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Turkish Journal of Analysis and Number Theory, 2015, Vol. 3, No. 4, 94-96

Available online at http://pubs.sciepub.com/tjant/3/4/1

© Science and Education Publishing

DOI:10.12691/tjant-3-4-1

Some Curvature Properties on a Special Paracontact

Kenmotsu Manifold with Respect to Semi-Symmetric

Connection

K. L. Sai Prasad1,*

, T. Satyanarayana2

1Department of Mathematics, Gayatri Vidya Parishad College of Engineering for Women, Visakhapatnam, Andhra Pradesh, India 2Department of Mathematics, Pragathi Engineering College, Surampalem, Near Peddapuram, Andhra Pradesh, India

*Corresponding author: [email protected]

Received April 28, 2015; Accepted July 03, 2015

Abstract The object of the present paper is to study some properties of curvature tensor R of a semi-symmetric

non-metric connection in a type of special paracontact Kenmotsu (briefly SP-Kenmotsu) manifold. We have

deduced the expressions for curvature tensor R and the Ricci tensor S of Mn with respect to semi-symmetric non-

metric connection . It is proved that in an SP-Kenmotsu manifold if the curvature tensor of the semi-symmetric

non-metric connection vanishes then the manifold is projectively flat.

Keywords: curvature tensor, ricci tensor, projective curvature tensor, non-metric connection, sp-kenmotsu

manifold

Cite This Article: K. L. Sai Prasad, and T. Satyanarayana, “Some Curvature Properties on a Special

Paracontact Kenmotsu Manifold with Respect to Semi-Symmetric Connection.” Turkish Journal of Analysis and

Number Theory, vol. 3, no. 3 (2015): 94-96. doi: 10.12691/tjant-3-4-1.

1. Introduction

Friedmann and Schouten [1,2] introduced the idea of

semi-symmetric linear connection on a differentiable

manifold. Hayden [3] introduced semi-symmetric metric

connection on a Riemannian manifold and it was further

developed by Yano [4]. Semi-symmetric connections play

an important role in the study of Riemannian manifolds.

There are various physical problems involving the semi-

symmetric metric connection. For example, if a man is

moving on the surface of the earth always facing one

definite point, say Jaruselam or Mekka or the North pole,

then this displacement is semi-symmetric and metric [1].

In 1975, Prvanovi c [5] introduced the concept of semi-

symmetric non-metric connection with the name pseudo-

metric, which was further studied by Andonie [6,7]. The

study of semi-symmetric non-metric connection is much

older than the nomenclature ‟non-metric‟ was introduced.

In 1992, Agashe and Chafle [8] introduced a semi-

symmetric connection satisfying 0X g on a

Riemannian manifold, and called such a connection as

semi-symmetric non-metric connection. Later, the

curvature properties of the connection in an SP-Sasakian

manifold were studied by Bhagwat Prasad [9], and many

others.

On the other hand, in 1976, Sato [10] defined the

notions of an almost paracontact Riemannian manifold.

After that, T. Adati and K. Matsumoto [11] defined and

studied para-Sasakian and SP-Sasakian manifolds which

are regarded as a special kind of an almost contact

Riemannian manifolds. Before Sato, in 1972, Kenmotsu

[12] defined a class of almost contact Riemannian

manifolds satisfying some special conditions. In 1995,

Sinha and Sai Prasad [13] have defined a class of almost

paracontact metric manifolds namely para Kenmotsu

(briefly P-Kenmotsu) and special para Kenmotsu (briefly

SP-Kenmotsu) manifolds.

In 1970, Pokhariyal and Mishra [14] have introduced

new tensor fields, called W and E-tensor fields in a

Riemannian manifold and studied their properties. In the

present paper, we consider the W-curvature tensor of a

semi-symmetric non-metric connection and obtained a

relation connecting the curvature tensors of Mn with

respect to semi-symmetric non-metric connection and the

Riemannian connection. It is proved that in an SP-

Kenmotsu manifold if the curvature tensor of the semi-

symmetric non-metric connection vanishes then the

manifold is protectively flat.

Let Mn be an n-dimensional differentiable manifold

equipped with structure tensors ( , ξ, η) where is a

tensor of type (1, 1), ξ is a vector field, η is a 1-form such

that

( ) = 1 (1.1)

2( ) = ( ) ; =X X X X X (1.2)

Then Mn is called an almost paracontact manifold.

Let g be the Riemannian metric in an n-dimensional

almost paracontact manifold Mn such that

( , ) = ( )g X X (1.3)

95 Turkish Journal of Analysis and Number Theory

= 0, ( ) = 0, rank = 1X n (1.4)

( , ) = ( , ) ( ) ( )g X Y g X Y X Y (1.5)

for all vector fields X and Y on Mn. Then the manifold Mn

[10] is said to admit an almost paracontact Riemannian

structure ( , ξ, η, g) and the manifold is called an almost

paracontact Riemannian manifold.

A manifold Mn with Riemannian metric „g‟ admitting a

tensor field of type (1, 1), a vector field ξ and 1-form η

satisfying equations (1.1), (1.3) along with

( ) ( ) = 0X YY X (1.6)

( ) = [ ( , ) ( ) ( )] ( )

[ ( , ) ( ) ( )] ( )

X Y Z g X Z X Z Y

g X Y X Y Z

(1.7)

2= = ( )X X X X (1.8)

is called a para Kenmotsu manifold or briefly P-Kenmotsu

manifold [13], where 𝛻 is the covariant differentiation

with respect to g.

It is known that [13] on a P-Kenmotsu manifold the

following relations hold:

( , ) = ( 1) ( )Ric X n X (1.9)

[ ( , ) , ] = [ ( , , )]

= ( , ) ( ) ( , ) ( )

g R X Y Z R X Y Z

g X Z Y g Y Z X

(1.10)

where R is the Riemannian curvature.

Let (Mn, g) be an n-dimensional Riemannian manifold

admitting a tensor field of type (1, 1), a vector field ξ

and 1-form η satisfying

( ) = ( , ) ( ) ( )X Y g X Y X Y (1.11)

( , ) = ( ) and ( ) = ( , ),

where is an associate of

Xg X X Y X Y

(1.12)

is called a special para Kenmotsu manifold or briefly SP-

Kenmotsu manifold [13].

A linear connection in a Riemannian manifold Mn is

said to be semi-symmetric connection if its torsion tensor

T satisfies

( , ) = ( ) ( ) .T X Y Y X X Y (1.13)

A semi-symmetric non-metric connection in an

almost paracontact metric manifold with torsion tensor

(1.13) is given by

= ( )X XY Y Y X (1.14)

where 𝛻 is a Riemannian connection with respect to

metric g [8].

Apart from conformal curvature tensor, the projective

curvature tensor is an other important tensor from the

differential geometric point of view. The Weyl-projective

curvature tensor W of type (1, 3) of a Riemannian

manifold Mn with respect to the Riemannian connection is

defined by [14]

( , )1

( , ) ( , ) ( , )1

Ric Y Z XW X Y Z R X Y Z

Ric X Z Yn

(1.15)

for X, Y, Z ,T M where R is the curvature tensor and

Ric is the Ricci tensor. If there exists a one-to-one

correspondence between each coordinate neighbourhood

of a Riemannian manifold Mn and a domain in Eucledian

space such that any geodesic of the Riemannian manifold

corresponds to a straight line in the Eucledian space, then

Mn is said to be locally projectively flat. For 3,n Mn is

locally projectively flat if and only if the projective

curvature tensor W vanishes. For n = 2, the projective

curvature tensor identically vanishes.

2. Curvature Tensor

The manifold Mn is considered to be an SP-Kenmotsu

manifold. Let us denote the curvature tensor of the semi-

symmetric non-metric connection by R and the

curvatre tensor of 𝛻 by R. By straight forward calculation,

we get

( , , )

= ( , , ) ( )( ) ( )( ) .X Y

R X Y Z

R X Y Z Z Y Z X (2.1)

As a consequence of equations (1.11) and (1.14),

equation (2.1) reduces to

( , , ) = ( , , ) ( , ) ( , )R X Y Z R X Y Z g X Z Y g Y Z X (2.2)

which is the relation between the curvature tensors of Mn

with respect to the semi-symmetric non-metric connection

and the Riemannian connection .

It is well known that a Riemannian manifold is of

constant curvature if and only if it is projectively flat or

conformally flat [15] and in general, the necessary and

sufficient condition for a manifold with a symmetric linear

connection to be projectively flat is that the projective

curvature tensor with respect to it vanishes identically on a

manifold [16].

As an example, if Mn is a Riemannian manifold with

vanishing curvature tensor with respect to semi-symmetric

non-metric connection, then Mn is projectively flat [8].

Analogus to this, we prove the following for an

SP-Kenmotsu manifold which is Riemannian.

Theorem 2.1: If in an SP-Kenmotsu manifold Mn the

curvature tensor of a semi-symmetric non-metric

connection vanishes, then the manifold is projectively

flat.

Proof: Since R = 0, then equation (2.2) gives

( , , ) = ( , ) ( , ) .R X Y Z g Y Z X g X Z Y (2.3)

On contracting the above equation, we get

( , ) = ( 1) ( , ).Ric Y Z n g Y Z (2.4)

Then, by (2.3) and (2.4), we get

1

( , , ) [ ( , ) ( , ) ] = 01

R X Y Z Ric Y Z X Ric X Z Yn

(2.5)

or W = 0 from (1.15), proves that the manifold is

projectively flat.

Theorem 2.2: If in an SP-Kenmotsu manifold the Ric

tensor of a semi-symmetric non-metric connection

vanishes, then the curvature tensor of is equal to the

projective curvature tensor of the manifold Mn.

Proof: From equation (2.2), we have

Turkish Journal of Analysis and Number Theory 96

( , , , )

= ( , , , ) ( , ) ( , ) ( , ) ( , ).

R X Y Z U

R X Y Z U g X Z g Y U g Y Z g X U (2.6)

On contracting the above equation, we get

' ( , ) = ( , ) ( 1) ( , ).Ric Y Z Ric Y Z n g Y Z (2.7)

Since 'Ric = 0, we have

1

( , ) = [ ( , )].1

g Y Z Ric Y Zn

(2.8)

From equations (2.2) and (2.8), we have R = W.

Theorem 2.3: In an SP-Kenmotsu manifold the projective

curvature tensor of a semi-symmetric non-metric

connection is equal to the projective curvature tensor

of the manifold.

Proof: From equations (2.2) and (2.7), we get

' ( , )1( , , ) = ( , , )

( , )1

1 [' ( , ) ( , )] .

1

Ric Y ZR X Y Z R X Y Z X

Ric Y Zn

Ric X Z Ric X Z Yn

(2.9)

The terms of the equation (2.9) can be rearranged as

1( , , ) [ ' ( , ) ' ( , ) ]

1

1= ( , , ) [ ( , ) ( , ) ]

1



(2.10)

which is „W = W, where „W is the Weyl projective

curvature tensor with respect to the semi-symmetric non-

metric connection.

Theorem 2.4: In an SP-Kenmotsu manifold with semi-

symmetric non-metric connection we have

a) ( , , ) ( , , ) ( , , ) = 0R X Y Z R Y Z X R Z X Y

b) ' ( , , , ) ' ( , , , ) = 0R X Y Z U R X Y U Z

Proof: Using the Bianchi‟s first identity with respect to

the Riemannian connection equation (2.2) gives (a). From

equation (2.6) we get (b).

Acknowledgement

The authors acknowledge Prof. Kalpana, Banaras

Hindu University, Dr. B. Satyanarayana of Nagarjuna

University and Dr. A. Kameswara Rao, G.V.P. College of

Engineering for Women for their valuable suggestions in

preparation of the manuscript. They are also thankful to

the referee for his valuable comments in the improvement

of this paper.

Competing Interest

The authors declare that there is no conflict of interests

regarding the publication of this paper.

References

[1] Friedmann and Schouten, J. A., Uber die Geometrie der

halbsymmetrischen Ubertragungen, Math Zeitschrift, 21, 211-223,

1924.

[2] Schouten, J. A., Ricci-calculus, Springer-Verlag. Berlin, 1954.

[3] Hayden, H. A., Subspaces of a space with torsion, Proc. London

Math. Soc., 34, 27-50, 1932.

[4] Yano, K., On semi-symmetric metric connection, Revue

Roumanine de Mathematiques Pures et Appliques, 15, 1579-1581,

1970.

[5] Prvanovi c, M., On pseudo metric semi-symmetric connections,

Pub. De L’Institut Math., N.S., 18(32), 157-164, 1975.

[6] Andonie, P.O.C., On semi-symmetric non-metric connection on a

Riemannian manifold. Ann. Fac. Sci. De Kinshasa, Zaire sect.

Math. Phys., 2, 1976.

[7] Andonie, P.O.C., Smaranda, D., Certains connections semi-

symmetriques, Tensor (N.S.), 31, 8-12, 1977.

[8] Agashe, N. S. and Chafle, M. R., A semi-symmetric non-metric

connection on a Riemannian manifold, Ind. J. of Pure and Appl.

Math., 23(6), 399-409, 1992.

[9] Prasad, On a semi-symmetric non-metric connection in an sp-

Sasakian manifold, Istambul Univ.Fen Fak. Mat. Der., 53, 77-80,

1994.

[10] Sato, I., On a structure similar to the almost contact structure,

Tensor (N.S.), 30, 219-224, 1976.

[11] Adati, T. and Matsumoto, K., On conformally recurrent and

conformally symmetric P-Sasakian manifolds, TRU Math., 13, 25-

32, 1977.

[12] Kenmotsu, K., A class of almost contact Riemannian manifolds,

Tohoku Math. Journal, 24, 93-103, 1972.

[13] Sinha, B. B. and Sai Prasad, K. L., A class of almost para contact

metric Manifold, Bulletin of the Calcutta Mathematical Society,

87, 307-312, 1995.

[14] Pokhariyal, G.P., Mishra, R.S., The curvature tensors and their

relativistic significance, Yokohoma Math.J., 18, 105-108, 1970.

[15] Yano, K. and Bochner, S., Curvature and Betti numbers, Annals of

Math Studies 32, Princeton University Press, 1953.

[16] Sinha, B. B., An introduction to modern Differential Geometry,

Kalyani Publishers, New Delhi, 1982.




DOI:10.12691/tjant-3-4-2

Study of Some Sequences of Prime Numbers Defined by Iteration

Idir Sadani*

Department of Mathematics, University of Mouloud Mammeri, Tizi-Ouzou. Algeria


Received May 11, 2015; Accepted July 22, 2015

Abstract We study the properties of prime number sequences obtained using a well-defined equivalence relation

. It will be seen that the elements of each class of are all prime numbers which constitute the fundamental

object of our study. The number of prime numbers of each class less than or equal to a given quantity , the number

of the different equivalence classes and some other results will be deduced.

Keywords: distribution of prime numbers, equivalence relation, iteration, asymptotic formula

Cite This Article: Idir Sadani, “Study of Some Sequences of Prime Numbers Defined by Iteration.” Turkish

Journal of Analysis and Number Theory, vol. 3, no. 4 (2015): 97-103. doi: 10.12691/tjant-3-4-2.

1. Introduction

Let be the set of prime numbers, and for all ,

let denote the number of prime numbers less than or

equal to . The prime number theorem which was shown

independently by de la Vallée Poussin [1], and Hadamard

[2] in , states that:

~ ,asln

xx x

x (1)

or

( ) ( ), ,x Li x as x

where is the logarithmic integral of defined by:

1

0 10lim ‍ .

ln ln

xdy dyLi x

y y

We can give an equivalent statement for this theorem as,

for example, let denote the n'th prime number. Then

1 1 ~ ln .nn p Li n n n as n (2)

One of our objectives here is to use a restriction of the

function to to study intrinsic properties of some

sequences of primes defined by iterations. The point of

departure for this study is the construction of an

equivalence relation that we denote by . The purpose of

this equivalence relation is to show first, that there is a

recurrence relation between prime numbers which can be

arranged in an infinity of well-defined classes dependent

on the initial value. Second, the use of its properties with

the famous prime number theorem is one way to find and

prove other results for the possible applications in number

theory. Part of our motivation came from the prime

number theorem and the Riemann hypothesis. Also, it was

an attempt to establish the relationship between these

prime numbers classes and several as-yet unproved

conjectures, such as Goldbach's Conjecture and Twin

Prime Conjecture.

The structure of this paper is as follows. In section 2,

we begin with the definition of our equivalence relation

and its equivalence classes where is a prime number,

which constitute the fundamental object of study, and we

propose some preliminary results. In section 3, we exhibit

the main results of this work by introducing and studying

the functions and . The methods used

here are part of elementary number theory and we have

attempted to present the ideas in as elementary a way as

possible. Finally, in section 4, we give some open

questions related this subject.

Notation

1) We set

1,

p x

x

this function count the number of primes less than .

2) We define the following functions:

times

( ) ( ) ,

n

n x x

times

1 1 1( ) ( ),

n

n y y

where is the composition operator.

3) In many situations, we search to estimate ∑ ,

where . Then, we use the following formula:

2

( ) ( )( ) .

ln ln

x

p x n x

f n f tf p dt

n t

(3)

2. Preliminary results

2.1. The Classes and Its Elements

We start with the following lemma:


Lemma 2.1. Let be the restriction of to . Then,

is a bijection and its inverse function is

.

Notation. Throughout this paper, we simply use the

notation to designate the restriction of to the set

instead of using .

The proof of the following theorem is obvious.

Theorem 2.1. We define the relation on the set of prime

numbers defined by: if and are two prime numbers,

if and only if:

1) for any prime number ,

2) there exists such that .

Then, is an equivalence relation.

Notation. The elements of the equivalence class are

defined by:

2 1 2, , , , , , .p p p p p p

The smallest element of , which we denote by , is

called the origin of the class . Then, in this case, we note

that the sets and have the same elements.

Example 1.

2 2,3,5,11,31,127, ,7 7,17,59, .

Notation. We denote by the set of all origins . The

set is given explicitly by:

0 {2,7,13,19,23,29,37,43 },

and we denote by the set of all origins less than or

equal to .

Theorem 2.2. Let be a prime number. Then, is not

a prime number implies that, for all , there is no an

integer such that .

Proof. Let . By the prime number theorem,

represents the -th prime number which we denote by

.

Next, , implies that

which is the -th prime number, so we obtain the

formula:

1 1 1 21p p .p p

p p

(4)

Now, we use the equation (4) which composed by

gives

2 1 1 21 1( ) ( )

p ( ) (p ) ( ( )),p p

p p

that is the -th prime number. We can write

1 1 2 31 2( ) ( )

(p ) ( ( )) p ( )p p

p p

Inductively, we obtain the following general formula:

( 1) ( )p ( ), 1.n

n pp n

(5)

To each number , we associate the set :

1 2{ ( ), 0} { , ( ), ( ), }npA p n p p p .

Now, suppose that is not a prime number, we

must prove that for all , there is no such that

. To show this, suppose that there exists a

positive integer such that , i.e.,

and we show that is a prime

number. Hence, we have, if and only if

, which implies that

But, for all is prime,

which proves what we wanted.

Remark. The cardinality of the set is infinite.

Theorem 2.3. There exists a partition of the set of

prime numbers defined by

1 2

{ , , , , },p p pnP A A A

such that

, where

and is not prime number.

Proof. On the one hand, according to the prime number

theorem, the number of prime numbers belonging to

is equal to . On the other hand, by

the Theorem 2.2, there are prime

numbers which do not belong to

. We denote these numbers

by , where . So, we

obtain of sets which are defined by:

1{ , ( ), , ( )},lpk

p k k kkA p p p

such that ∑ and

. Finally, it is not difficult to see that the sets

constitute a partition of the set of prime

numbers less than or equal to . So, since is arbitrary in

, letting tend to we obtain an infinity of sets

which then form a partition of the set .

Theorem 2.4. Let be a finite set of prime numbers

defined by

1 2{ , , , },nA p p p

such that are consecutive. Then,

there exists at least one set where , such that

. Before giving the proof of this result, we give an

illustrative example.

Example 2. Let be a set which defined by

{5,7,11,13,17}.A

• The class of the integer is and its

cardinality is greater than .

• We notice that , therefore the integer constitutes

the origin of a new class which is and its

cardinality is greater than . It only remains to see that the

prime number does not belong to the two classes and

. Thus the prime number constitutes the origin of a

new class and clearly its cardinality is .

Proof of theorem 2.4. We suppose that each class

containing at least two elements i.e., the

cardinality of is equal or greater than , and we suppose

that is prime. According to the theorem of prime

numbers, the interval contains prime numbers. This leads to the two

following cases:

• If , i.e., there exists only one prime number in

, namely . Therefore, we obtain

and is not a prime number since and are


consecutive. Then contains only one element in which

is the prime number .

• If , i.e., there exist at least two prime numbers in

, namely . We suppose that where

, is a prime number. Then is not prime

and since , then, the unique element of

is .

Finally, in both cases, there exist at least one class

where , such that is a unique element of this class

in , i.e., . The proof now is completed.

We have the following definition:

Definition 1. Let be the set defined as in the Theorem

2.4 and is a prime number belonging to . We say that

is an outside class of , if

1 ( ) . .,| | 1 .p is not prime in A and p A i e p in A

if in , we say that this class is an inside class of

.

Remark. According to the Definition 1, the number is

the smallest element of the class in , therefore, it is the

origin of this class i.e., .

2.2. Study of

The function

, , is a contraction.

Therefore, the sequence defined by:

1 ( ) ,ln

nn n

n

xx p x

x

admits a fixed point. Since decreasing and is bounded

below by , then it converges to the single

fixed point .

Notation. ⏞

, where is the

function composition operator.

Theorem 2.5. Let be an initial value. Then

10 0

0 0

( ) ln ( ) ln ( ),ii

i i

p x e p x e p x

(6)

Proof. We set and we have

01 0 1 0 2 0

0

12 1 2 1 2 1

1

( ) ln ln lnln

( ) ln ln lnln

xx p x x x x

x

xx p x x x x

x

1 1 2( ) ln ln lnln

nn n n n n

n

xx p x x x x

x .

And combining all these, we obtain

1 0 2

0

ln ln ( ) ln ln .n

n n i

i

x p x x x

(7)

Which is equivalent to

0

1 0

l .nn

in i

xx

x

(8)

Then, passing to the limit, we obtain

0 0

1 0

1

0

0

lim lim lim ln

lim ln ln ln ( ).

n

in n nn n i

n

in

i

x xx

x p x

x p x

(9)

Consequently,

00

0 0

ln ln ln ( ),i i

i i

xx x p x

e

(10)

and since

lim ,nn

x e

the formula (10) is obviously equivalent to

0

0

( ) ln ( ).i

i

p x e p x

Which is the desired result.

Lemma 2.2. Let be the sequence which is

decreasing and bounded below by , and let be a

real number. Then, the number of iterations, which denote

by , is depend on and the initial value , and

given by :

0 0

ln ln( , ) ~ ,as ,

lnln lnlniter

c xn x x

(11)

where is a bounded value, and , i.e.,

depend on .

Proof. We have

2 1 1 1 0 1 1 0'( ) , [ , ],x x p x x x x

where is the derivative of the function

'3 2 2 2 1 2 1 1 0'( ) '( ) ,x x p x x p p x x

where, Inductively, we obtain

1 1 0 1

1

'( ) . , [ , ].n

n n i n n n

i

x x p x x x x

Next, there exists a real number , such that

1 0 1 0'( ) .n

n nx x p x x

Or in an equivalent way, since, and

,

1 0 0 1'( ) . .n

n nx x p x x

We can extract the value of from the above equality:

1 0 1

0 0

ln( ) ln( ),

ln '( ) ln '( )

n nx x x xn

p p

and replacing by its value, we get

1 0 1

0 0 0 0

ln( ) ln( )

ln(ln 1) 2lnln ln(ln 1) 2lnln

n niter

x x x xn

Now, according to the definition of , the sequences

and are bounded, then it holds for the difference


which we denote it by . Then, we have, by

setting and :

0 0

1

0 0

ln,

ln ln 1 2lnln

ln( )

ln(ln 1) 2lnln

iterc

n x

x x

(12)

Finally, we have the following limits:

0 as ,x

0 0 0ln(ln 1) 2lnln lnln as ,x

1 as ,x x x x

then,

0 0

ln ln( , ) ~ , as .

lnln lnlniter

c xn x x

3. Principal Results

3.1. Definition and Estimate of the Functions

and

We set the following definition:

Definition 2. Let . We define

0

0 0

0 ( )0

( , ) 1 1, .iterp p x p p x

p p lp p

x p l

Then counts the number of prime numbers

less than or equal to belonging to the class .

Lemma 3.1. We have

0 0( , )~ ( , )~ ( , ), as .iter iter iterx p n x n x p x

Proof. According to the definition of , there exists

such that . Next, from the prime number

theorem, we have

( )~ ( ) , ( )~ ( ), .ln( )

n nxx p x x x p x x

x

Moreover, supposing that with not a

prime number, it follows that

0( ( ))~( ( ) ), .n np x p x x

And,

0

0 0

0 0( ) ~ ( )0

( , )

0

1

( , ) 1 1~ 1~ 1

1 ( , )~ ( , ),

iterp p x p p x p x p x

p p l p p lp p p p

n x

iter iter

l

x p

n x n x p

as Definition 3. Let . We define the functions

and as follows:

0

0

0

, ln ln .p p x

p p

x p p

0

, 0

lnln( , ) ln ln .

lnlnp x p p

xx p p

p

Theorem 3.1. We have

0( , )~ln (ln ).x p x o x

Proof. In view of the proof of Lemma 3.1, we have,

0 00 0

0 00 0

ln ln ( ) ~ ln ln ~ ,n n

i i

i i

x xx x x p x

p p

which is equivalent to

0

0 0

0

0

0

ln ln ln ln ( ) ln ln ln ln ( )

ln ln ~ ln .ln

n n

i i

i i

p p x

p p

x x x x

p x p

Finally, for all fixed and tend to infinity, we have

, then

0

0

ln ln ~ ln ln .p p x

p p

p x o x

i.e., Theorem 3.2. We have,

0

0

,1) , ~ ,

ln lniter

x px p x

x

0ln ln

2) , ~ ~ ,ln ln ln ln

iter iterx x

x p x o xx x

0

0

, ln3) , , .

ln ln ln lniter

x p xx p O x

x x

Proof.

1) For the proof of the first formula, we have, on the one

hand

0

0 0

0 0

0

, ln ln ln ln 1

, ln ln .

p p x p p x

p p p p

iter

x p p x

x p x

Then we obtain

0

0

,, .

ln lniter

x px p

x

On the other hand, for all with , we have

0

0

00

0 0

0

, ln 1 ln n ln 1

ln ln ln , ,

( , )ln

ln ln ln

p p xx p xp pp p

iter iter

x p x x

x x p x p

x px

x

Then


0

0

,, ln .

ln δ ln lniter

x px p x

x

Now, according to Lemma 3.1,

( ) , and then for x sufficiently

large (depending on ), ,

and thus

0

0

,, .

ln δ ln lniter

x px p

x

Now, for all , we can choose more near to 1.

For this , so that

, and for sufficiently large,

we have

0

0

,, 1 .

ln lniter

x px p

x

2) According to Theorem 3.1, we have

00

, ln ln, ~ ~

ln ln ln ln ln ln

~ , .

iter

iter

x p x xx p o

x x x

x x

3) Concerning the third equality, we have

0 0, , ln ln .iterx p x p x

Evaluate now the difference

0 , 0

lnln, ln ln ln ln .

lnlniter p x p p

xx p x p

p

Then

0

, 0

, 0

, 0

, 0

, 0

lnln, ln ln ln ln

lnln

lnlnln ln ln ln

lnln

lnlnln ln ln ln

lnln

lnln+ ln ln ln ln

lnln

lnln

lnln

iter

p x p p

p x p p

p x p p

x p x p p

p x p p

xx p x p

p

xx p

p

xx p

p

xx p

p

x

, 0

0

, 0

0

, 0

ln ln

ln ln ln ln

,ln

, 1ln

x p x p p

x

px p x p p

x

p txx p x p p

pp

x p

dtx p

t t

dtx p

t t

However, for [√ ]

,, ,00 0

1 1 1p t p t p tp x p px p x p p p x p p

Then,

0 0 01 1 , , , ,iter iter iter

p t p x

t p x p t p

thus

0

00

0

0

, ln ln

,,

ln ln

2ln,

ln ln ln

2, .

ln ln

iter

x iter

x

x

x

x

x

x p x x

t px p dt

t t

tx p dt

t t t

x p dtt t

By using , since

0 0ln

, , ln ln ln ln ln .ln ln

iterx

x p x p x x xx

we obtain

0 0ln

, ln ln , lnln ln

iterx

x p x x p O x Ox

Then,

(

)

3.2. Definition and Estimate of and

3.2.1. Definition and Estimate of

Definition 4.

1. Let be a positive real number. We denote by

the number of different classes such that .

Precisely,

0 0

0

( ) : 1, .c

p x

x p

2. We denote by the function defined by

0 0

0

( ) n .l

p x

x p

Example 3. In , the value represents the origin

of the class but the values do not, since they

belong to the same class . The value represent the

origin of the class . Thus in this case, we have .

We have the following result:

Theorem 3.3. We have,

lim ( ) .cx

x

Proof. Suppose that the number of different classes is

finite as . We know that

0

1.

iip

(13)

Next, let , where is finite by hypothesis. We

obtain

0 0

1 1.

kii k i i

p p

Therefore, we have

0

1

ki ip

, and since the second

sum has a finite number of terms, we deduce


0

1,

kk i ip

which contradicts formula (13).

Theorem 3.4. Let . Then

( ) ( ) ( ( )).c x x x (14)

Proof. To find the value of means that we estimate

the number of origins . Clearly, the prime number is

an origin that means is not a prime number. Then,

let be between and , and let be the

greatest prime number in , therefore is not a

prime number and for all integer ,

. So, we only have to search the numbers which

are not primes and less than . Thus, we have

1° The number of the even numbers less than or equal

to equal to

.

2° The number of the odd numbers less than or equal to

equal to

such that is

the number of the prime numbers less than or equal to .

Next, we add the two quantities, we obtain, since

, the quantity which is equal to

( ) ( ) ( ( )).c x x x

3.2.2. Estimate of

For all initial value , we define the following

sequence:

1 ln .n n ny y y (15)

It is not difficult to see that, this sequence is stationary

for and increasing divergent to infinite for

and as . It is clear that, inductively,

, then we have the

following consequence:

Lemma 3.2. We have,

0

0 0

2 2

ln ( )lnln lnln( ) ,

ln lnln ln

p xc

p xx x x xx

x xx x

where represents the origin of the classes .

Proof.. Since for all , we have ,

then

0 0ln ln 0ln , ,0 0 0 lnln 0

00 0

00 0

0

0

lnln l‍nln lnln

ln lnlnln ~ ln ln

lnln

l .n ln

n x pp x p p x n p np

p x p x

iter

p x

p p p

x pp x p

p

x x p

In the expression:

0

ln ,0 0

,ln lnnp p x n

p

the value of obviously depends on and if we denote

by , then, we have

0

0

~ ( ),p

p

n x

i.e., the number of elements of the form must

be equal to number of prime numbers .

According to the formula (3), we have

lnln

lnln ,ln

p x

x xp

x

therefore, the inequality is obtained directly by

substitution.

Proposition 3.1. We have,

0

1 lnln1

ln lnln ln

1( ) 1

ln lnln

xx

x x x

x xx x

(16)

0( )~ as ..x x x

Proof.

1. We have

00

2

0

0

lnlnln

( ) ( ( ))lnln

lnlnln ( ( ) ( ( ))) ln .

ln

p x

p x

px x

x xxx

x xp x x x

x

Moreover, since

0

0 0

ln ln 1 ( ) ln .c

p x p x

p x x x

And recalling that

( )~ .ln

xx

x

We obtain

0

0

lnlnln ln

ln ln (ln lnln ) ln

1 lnln1 .

ln lnln ln

p x

x x x xp x

x x x x x

xx

x x x

The second inequality is obtained in the same way,

0

0

ln lnln ln (ln lnln )

11 .

ln lnln

p x

x xp x

x x x x

xx x

2. It is enough to tend to in the inequality (16).

4 Conclusion and Future Work

There is a lot of results on properties of prime numbers.

There are innumerable ideas in this field regarding the

randomness of prime numbers. However, it turns out that

prime numbers do not appear absolutely randomly,

meaning, it is not entirely true that there is no way

whatsoever to see some relations and find some functions


to generate a lenght of few prime numbers. Patterns do

emerge in the distribution of primes over varied ranges of

number sets. In this paper we have investigated sequences

of primes obtained by the equivalence relation , from

which we have associated various arithmetic functions. As

we have seen in the sections above, many classical results

stated in elementary number theory can be restated with

our sequences. Also, there are many other asymptotic

formulas can be deduced, we have just presented several

of them as examples.

As far as motivation, the concern relates to the

developing of new ideas for solving great unsolved

problems and conjectures encountered in this field.

A highly intriguing area in primes is the concept of twin

primes. These are prime numbers which differ by the

number for example: and , and , and , etc.

There is an attempt being made to prove that there are

infinitely many twin primes that exist in the natural

number system. These are the concepts that come to mind.

Others are pretty much minor results.

Yitang Zhang, in his paper [3], attacked the problem by

proving that the number of primes that are less than

million units apart is infinite. While million is a long,

long way away from , Zhang's work marked the first

time anyone was able to assign any specific proven

number to the gaps between primes.

In November , James Maynard, in [4] , introduced

a new refinement of the GPY sieve, allowing him to

reduce the bound to and show that for any there

exists a bounded interval containing prime numbers.

At the present time, let me explain and share some

general ideas and questions about my future work. Firstly,

the questions raised are very broad in scope and cannot be

addressed directly. This means that, it is preferable to

resort to a methodology (plan in stages) to tackle the great

problems in a structured manner. Secondly, for that reason,

we have proposed and developed the results of this paper

(see the problem 1 and 2).

Finally, here are just a few questions and conjectures a

little more direct, I think are important.

Problem 1 The twin prime conjecture is equivalent to

conjecturing the translates of the (the

values of the pair ) are simultaneously prime

values infinitely often. The question is: show that if the

are simultaneously prime values

infinitely often then the are

simultaneously twin primes infinitely often, and then the

twin prime conjecture is true. We have posed this question

from the perspective of finding recurrence relation

between primes.

Problem 2 Let and be twin primes. Show that

2

1 1 3 22

1

( ) ( ) ~ ( ln ).i n

i i

i

p p O n n

Problem 3

1. Does the set contain an infinity of twin primes?

2. Let fixed. Are there an infinity of primes of

the form such that is prime?

3. Study of

0

1, ,

11p

s

s is a complex number

p

There is continuing research to prove these conjectures

and questions rigorously, using the results of this paper

and advanced techniques in number theory in the next

work.

Acknowledgement

We sincerely thank the editor and the referees for their

valuable comments and suggestions.

References

[1] C. J. de la Vallée Poussin, “Recherche analytique sur la théorie

des nombres”, Ann. Soc. Sci. Bruxelle, 20. 183-256. 1896.

[2] J. Hadamard, “Sur la distribution des zéros de la fonction et

ses conséquences arithmétiques”, Bull. SOC. Math. France, 24.

199-220. 1896.

[3] Y. Zhang, “Bounded gaps between primes”, Ann. of Math, 179,

1121-1174. 2014.

[4] J. Maynard, “Small gaps between primes”, Ann. of Math, 181.

383-414. 2015.




DOI:10.12691/tjant-3-4-3

On the Generalized Degenerate Tangent Numbers and

Polynomials

Cheon Seoung Ryoo*

Department of Mathematics, Hannam University, Daejeon, Korea


Received May 24, 2015; Accepted August 01, 2015

Abstract In [7], Ryoo introduced the generalized tangent numbers and polynomials. In this paper, our goal is to

give generating functions of the degenerate generalized tangent numbers and polynomials. We also obtain some

explicit formulas for degenerate generalized tangent numbers and polynomials.

Keywords: generalized tangent numbers and polynomials, degenerate generalized tangent numbers and

polynomials

Cite This Article: Cheon Seoung Ryoo, “On the Generalized Degenerate Tangent Numbers and

Polynomials.” Turkish Journal of Analysis and Number Theory, vol. 3, no. 4 (2015): 104-107. doi:

10.12691/tjant-3-4-3.

1. Introduction

In [2], L. Carlitz introduced the degenerate Bernoulli

polynomials. Recently, Feng Qi et al. [3] studied the

partially degenerate Bernoull polynomials of the first kind

in p-adic field. In [9], Ryoo constructed the degenerate

tangent numbers and polynomials. In this paper, we

introduce the degenerate generalized tangent numbers and

polynomials. We also obtain some interesting properties

for degenerate generalized tangent numbers and

polynomials. Throughout this paper, we always make use

of the following notations: denotes the set of natural

numbers and 0 , denotes the set of

complex numbers. The tangent numbers are defined by

means of the following generating function:

2

0

2.

! 21

n

n tn

tT t

n e

(1.1)

Ryoo [6] defined tangent polynomials by multiplying xte on the right side of the Eq. (1.1) as follows:

2

0

2.

! 21

nxt

n tn

tT x e t

n e

(1.2)

Let be Dirichlet's character with conductor d

with 1 mod 2 .d Then the generalized tangent

numbers associated with associated with ,, nT , are

defined by the following generating function

1 2

0,2

0

2 1.

!1

d a at na

ndtn

a e tF t T

ne

(1.3)

We now consider the generalized tangent polynomials

associated with ,, ,nT x are also defined by

1 2

0

2

,0

2 1,

1

.!

d a atxta

dt

n

nn

a eF x t e

e

tT x

n

(1.4)

When 0 , above (1.3) and (1.2) will become the

corresponding definitions of the tangent numbers nT and

polynomials nT x (see [6]). For more theoretical

properties of the generalized tangent numbers and

polynomials, the readers may refer to [7].

For a variable t, we consider the degenerate tangent

polynomials which are given by the generating function to be

/

,2/0

21 .

!1 1

nx

nn

tt T x

nt

(1.5)

When 0,x , ,0n nT T are called the

degenerate tangent numbers(see [8]).

We recall that the classical Stirling numbers of the first

kind 1 ,S n k and 2 ,S n k are defined by the relations

(see [9,10])

1 20 0

, , ,n n

k nn k

k k

x S n k x and x S n k x

respectively. Here 1 1n

x x x x n denotes

the falling factorial polynomial of order .n The numbers

2 ,S n m also admit a representation in terms of a

generating function

2

1, .

! !

mt

n

n m

etS n m

n m

(1.6)


We also have

1

log 1, .

! !

mn

n m

ttS n m

n m

(1.7)

The generalized falling factorial n

x with increment

is defined by

1

0

n

nk

x x k

(1.8)

for positive integer n, with the convention 0

1.x We

also need the binomial theorem:

for a variable x ,

/

0

1 .!

nx

nn

tt x

n

(1.9)

2. On the Degenerate Generalized

Tangent Polynomials

In this section, we define the degenerate generalized

tangent numbers and polynomials, and we obtain explicit

formulas for them. Let be Dirichlet's character with

conductor d with 1 mod 2 .d Then the

degenerate generalized tangent numbers associated with , , ,nT are defined by the following generating

function

1 2 //0

2 /

, ,

0

2 1 11

1 1

.!

d a axa

d

n

nn

a tt

t

tT x

n

(2.1)

When 0,x , , , ,0n nT T are called the

generalized degenerate tangent numbers.

From (2.1) and (1.4), we note that

, ,0

0

1 2 //0

2 /0

1 2

0

2

,0

lim!

2 1 1lim 1

1 1

2 1

1

.!

n

nn

d a axa

d

d a atxta

dt

n

nn

tT x

n

a tt

t

a ee

e

tT x

n

Thus, we get

0

, , ,l 0im , .n nT x T x n

From (2.1) and (1.5), we have

, ,0

1 2 //0

2 /

, ,0 0

,0 0

!

2 1 11

1 1

! !

.!

n

nn

d a axa

d

m l

m lm t

nn

l n ln l

tT x

n

a tt

t

t tT x

m l

n tT x

l n

(2.2)

Therefore, by (2.1) and (2.2), we obtain the following

theorem.

Theorem 1. For 0n , we have

, , , ,0

.n

n l n ll

nT x T x

l

For d with 1 mod 2 ,d we have

1/ 2 /

2 /0

, ,0

1 2 //0

2 /

1

, /0 0

21 1 1

1 1

!

2 1 11

1 1

21 .

!

dx l l

dl

n

nn

d a axa

d

ndln

n dn l

t l tt

tT x

n

a tt

t

l x td l T

d n

(2.3)

By comparing coefficients of !

mt

min the above

equation, we have the following theorem.

Theorem 2. Let be Dirichlet's character with

conductor d with 1 mod 2 .d Then we have

(1) 1

, , , /0

21 ,

dam

m m da

a xT x d a T

d

(2) 1

, , , /0

21 .

dam

m m da

aT d a T

d

From (2.1), we can derive the following relation:

, , , ,0

1 2 //0

2 /

1 2 /

0

2 /

12 /

0

1

0 0

2!

2 1 11

1 1

2 1 1

1 1

2 1 1

2 1 2 .!

m

m mm

d a axa

d

d a a

a

d

da a

a

mda

mm a

tT d T x

m

a tt

t

a t

t

a t

ta a

m

(2.4)


By comparing of the coefficients !

mt

m on the both sides

of (2.4), we have the following theorem.

Theorem 3. For ,n we have

1

, , , ,0

2 2 1 2 / .d

am m m

a

T d T a a

By (2.1), we have

1

0 0 0

, , , ,0

12 / /

0

2 1 2 .!

2!

2 1 1 1

md ma

l m lm a l

m

m mm

da a x

a

m ta x a

l m

tT x d T x

m

a t t

(2.5)

By comparing of the coefficients !

mt

m on the both sides

of (2.5), we have the following theorem.


, , , ,

1

0 0

2

2 1 2 .

m m

d ma

l m la l

T x d T x

ma x a

l

From (2.1), we have

, ,0

1 2 //0

2 /

1 2 /

0

2 /

/ /

, ,0 0

, ,0 0

!

2 1 11

1 1

2 1 1

1 1

1 1

! !

.!

n

nn

d a ax ya

d

d a a

a

d

x y

n n

n nn n

nn

l n ln l

tT x y

n

a tt

t

a t

t

t t

t tT x y

n n

n tT y

l n

(2.6)

Therefore, by (2.6), we have the following theorem.


, , , ,0

.n

n k n kk

nT x y T x y

k

From Theorem 5, we note that , ,nT x is a Sheffer

sequence.

By replacing t by 1te

in (2.1), we obtain

1 2

0

2

, ,0

, , 20

, , 20 0

2 1

1

1 1

!

,!

, .!

d a atxta

dt

nt

nn

mn m

nn m n

mmm n

nm n

a ee

e

eT x

n

tT x S m n

m

tT x S m n

m

(2.7)

Thus, by (2.7) and (1.4), we have the following theorem.


, , , 20

, .m

m nm n

n

T x T x S m n

By replacing t by 1/

log 1 t

in (1.4), we have

1/,

0

1 2 //0

2 /

, ,0

1log 1

!

2 1 11

1 1

,!

n

nn

d a axa

d

m

mm

T tn

a tt

t

tT x

m

(2.8)

and

1/,

0

, 10 0

1log 1

!

, .!

n

nn

mm n

nm n

T tn

tT x S m n

m

(2.9)

Thus, by (2.8) and (2.9), we have the following theorem.


, , , 10

, .m nm n

n

T x T x S m n

References

[1] L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers,

Utilitas Math. 15(1979), 51-88.

[2] L. Carlitz, A degenerate Staudt-Clausen theorem, Arch. Math.

(Basel) 7(1956), 28-33.

[3] F. Qi, D. V. Dolgy, T. Kim, C. S. Ryoo, On the partially

degenerate Bernoulli polynomials of the first kind, Global Journal

of Pure and Applied Mathematics, 11(2015), 2407-2412.

[4] T. Kim, Barnes' type multiple degenerate Bernoulli and Euler

polynomials, Appl. Math. Comput. 258(2015), 556-564.

[5] H. Ozden, I. N. Cangul, Y. Simsek, Remarks on q-Bernoulli

numbers associated with Daehee numbers, Adv. Stud. Contemp.

Math. 18(2009), no. 1, 41-48.

[6] C. S. Ryoo, A Note on the tangent numbers and polynomials, Adv.

Studies Theor. Phys., 7(2013), no. 9, 447-454.

[7] C. S. Ryoo, Generalized tangent numbers and polynomials

associated with p-adic integral on p, Applied Mathematical

Sciences, 7(2013), no. 99, 4929-4934.


[8] C. S. Ryoo, Some identities on the (h; q)-tangent polynomials

and Bernstein Polynomials, Applied Mathematical Sciences,

8(2014), no. 75, 3747-3753.

[9] C. S. Ryoo, Notes on degenerate tangent polynomials, to appear in

Global Journal of Pure and Applied Mathematics, Volume 11,

number 5(2015), pp. 3631-3637.

[10] P. T. Young, Degenerate Bernoulli polynomials, generalized

factorial sums, and their applications, Journal of Number Theorey,

128(2008), 738-758.




DOI:10.12691/tjant-3-4-4

Generalized s-topological Groups

Rehman Jehangir1,*

, Moizud Din Khan2,*

1Department of Mathematics, Preston University Kohat (Islamabad), Pakistan 2Department of Mathematics, COMSATS institute of information technology, Chak Shehzad Islamabad, Pakistan

*Corresponding author: [email protected], [email protected]

Received June 04, 2015; Accepted August 09, 2015

Abstract In this paper, we explore the notion of generalized semi topological groups. This notion is based upon

the two ideas, generalized topological spaces introduced by Csaszar [2,3] and the semi open sets introduced by

Levine [7]. We investigate on the notion of generalized topological group introduced by Hussain [4]. We explore the

idea of Hussain by considering the generalized semi continuity upon the two maps of binary relation and inverse

function.

Keywords: generalized semi open sets, generalized compact sets, generalized continuity, generalized continuity,

generalized discrete sets

Cite This Article: Rehman Jehangir, and Moizud Din Khan, “Generalized s-topological Groups.” Turkish

Journal of Analysis and Number Theory, vol. 3, no. 4 (2015): 108-110. doi: 10.12691/tjant-3-4-4.

1. Introduction

Let denotes the generalized topological space ( , ).

In accordance with [3], let A be generalized semi open

if and only if there exists a generalized open set ( -open

set) such that , where

denotes the generalized closure of the set O in . For more

details on generalized topological spaces, we refer to [2,3].

In 2013, Murad et al. [4], defined and studied the concept

of generalized topological groups ( -topological groups).

This study was further extended and published in [5] and

[6]. In 2015, C. Selvi and R. Selvi [10] were motivated by

-topological groups [4] and S-topological groups [9], and

defined on new notion with the name of generalized S-

topological groups.

In this paper, we intend to generalized further the

notion of -topological groups and -S-topological groups

by using -semi continuity. -semi continuity is a

generalization of -continuity and it was defined by Á.

Császár in [3].

2. Generalized Semi Topological Group

In this section, we will explore the notion of

generalized semi topological group. Generalized semi

topological groups contains the structure of generalized

topology and groups. The whole idea is backed by the

generalized semi continuity, as the binary operation and

the inverse map undergo the process of generalized semi

continuity. We will study the basic definitions and gradual

development of the phenomenon.

Let ( ) and ( ) be generalized

topological spaces and is generalized semi

continuous, then for any subset of ,

( )

Let ( , ) and ( , ) be generalized topological spaces and let ( , ) = ( , ) be their product generalized topological space if is

generalized semi open set in and is generalized semi

open in then is generalized semi open

in ( ) Assume that = , where is

generalized open in and for

. This is nowhere dense set as well. Then,

1 2 2

1 2

1 2 1

1 2 2 1 21

A A O B O B

O O B O O B B B

But is generalized open set in and

1 1 2 1 2

1 2

1 2 2 1

2

2

1

B O O B B B cl O O

cl O O

Hence, 1 2 ( ) ( )

= ( )

1 2 1 2 1 2 1 2O O A A cl O O

This proves that 1 2 is generalized semi open set

in .

. Let: ( , ) ( , ) semi generalized

continuous map between two generalized topological

spaces. Let be semi generalized compact set relative to

( , ) then is semi generalized compact in ( , ).

Let be any collection of generalized

open set of ( , ), such that : i . Then

{ ( ): holds by hypothesis and there

exists a finite subset of of such that { ( ):

which shows that is semi generalized

compact in ).


3. Semi Generalized Topological Group

In this section, we will define semi generalized

topological groups ( -s-topological groups) and

investigate its basic properties.

is said to be a - s-topological

group if

(1 is generalized topological space;

(2). is a group;

(3). The multiplication map , defined by

and the inverse map defined by

; are the generalized semi-continuous.

Equivalently, is semi generalized topological

group if and for each generalized open set

containing , there exist generalized semi

open sets containing and containing y, such that,

. Since every - is continuous therefore, every -topological group is

- -topological groups. And - -topological group may

not be a -topological group. Further, we note that, every

-topological group is -s-topological group and every -

s-topological group is - -topological group. However,

converses may not be true in general. It is evident from the

following example:

. Let = = {0 1} be the two-

element (cyclic) group with the multiplication mapping

= +2 the usual addition modulo 2. Equip with

the Sierpinski topology = { , {0} }. Then

the collection of all the semi open sets

SO

0,0 , 0,0 , 0,1 ,

0,0 , 1,0 , 0,0 , 0,1 , 1,0 ,

0,0 , 0,1 , 1,0 , 1,1 ,

0,0 , 1,1 , 0,0 , 0,1 , 1,1 ,

0,0 , 1,0 , 1,1

and that : is continuous at

(0 0) (1 0) (0 1), but not continuous at

(1 1). However, is semi-continuous at (1 1).

For this, let us take the open set = {0} in

containing (1 1) = 0. Then the semi-open set

= {(0 0) (1 1)} contains

(1 1) . The inverse mapping : is

continuous and hence semi-continuous. Therefore,

( +2 ) is a - -topological group which is

not a topological group. It was noticed in [1] that

( +2 ) is not a - -topological group.

Let , , ) be a generalized s-

topological group. Let be an inverse

mapping defined by = ; . Then is

generelized semi continuuous mapping.

Let . Let W be a generalized open set in

containing . Then by hypothesis, there exist

generalized semi open sets U and V containing e and x,

respectively, such that In

particular = e .

. If is semi generalized compact, then

is semi generalized compact in a semi generalized

topological group ( ).

Let { : I} be a cover of

⋃ This implies that ⋃ =

⋃ . This implies that ⋃

.

Since is semi generalized compact, then there

exists a finite set of such that

⋃ . This implies that y ⋃ That

is has a finite subcover of X. Hence is

semi generalized compact.

. A non empty subgroup of a semi

generalized topological group is semi open if and only if

its semi interior is non empty.

Assume that (semi generalized

interior). Then by definition there is a semi generalized

open set V such that : For every , we

have y Since V is semi generalized

open so is , we conclude that is a semi generalized open set as the union of semi

generalized open sets is semi generalized open. Converse

of this theorem is quite simple.

Let ( ) and ( ) be generalized

topological spaces and is generalized semi

continuous, then for any subset of ,

( ) Further theorem is the extension of the work presented

by Bohn Lee [1].

Let be a semi generalized

topological group. Then for each generalized open set

subset of ; is semi generalized open.

Let be generalized open in , there exists a

generalized open set in , such that,

( )U A cl U (By [5])

1 1 1[cl(U)]U A

Because, is semi generalized topological group.

Let be -topological space. If is -

semi open and , then is -semi open.

Let . Since is -open

therefore there exists a -open set such that .

implies that .

This proves that is semi -open in .

Let ) be a -s-topological group.

Then the multiplication mapping

: defined by

is semi -continuous for each Let and W be a -open set containing

, since X is -semi open sets and containing

and , such that

1

1

1

1 1

*

* *

, * *

*

U V W

x y U V W

m x y x y U V W

m U V U V W


Since is -open set containing , therefore,

by Theorem-1.6, is -semi open set containing y.

Moreover, by Theorem 2, is -semi open set

containing . Hence „ ‟ is semi -continuous for

each By Theorems 2.5 and 2.7, it is clear that every -s-

topological group is - -topological group.

Let ) be a -s-topological group

and be any -open set in . Then, for each ,

both are -semi open in .

Proof: Let . This gives for some .

1 1.* * *y z x A x x

Since, is -s-topological group, therefore, for -open

set A containing , there exist -open set and

containing and respectively, such that

1* .U V A

Or

1 1 1* * * .z x U x U V A

This gives This proves that is -

semi open set.

Let . be semi -topological

group. If is generalized open and , then is generalized semi open in .

Let and

* z y x

or, for some = ( )

Now, , implies,

z

Where is generalized open set in , therefore, by the

hypothesis, i.e., is semi generalized topological group,

there exist generalized semi open set in X containing z and

containing such that,

1*VU A

or

1 1* *VU x U A

or

* .U A x

This implies that for each point z x, we can find a

generalized semi open set U containing z such that

x. This means x is generalized semi open. Since the

union of semi open sets is generalized semi open,

therefore,

* *xx BA B A

is generalized semi open.

Let be a semi -topological group

and let be the base at identity element e of . Then, for

every , there is an element ; so that

following holds,

1) U.

2) U.

3) .

. Let be a semi -topological

group. Then each left(right) translation :

( : ) is -semi homeomorphic.

It is obviously bijective map and : is

semi -continuous containing , there exists -semi

open set containing x such that Again, let

be a -open set in , then is semi -open.

That is the image of -open set is semi -open. This

proves that is -semi homeomorphic.

Let be semi -topological group and

. Then for any local base and e , then each of

the families and { is a semi -open neighborhood system of .

References

[1] E. Bohn, J. Lee, Semi-topological groups, Amer. Math. Monthly

72 (1965), 996.998.

[2] Á. Császár, Generalized open sets, Acta Math. Hungar., 75 (1997),

65-87.

[3] Á. Császár, Generalized topology, generalized continuity, Acta

Math.Hungar., 96 (2002), 351-357.

[4] M. Hussain, M. Khan and C. Ozel, On generalized topological

groups, Filomat, 2013 27(4):567-575.

[5] M. Hussain, M. Khan and C. Ozel, More on generalized

topological groups, Creative Math. & Inf. 22(2013)(1), 47-51.

[6] M. Hussain, M. Khan, and C. Ozel, Extension closed

properties on generalized topological groups, Arab J Math (2014)

3:341-347.

[7] N. Levine, Semi-open sets and semi-continuity in topological

spaces, Amer. Math. Monthly 70 (1963), 36.41.1.

[8] Michel Coste, Real Algebraic Sets, March 23, 2005.

[9] J. Cao, R. Drozdowski, Z. Piotrowski, Weak continuity properties

of topologized groups, Czech. Math. J., 60 (2010), 133-148.

[10] C. Silva and R. Silva, On generalized S topological group,

International journal of science and research, 6, (2015), 1-4.




DOI:10.12691/tjant-3-4-5

Further Inequalities Associated with the Classical Gamma Function

Kwara Nantomah*

Department of Mathematics, University for Development Studies, Navrongo Campus, Navrongo UE/R, Ghana

*Corresponding author: [email protected], [email protected]

Received June 15, 2015; Accepted August 20, 2015

Abstract In this paper, the authors present some double inequalities associated with certain ratios of the Gamma

function. The results are further generalizations of several previous results. The approach is based on some

monotonicity properties of some functions involving the generalized Gamma functions. At the end, some open

problems are posed.

Keywords: Gamma function, Psi function, inequality, generalization

Cite This Article: Kwara Nantomah, “Further Inequalities Associated with the Classical Gamma Function.”

Turkish Journal of Analysis and Number Theory, vol. 3, no. 4 (2015): 111-115. doi: 10.12691/tjant-3-4-5.

1. Introduction

Inequalities involving the classical Euler’s Gamma

function has gained the attention of researchers all over

the world. Recent advances in this area include those

inequalities involving ratios of the Gamma function. In

[1,5,6,10] and [11-17], the authors established some

interesting inequalities concerning such ratios, as well as

some generalizations. By utilizing similar techniques, this

paper seeks to present some new results generalizing the

results of [11-17]. At the end, we pose some open

problems involving the generalized Psi functions. In the

sequel, we recall some basic definitions concerning the

Gamma function and its generalizations. These definitions

are required in order to establish our results.

The well-known classical Gamma function, ( )t and

the classical Psi or Digamma function ( )t are usually

defined for 0t as:

1

0

( )( ) and ( ) .

( )

x t tt e x dx t

t

The p-Gamma function, ( )p t and the p-Psi function

( )p t are defined for p N and 0t as:

( )!

( ) and ( )( 1)...( ) ( )

tp

p pp

tp pt t

t t t p t

where ( ) ( )p t t and ( ) ( )p t t as p . For

more information on this function, see [9] and the

references therein.

Also, the q-Gamma function, ( )q t and the q-Psi

function ( )q t are defined for (0,1)q and 0t as:

1

1

( )1( ) (1 ) and ( ) .

( )1

nqt

q qt nqn

tqt q t

tq

where ( ) ( )q t t and ( ) ( )q t t as 1q .

See also [4,5] and the references therein.

Similarly, the k-Gamma function, ( )k t and the k-Psi

function ( )k t are defined for 0k and 0t as (see

[2,7]):

1

0

( )( ) and ( )

( )

kxt kk

k kk

tt e x dx t

t

where ( ) ( )k t t and ( ) ( )k t t as 1k .

Also, the (q,k)-Gamma function ( , ) ( )q k t and the

(q,k)-Psi function ( , ) ( )q k t are defined for (0,1)q ,

0k and 0t as [3]:

1

, ( , )( , ) ( , )

1 ( , )

(1 ) ( )( ) and ( )

( )(1 )

tk k

q k q kq k q kt

q kk

q tt t

tq

where 1

,

0

( ) ( )n

n k

j

t t jk

is the k-generalized Pochhammer

symbol and ( , ) ( ) ( )q k t t , ( , ) ( ) ( )q k t t as 1q ,

1k .

Furthermore, the (p,q)-Gamma function ( , ) ( )p q t and

the (p,q)-Psi function ( , ) ( )p q t are defined for p N ,

(0,1)q and 0t as [8]:

( , )

[ ] [ ] !( )

[ ] [ 1] [ ]

tq q

p qq q q

p pt

t t t p


and

( , )

( , )( , )

( )( )

( )

p qp q

p q

tt

t

where 1

[ ]1

p

q

qp

q

, and ( , ) ( ) ( )p q t t , ( , ) ( ) ( )p q t t

as p , 1q .

As defined above, the generalized Psi functions: ( )p t ,

( )q t , ( )k t , ( , ) ( )q k t and ( , ) ( )p q t possess the

following series forms (see [16,17] and the references

therein):

0

1( ) In

p

pn

t pn t

(1)

1

( ) In(1 ) (In )1

nt

q nn

qt q q

q

(2)

1

In - 1( )

( )k

n

k tt

k t nk nk t

(3)

( , )1

( ) In[ ] (In )1

p nt

p q q nn

qt p q

q

(4)

( , )1

-In(1 )( ) (In )

1

nkt

q k nkn

q qt q

k q

(5)

with

1

1lim In 0.5721566...

n

nk

nk

denoting the

Euler-Mascheroni’s constant.

2. Results

We now present our results. Let us begin with the

following Lemmas pertaining to the results.

Lemma 2.1. Assume that 0 , p N , (0,1)q

and ( ) 0g t . Then,

( , )

In(1 ) In[ ]

( ( )) ( ( )) 0.

q

q p q

q p

g t g t

Proof. By using equations (2) and (4) we obtain,

( , )

( ) ( )

1 1

In(1 ) In[ ] ( ( )) ( ( ))

(In ) 0.1 1

q q p q

png t ng t

n nn n

q p g t g t

q qq

q q

concluding the proof.

Lemma 2.2. Assume that 0 , (0,1)q , 1k

and ( ) 0g t . Then,

( , )

In(1- )In(1 )

( ( )) ( ( )) 0.q q k

qq

k

g t g t


( , )

( ) ( )

1

In(1- )In(1 )

( ( )) ( ( ))

(In ) 01 1

q q k

ng t nkg t

n nkn

qq

k

g t g t

q qq

q q


Lemma 2.3. Assume that 0 , 0 , 0k , p N ,

(0,1)q and ( ) 0g t . Then,

( , )

InIn[ ]

( )

( ( )) ( ( )) 0.

q

k p q

kp

k k g t

g t g t


( , )

( )

1 1

InIn[ ]

( )

( ( )) ( ( ))

( )(In ) 0

( ( )) 1

q

k p q

p ng t

nn n

kp

k k g t

g t g t

g t qq

nk nk g t q


Lemma 2.4. Assume that 0 , 0 , (0,1)q ,

0k and ( ) 0g t . Then,

( , )

In( (1 ) )

( )

( ( )) ( ( )) 0.k q k

k q

k g t k

g t g t


( , )

( )

1 1

In( (1 ) )

( )

( ( )) ( ( ))

( )(In ) 0

( ( )) 1

k q k

nkg t

nkn n

k q

k g t k

g t g t

g t qq

nk nk g t q


Theorem 2.5. Let ( )g t be a positive, increasing and

differentiable function, p N and (0,1)q . Then for

positive real numbers and such that , the

inequalities:

( (0) ( ))

( (0) ( ))( , )

( , )

( ( ) ( ))

( ( ) ( ))( , )

(1 ) ( (0))

[ ] ( (0))

( ( ))

( ( ))

(1 ) ( ( ))

[ ] ( ( ))

g g xq

g g xq p q

q

p q

g y g xq

g y g xq p q

q g

p g

g x

g x

q g y

p g y

(6)

hold true for 0 x y .

Proof. Define a function G for p N and (0,1)q

by


( )

( )( , )

(1 ) ( ( ))( ) , (0, )

[ ] ( ( ))

g tq

g tq p q

q g tG t t

p g t

.

Let ( ) InG( )u t t . Then,

( )

( )( , )

( , )

(1 ) ( ( ))( ) In

[ ] ( ( ))

( )In(1 ) ( )In[ ]

In ( ( )) In ( ( )).

g tq

g tq p q

q

q p q

q g tu t

p g t

g t q g t p

g t g t

Then,

( , )

( , )

( ) ( )In(1 ) ( )In[ ]

( ) ( ( )) ( ) ( ( ))

( ) In(1 ) In[ ]

( ( )) ( ( )) 0

q

q p q

q

q p q

u t g t q g t p

g t g t g t g t

g t q p

g t g t

as a consequence of Lemma 2.1. That implies u is non-

increasing on (0, )t . Hence ( )u tG e is non-

increasing and for 0 x y we have,

(0) ( ) ( )G G x G y

establishing the inequalities in (6).


differentiable function, (0,1)q and 1k . Then for

positive real numbers and such that , the

inequalities:

( (0) ( ))

( (0) ( ))

( , )

( , )

( ( ) ( ))

( ( ) ( ))

( , )

(1 ) ( (0))

(1 ) ( (0))

( ( ))

( ( ))

(1 ) ( ( ))

(1 ) ( ( ))

g g xq

g g xk

q k

q

q k

g y g xq

g y g xk

q k

q g

q g

g x

g x

q g y

q g y

(7)


Proof. Define a function H for (0,1)q and 1k by

( )

( )

( , )

(1 ) ( ( ))( ) , (0, )

(1 ) ( ( ))

g tq

g t

kq k

q g tH t t

q g t

.

Let ( ) In ( )v t H t . Then,

( )

( )

( , )

( , )

(1 ) ( ( ))( ) In

(1 ) ( ( ))

( ) ( )In(1 ) In(1- )

In ( ( )) In ( ( )).

g tq

g t

kq k

q q k

q g tv t

q g t

g tg t q q

k

g t g t

Then,

( , )

( , )

( )( ) ( )In(1 ) In(1 )

( ) ( ( )) ( ) ( ( ))

In(1 ) ( ) In(1 )

( ( )) ( ( )) 0

q q k

q q k

g tv t g t q q

k

g t g t g t g t

qg t q

k

g t g t

as a consequence of Lemma 2.2. That implies v is non-

increasing on (0, )t . Hence ( )v tH e is non-

increasing and for 0 x y we have,

(0) ( ) ( )H H x H y



differentiable function, 0k , p N and (0,1)q . Then

for positive real numbers and , the inequalities:

( (0) ( )) ( (0) ( ))

( (0) ( ))( , )

( , )

( ( ) ( )) ( ( ) ( ))

( ( ) ( ))( , )

( (0)) ( (0))

( ( )) [ ] ( (0))

( ( ))

( ( ))

( ( )) ( ( ))

( ( )) [ ] ( ( ))

g g x g g xk k

k

g g xq p q

k

p q

g y g x g y g xk k

k

g y g xq p q

g k e g

g x p g

g x

g x

g y k e g y

g x p g y

(8)


Proof. Define a function S for 0k , p N and

(0,1)q by

( ) ( )

( )( , )

( ( )) ( ( ))( ) , (0, )

[ ] ( ( ))

g t g t

k kk

g tq p q

g t k e g tS t t

p g t

.

Let ( ) In ( )w t S t . Then,

( ) ( )

( )( , )

( , )

( ( )) ( ( ))( ) In

[ ] ( ( ))

( ) ( )( )In[ ] In In( ( ))

In ( ( )) In ( ( )).

g t g t

k kk

g tq p q

q

k p q

g t k e g tw t

p g t

g t g tg t p k g t

k k

g t g t

Then,

( , )

( , )

( )In ( )( ) ( )In[ ]

( ) ( ) ( ( )) ( ) ( ( ))

( )

In ( ) In[ ]

( )

( ( )) ( ( )) 0

q

k p q

q

k p q

g t k g tw t g t p

k k

g tg t g t g t g t

g t

kg t p

k k g t

g t g t

as a result of Lemma 2.3. That implies w is increasing on

(0, )t . Hence ( )w tS e is increasing and for

0 x y we have,


(0) ( ) ( )S S x S y



differentiable function, 0k and (0,1)q . Then for

positive real numbers and , the inequalities:

( (0) ( ))

( (0) ( )) ( (0) ( ))

( , )

( , )

( ( ) ( ))

( ( ) ( )) ( ( ) ( ))

( , )

( ( )) ( (0)) ( (0))

(1 ) ( (0))

( ( ))

( ( ))

( ( )) ( ( )) ( ( ))

(1 ) ( ( ))

g g xk

k

g g x g g xk k

q k

k

q k

g y g xk

k

g y g x g y g xk k

q k

g x g e g

k q g

g x

g x

g x g y e g y

k q g y

(9)


Proof. Define a function T for 0k and (0,1)q by

( )

( ) ( )

( , )

( ( )) ( ( ))( ) , (0, )

(1 ) ( ( ))

g t

kk

g t g t

k kq k

g t e g tT t t

k q g t

.

Let ( ) In ( )t T t . Then,

( )

( ) ( )

( , )

( , )

( ( )) ( ( ))( ) In

(1 ) ( ( ))

( ) ( ) ( )In ( ) In In(1 )

In ( ( )) In ( ( )).

g t

kk

g t g t

k kq k

k q k

g t e g tt

k q g t

g t g t g tg t k q

k k k

g t g t

Then,

( , )

( , )

( ) ( ) ( )In( (1 ) )( )

( )

( ) ( ( )) ( ) ( ( ))

In( (1 ) ) ( )

( )

( ( )) ( ( )) 0

k q k

k q k

g t g t g t k qt

k g t k

g t g t g t g t

k qg t

k g t k

g t g t

as a result of Lemma 2.4. That implies is -increasing on

(0, )t . Hence ( )tT e is increasing and for

0 x y we have,

(0) ( ) ( )T T x T y


3. Concluding Remarks

In particular, if we let ( )g t t for 0 and

0 on the interval 0 1t , then we recover the entire

results of [17]. Also, by setting ( )g t t and

1 on the interval 0 1t , we obtain the results

of [16]. The results [11] – [17] are therefore special cases

of the results of this paper. For example, let ( )g t t

for , 0 on the interval

0 1t . Then;

(i) by allowing 1q in Theorem 2.5, we recover

Theorem 3.7 of [13].

(ii) by allowing 1k in Theorem 2.8, we recover


(iii) by allowing 1q in Theorem 2.6, we recover


(iv) by allowing 1k in Theorem 2.7, we recover


This paper is a slightly modified version of preprint

[18].

4. Open Problems

For 0k , p N and (0,1)q , let ( )p t , ( )q t ,

( , ) ( )p q t and ( , ) ( )q k t be the generalized Psi functions

as defined in equations (1) – (5).

Problem 1: Under what conditions will the statements:

0 1

In In(1 ) ( ) ( )

1(In ) ( )0

1

q p

p nt

nn n

p q t t

qq

n t q

be valid?

Problem 2: Under what conditions will the statements:

( , ) ( , )

1 1

In(1 )In[ ] ( ) ( )

(In ) ( )01 1

q p q q k

p nt nkt

n nkn n

qp t t

k

q qq

q q

be valid?

Competing Interests

The authors declare that there is no competing interest.

Acknowlegement

The authors are very grateful to the anonymous

reviewers for their useful comments and suggestions

which helped in improving the quality of this paper.

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Turkish Journal of - sciepubpubs.sciepub.com/tjant/TJANT-3-4.pdf · 2020-05-30 · Turkish Journal of Analysis and Number Theory, vol. 3, no. 3 (2015): 94-96. doi: 10.12691/tjant-3-4-1