ 0  σ  lim lim n λn

____

n 1



Corollary 1.1: For a Dirichlet Series f (s) 

σ 0  σ  lim

A Dirichlet series of type

log n is the Generalized 

Riemann Zeta function is given by

f (s)   a n n

 s



that the Dirichlet series

f (s)   a n e

ns

Hence,

σ 0  σ  lim

Corollary

1.2:

log a n n

The

f (s)   a n e  n s

Complex

Dirichlet

Series

represents an Entire function iff

n 1 1

an

n

 0 as n   .

Proof : 

converges for

For entireness of

n 1

σ 0 is called the abscissa of convergence of the series, and the Re(s)  σ 0 is called the line of convergence.

f (s)   a n e  n s n 1

lim

log a n n

 lim log a n  an

Integrated Research Advances

n n  lim  1   n n



Re (s)  σ 0 and diverges for Re (s)  σ 0 . The number line

n

 n  n  lim

n 1

Abscissae of convergence and absolute convergence: To every Dirichlet series, there exists a number σ 0 such

log a n

Proof:



n 1

 a n e n s n 1

e  s is

  a n (e  s ) n

.

1

n

    lim 1

n

log a n n

    lim a n

  1

n

0

 0 as n  

Int. Res. Adv., 2018, 5(2), 46-51

47

Kumar 

f (s)   a n e  n s represents an Entire function if

Hence

n 1

an

1

n

1

 n  2  n  (1n ) 2  ( 2 n ) 2  ( 3n ) 2  ( 4 n ) 2

 1n  4 n   2 n  3n Also

 0 as n   .

, 2

2

2

 n  (1n )  ( 2 n )  ( 3n )  ( 4 n ) 2. BICOMPLEX DIRICHLET SERIES:

In this paper we discuss a Bicomplex Dirichlet Series of type n, which is a Bicomplex Power Series in 

f ( )    n e

n 1

where { α n } is a sequence of bicomplex numbers and a bicomplex variable. Note that,





1

ξ is









 n 1

2

  n e  n    1 n e  n   e1   2  n e  n  n 1

n 1



 e2 

2

 n iff 1n  4 n   2 n  3n

1  1  lim

As,  n

   n (e )

n 1



 n  1 n 

e 



n

Hence,

 2   2  lim Hence,

2

log 2  n

log 1 n

and

n [cf. Cor. 1.1]

n

1  1   2   2  lim

log  n n

THEOREM 2.2: 

1

1

2

The Bicomplex Dirichlet series

2

 f ()  f ( ) e1  f ( ) e 2 

f ( )    n e

Where,

n

is a Bicomplex Dirichlet 

f ( )   1ne n

1



,

2

f ( 2 )   2  n e  n

2



are Complex Dirichlet Series.

Throughout, We denote the abscissae of convergence of the 





1

n en 

1

and

n1

by

σ1

and

2

 2  n en  n 1

σ 2 and abscissae of their absolute convergence by

 2   2  lim

n 1 , 2

 (  n) k α n e  n ξ . n 1

T HEOREM 2.1:

The,



For the Bicomplex dirichlet Series

f ( )    n e  n  n 1

1  1   2   2  lim

k

log  n 

n

 lim

1n  4 n   2 n  3n

 lim

 n  1n  i1 2 n  i 2  3n  i1i 2  4 n  n  1 n e1  2  n e 2 1

 n  (1n   4 n )  i1 ( 2 n   3n )

and

log  n  1 n

1  1  lim

log  n

Proof:

2

and

n 2 log  n



Dirichlet series

Where,

log 1  n

Let 1 ,  2 and are the associated abscissae of convergence and absolute convergence of the Bicomplex

1 and 2 , respectively.

if ,

have the same

region of convergence.

1  1  lim

n 1

Complex Dirichlet series

(  n)k αn e n ξ

derivative defined by

Proof:

n1 



n1

1 1

and

and the kth

n 1

n 1

Series

αn enξ

k

n  log  n 1

n k log n  log 1  n n

log 1  n log n  k lim  lim n n

 n  (1n   4 n )  i1 ( 2 n   3n )

Integrated Research Advances

Int. Res. Adv., 2018, 5(2), 46-51

48

Kumar

 1

log 1  n log 1  n log n  k lim  lim  0  lim n n n

 lim

Similarly, 2

log

log 1  n

 2  2  lim

 2  2  lim

log  n k 2  n

 lim

n

log 2  n

 αn e

nξ

and the

α

 (  n)n k

e  n ξ obtained after

n 1



k–times term-by-term integration of

 αn e

nξ

have the

n 1

1 n

n

n 1 

0

n 

 2   2  lim



e n ξ .

log 1  1  lim 1

 lim

 lim

n

0

n  ,

as

where

1 n 

n

 m N , such that

2

2

 k lim

 n k

n

Integrated Research Advances

 1n  2

n

2

 2

  n  

2

2n

1 2

n

2

2

 2 2 n  1  n  2 2 n and

1

 n

1 n

n

log n n

log 1  n log n  k lim  lim 0 n n

log 1  n

2

 1  2  n n   2 

n

Conversely 2

n

n  n 

 n m ,

Now

1

n log 1  n

0

as

 0 as n  

n

α

n

0

 n m

n 1

log  n

1 n

n

Given

and

Let 1 ,  2 and 1 , 2 are the associated abscissae of convergence and absolute convergence of the Bicomplex

 (  n)n k

1 n

n

1 n

Proof:

Proof:

n 2 log  n

2

1

if and only if

 n  1 n e1  2  n e 2 .

Let

1  1  lim

n 

as

and

same region of convergence.

log 1  n

is

THEOREM 2.4:



Bicomplex Dirichlet series

n

C 2 –space.

convergent in the entire

The Bicomplex Dirichlet series

f      n e 

said to be an entire Bicomplex Dirichlet Series if it is

T HEOREM 2.3:

 lim

 2 .

n

DEFINITION 2.1:

The Bicomplex Dirichlet series

n

 2 .

Dirichlet series

 lim

n

log 2  n

ENTIRE BICOMPLEX DIRICHLET SERIES

Similarly,

Then,

 n k

n

 1

n

1 n

0

i.e. Given Such that

1

let as

n

0 1 n

2

and

0

n

1 n

0

n 

as

and

n 

  0  m1 , m 2  N 1

n

1 n



 n  m1

and

2

n

1 n

 n  m2

Int. Res. Adv., 2018, 5(2), 46-51

49



Kumar Proof: 

m  max (m1 , m 2 )

Let Then

1

 n  m , n

 1 n n    

2

1 n



and

n

1 n

 n



i.e. given

 0 as n  

f      n e 

The Bicomplex Dirichlet series

n

 

f    n e

 n 1

and

2

f

    2

2

n e

 n 2

2.1:

The

Bicomplex

Dirichlet

1  

and

1 n

n n

2.2:

series

f      n e  n  if and only if

1

n

1 n

Dirichlet

series

is an entire Bicomplex Dirichlet series

0

and

n

2.3:

f      n e  n iff

n

1 n

The 

Bicomplex

 

n

1 n

 and  n

1 n

n

1 n



n





1 2 (2) n

() ()

1

1 n

 (2) 2 n  2   n  n

1 n

 0 as

1 n

is an entire Bicomplex

Dirichlet Series. THEOREM 2.7:

 0. Dirichlet

h ()   α n  n  e  n ξ n 1



If Corollary





Bicomplex

2

1 n

 2

 n n n 

2   .

The

1 n

1

Hence Corollary

n

and

 ( 2) 2 n  2

f      n e  n  is an entire Bicomplex Dirichlet series if and only if

1 n

 n n  2  n

Now,

are entire Complex Dirichlet series. Corollary

 n  m1



 n  m , n

is an

entire Bicomplex Dirichlet series if and only if both 1

1 n

n

 n  m2 Let m  max ( m1 , m 2 )

T HEOREM 2.5:

1

 0 as

  0  m1 , m 2  N

Such that

Now

1

1 n

 0 as n   and  n

2

 2n  2n  2 2n 1 n

are two

n

2

 n

and

n 1

1 n

 n

2 n 2 1n  2n 

2 n

g ( )    n e  n ξ

Entire Bicomplex Dirichlet Series

   

2





n 1

1 2 2

2

2

2

f ( )   α n e 

series

is an entire Bicomplex Dirichlet series

f ()   αn e n ξ



and

g ( )    n e  n ξ

n 1

entire

be two

n 1

Bicomplex

Dirichlet

series,

then

the

series



 α n  n e  n ξ

 0.

is also an entire Bicomplex Dirichlet

n 1

series. T HEOREM 2.6: THEOREM 2.8:



Let

h ()   α n  n e

 nξ

If

n 1 

of

f ()  αn e n ξ



and

g()   n e n ξ . If f and g are

n1

n1

entire Bicomplex Dirichlet series, then h is also an entire Bicomplex Dirichlet series.

Integrated Research Advances



be the Hadamard product

 α n e n ξ

is an Entire Bicomplex Dirichlet Series,

n 1 

then kth derivative defined by

 (  n)k α n e  n ξ

is also

n 1

an Entire Bicomplex Dirichlet Series.

Int. Res. Adv., 2018, 5(2), 46-51

50

Kumar Proof: 

f ( )   α n e  n ξ

is an Entire Bicomplex Dirichlet

Now,

n 1

1 n

αn



(  n) k

Series i.e.

1 n

n

1 n

(  n) k α n

Now, k

(  n) α n n 

αn (  n) k

 0 as n  

1 n

0

as



k (n) n

1 αn n



k  (n) n

n 

0

as



If the Bicomplex Dirichlet series

 α n e  n ξ is an entire n 1

Bicomplex Dirichlet series then the Bicomplex Dirichlet series

 n 1

αn e  n ξ obtained k (  n)

after

k–times

term-by-term



integration of

 α n e n ξ

is also an entire Bicomplex

n 1

Dirichlet series. Proof: 

Let

f ()   α n e  n ξ

series i.e.

n

n 1

1 n



1 k n (  n)

αn

1 n



k (n) n

k

0

as

n    (n ) n  0

αn e nξ k (  n)

as n



is an entire Bicomplex Dirichlet

series. ACKNOWLEDGMENTS

T HEOREM 2.9:







1 n

αn

n 1 1 n 

is an entire Bicomplex Dirichlet

0 as n  

Integrated Research Advances

I am heartily thankful to Mr. Sukhdev Singh, Assistant Professor-Mathematics, Lovely Professional University, Punjab and Dr. Mamta Nigam, Assistant ProfessorMathematics,University of Delhi for their encouragement and support during the prepration of this paper. REFERENCES AND NOTES 1. M.E. Luna-Elizarrarás, M. Shapiro, D. C. Struppa, A. Vajiac, Bicomplex Holomorphic Functions: The Algebra, Geometry and Analysis of Bicomplex Numbers, Springer International Publishing, (2015) 2. G. B. Price, An int. to multicomplex space and Functions, Marcel Dekker (1991). 3. Rajiv K. Srivastava,Certain Topological aspects of Bicomplex Space, Bull. Pure and Appl. Math., 2(2),(2008), 222–234. 4. G. H. Hardy and M. Riesz, The General Theory of Dirichlet Series, Cambridge Univ. Press (1915). 5. E. C. Titchmarsh, The Theory of functions, Oxford University press (1960). 6. S. Mandelbrojt, Dirichlet series: Principles and Methods, Dordrecht Holland (1969).

.

Int. Res. Adv., 2018, 5(2), 46-51

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