ON GENERALIZATION OF K-WRIGHT FUNCTIONS ...

ON GENERALIZATION OF K-WRIGHT FUNCTIONS AND ITS PROPERTIES KULDEEP SINGH GEHLOT AND JYOTINDRA C. PRAJAPATI

Dedicated to great mathematician Srinivasa Ramanujan on his 125th birthday Abstract. In present paper, authors discuss about generalized K-Write Function p Ψkq (z) with its convergence condition. We obtain functional relation between generalized K-Wright function and generalized Wright function, recurrence relation, and integral representation. Some special cases have also been discussed.

1. Introduction Generalized K-Gamma Function Γk (x) defined as (Diaz and Pariguan [1]) x

− 1

n! k n (nk) k Γk (x) = lim n→∞ (x)n,k

, k > 0, x ∈ C\kZ− ,

(1)

where (x)n,k is the k-Pochhammer symbol and is given by (x)n,k = x (x + k) (x + 2k) . . . . . . . . . . . . (x + (n − 1) k) ,

x ∈ C, k ∈ R, n ∈ N+ (2)

For Re(x) > 0, Γk (x) is defined as the integral Z∞ Γk (x) =

tx−1 e

−tk k

dt

(3)

0

It is easy to prove following results x

Γk (x) = k k −1 Γ

x k

Γk (x + k) = xΓk (x) γ  (γ)r,k = (k)r k r (γ)r,k =

Γk (γ + rk) Γk (γ)

(4) (5) (6) (7)

rk(x)r−1,k = (x)r,k − (x − k)r,k

(8)

(δ)n+j,k = (δ)j,k (δ + jk)n,k

(9)

2000 Mathematics Subject Classification. 26A33, 33B15, 33C20, 33E12. Key words and phrases. Generalized K-Wright function, K-Gamma function, Mittag-Leffler function. 1

2

KULDEEP SINGH GEHLOT AND JYOTINDRA C. PRAJAPATI

2. Generalized K-Wright Function Generalized K- Wright Function is defined as p Ψkq (z) for k ∈ R+ ; z ∈ C, αi , βj ∈ R(αi , βj 6= 0; i = 1, 2, ..., p; j = 1, 2, ..., q) and (ai + αi n), (bj + βj n) ∈ C\kZ− ,

k p Ψq

(z) =

k p Ψq





(ai , αi )1,p z = (bj , βj )1,q

∞ X n=0

p Q

Γk (ai + αi n)z n

i=1 q Q

(10) Γk (bj + βi n)n!

j=1

We use following notations for describing convergence condition, β  X  X q  q  p  p p  −αi q j X X Y αi  bj ai  p − q βj αi k Y βj k ; µ = − ;δ = − + . ∆= k k k k k k 2 j=1 j=1 i=1 i=1 j=1 i=1

Theorem 2.1. For k ∈ R+ ; z ∈ C, αi , βj ∈ R(αi , βj 6= 0; i = 1, 2, ..., p; j = 1, 2, ..., q) and (ai + αi n), (bj + βj n) ∈ C\kZ− , (a) if ∆ > −1 then series (10) is absolutely convergent for all z ∈ C and Generalized K-Wright function p Ψkq (z) is an entire function of z. (b) if ∆ = −1 then series (10) is absolutely convergent for all |z| < δ and of |z| = δ, Re(µ) > 21 . Above theorem can prove easily by using results of Diaz and Pariguan [1], Kilbas [4].

3. Special Cases For the appropriate values of the parameters, authors obtain some special cases of Generalized K-Wright function p Ψkq (z) in terms of family of Mittag-Leffler function defined as [2], [3], [5], [6], [8], [9] and [10].   ∞ P (γ, k) (γ)n,k z n γ k k z = (i) 1 Ψ2 (z) = 1 Ψ2 Γk (nα+β)(n!) = Ek,α,β (z) (β, α), (γ, 0) n=0   ∞ P (γ, rk) (γ)nr,k z n γ,r (ii) 1 Ψk2 (z) = 1 Ψk2 z = Γ (nα+β)(n!) = GEk,α,β (z) k (β, α), (γ, 0) n=0   ∞ P (1, 1) 1 1 zn (iii) 1 Ψ2 (z) = 1 Ψ2 z = Γ(nα+1) = Eα (z) (1, α), (γ, 0)   n=0 ∞ P (γ, 1) (γ)n z n γ 1 1 (iv) 1 Ψ2 (z) = 1 Ψ2 z = Γ(nα+β)(n!) = Eα,β (z) (β, α), (γ, 0) n=0   ∞ P (γ, r) (γ)nr z n γ,r (v) 1 Ψ12 (z) = 1 Ψ12 z = = Eα,β (z) Γ(nα+β)(n!) (β, α), (γ, 0) n=0   ∞ P (1, 1) zn z = (vi) 1 Ψ12 (z) = 1 Ψ12 Γ(nα+β) = Eα,β (z) (β, α), (γ, 0) n=0   (ai , αi )1,p 1 1 (vii) p Ψq (z) = p Ψq z = p Ψq (z) (bj , βj )1,q

ON GENERALIZATION OF K-WRIGHT FUNCTIONS AND ITS PROPERTIES

3

4. Recurrence relations Theorem 4.1. The relationship between generalized K-Wright function and generalized Wright function is given by the relation  

k p Ψq





p P i=1

k (ai , αi )1,p z = (bj , βj )1,q

 q P ai − bj  j=1 k

p Ψq

k p−q



 ai k , bj (k,

 

αi k 1,p βj k )1,q






p P i=1

ai −

zk

q P j=1

 bj 

   (11)

k

or counterpart q P

 p Ψq



k (ai , αi )1,p z = (bj , βj )1,q

j=1

!

p P

bj −

ai



i=1

k p Ψq

k q−p

 (kai , kαi )1,p zk (kbj , kβj )1,q

q P

βj −

j=1

p P

αi

!

i=1

 (12)

Proof: Using definition of generalized K-Wright function (10), we have p Q  X  Γk (ai + αi n)z n ∞ (ai , αi )1,p i=1 k k z = , A ≡ p Ψq (z) = p Ψq q Q (bj , βj )1,q n=0 Γk (bj + βi n)n! j=1

using the equation (4), we get

A≡

∞ X n=0

p Q

k

ai +αi n −1 k

Γ

i=1 q Q

k

bj +βj n −1 k

Γ

ai +αi n k



j=1




zn

bj +βj n k







n!

This reduces to

 

A≡

p P i=1

k

 q P ai − bj  j=1 k

k p−q

p Q ∞ X



Γ

i=1

ai k

n=0

+

αi n k

q Q

Γ


 zk



j=1

bj k

+

p P i=1

ai −

q P j=1

 bj 

k

βj n k



n  

n!

Finally, we arrived at  

A≡

k

 q P ai − bj  i=1 j=1 k p P

k p−q



 p Ψq

 

ai k , bj (k,

αi k 1,p βj k )1,q




zk

 q p P P  ai − bj  i=1 j=1 k

  

Counterpart (12) can be proving in similar manner. Theorem 4.2. Let k ∈ R+ ; z ∈ C; ai , bj ∈ C; αi , βj , β ∈ R (αi , βj 6= 0, i = 1, 2,  ..., p; j = 1, 2, ..., q) then (ai , αi )1,p k z = p Ψq+1 (b j , βj )1,q , (b, β)     (ai , αi )1,p (ai , αi )1,p k k d b p Ψq+1 z + (βz) dz p Ψq+1 z (13) (bj , βj )1,q , (b + k, β) (bj , βj )1,q , (b + k, β)

4

KULDEEP SINGH GEHLOT AND JYOTINDRA C. PRAJAPATI

Proof: Using the definition of generalized K-Wright function (10), we have p p Q Q n Γk (ai + αi n)z n Γ (a + α n)z ∞ ∞ k i i X d X i=1 i=1 B≡b + (βz) , q q Q dz n=0 Q n=0 Γk (bj + βj n)Γk (b + k + βn)n! Γk (bj + βj n)Γk (b + k + βn)n! j=1

j=1

using equation (5), we get

B≡

p Q

∞ X n=0

Γk (ai + αi n)(b + βn)z n

i=1 q Q

Γk (bj + βj n)(b + βn)Γk (b + βn)n!

j=1

This immediately leads to k p Ψq+1



(ai , αi )1,p z (bj , βj )1,q , (b, β)



Theorem 4.3. Let k ∈ R+ ; z ∈ C; a, ai , bj ∈ C; α, αi , βj , β ∈ R (αi , βj 6= 0, i = 1, 2, ..., p; j = 1, 2, ..., q) then     (ai , αi )1,p , (a + k, αk) (ai , αi )1,p , (a, αk) k z − p+1 Ψkq+1 z p+1 Ψq+1 (bj , βj )1,q , (a + k, 0) (bj , βj )1,q , (a, 0)   (ai + αi , αi )1,p , (a + ak, αk) k = zak(a + k)α−1,k p+1 Ψq+1 z (14) (bj + βj , βj )1,q , (a + αk, 0) Proof: Using the definition of generalized K-Wright function (10), we have p Q ∞ X i=1 Γk (ai + αi n) z n Γk (a + k + akn) Γk (a + akn) { − }, D≡ q Q n! Γk (a + k) Γk (a) n=0 Γk (bj + βj n) j=1

using equations (7) and (8), we get

D≡

∞ X n=0

p Q i=1 q Q

Γk (ai + αi n) Γk (bj + βj n)

zn {αnk(a + k)αn−1,k }. n!

j=1

This yield

D≡

∞ X n=0

p Q i=1 q Q

Γk (ai + αi (n + 1)) Γk (bj + βj (n + 1))

z n+1 {αk(a + k)αn+(α−1),k }. n!

j=1

This leads to

D≡

∞ X n=0

p Q i=1 q Q j=1

Γk (ai + αi (n + 1)) Γk (bj + βj (n + 1))

z n+1 {αk(a + k)α−1,k (a + k + (α − 1)k)αn,k }. n!

ON GENERALIZATION OF K-WRIGHT FUNCTIONS AND ITS PROPERTIES

5

Finally, using (7), we arrived at

D

≡ zak(a + k)α−1,k

∞ X

p Q

Γk (ai + αi + αi n) Γk (a + αk + αkn)

i=1

n=0

q Q

Γk (bj + βj + βj n) Γk (a + αk)

zn n!

j=1

≡ zak(a +

k)α−1,k p+1 Ψkq+1



(ai + αi , αi )1,p , (a + ak, αk) z (bj + βj , βj )1,q , (a + αk, 0)



5. Integral representations Theorem 5.1. For k ∈ R+ ; z ∈ C, ai , bj ∈ C; αi , βj ∈ R (αi , βj 6= 0, i = 1, 2, ..., p; j = 1, 2, ..., q) and (ai + αi n), (bj + βj n) ∈ C\kZ− , then k p Ψq



 (ai , kαi )1,p z = (bj , kβj )1,q

p Q i=1 q Q

βj Γk (ai ) Q αi Q Γk (bj ) r=1 s=1

1 B(µr ,ϑs −µr )

j=1 (kαi )αi tz β (kβj ) j

R1

tµr −1 (1 − t)ϑs −µr −1 e dt,    ai  bj k +r−1 k +s−1 where = µ and = ϑs r αi βj ×

(15)

0

Proof: Using the definition of generalized K-Wright function (10), we have p Q   Γk (ai + kαi n) n ∞ X zn z (ai , kαi )1,p i=1 k z = D , Ψ = (16) p q q Q (bj , kβj )1,q n! n! n=0 Γk (bj + kβj n) j=1

where

p Q

D≡

i=1 q Q

p Q

Γk (ai + kαi n) ≡ Γk (bj + kβj n)

j=1

i=1 q Q

Γk (ai )(ai )αi n,k , Γk (bj )(bj )βj n,k

j=1

using (6), above equation reduces to p
Q Γk (ai )k αi n aki α n i , D ≡ i=1   q Q b Γk (bj )k βj n kj j=1

(17)

βj n

using generalized Pochhammer symbol(Shukla and Prajapati [8]),  q  Y Γ(γ + nq) γ+r−1 (γ)nq = = q qn , if q ∈ N; Γ(γ) q n r=1 Equation (17), immediately leads to,  p αi  ai Q Q k +r−1 γk (ai ) αi n αi n α i k (αi ) n D ≡ i=1 × r=1 b  q β Q j j k βj n (βj )βj n Q k +s−1 γk (bj ) j=1

s=1

βj

n

(18)

6

KULDEEP SINGH GEHLOT AND JYOTINDRA C. PRAJAPATI

Let

 ai k

+r−1 αi



p Q

D

≡

i=1 q Q j=1 p Q

≡

i=1 q Q

b

j k

= µr and

+s−1 βj

 = ϑs then above equation follows αi Q

γk (ai )

(µr )n k αi n (αi )αi n r=1 × βj k βj n (βj )βj n Q γk (bj ) (ϑs )n s=1

γk (ai ) γk (bj )

αi Q

αi n

αi n

k (αi ) × β k βj n (βj )βj n Qj

j=1

Γ(ϑs )

r=1

×

Γ(ϑs − µr )Γ(µr )

Γ(µr + n)Γ(ϑs − µr ) , Γ(ϑs − µr + µr + n)

s=1

above equation can be written in the form of Beta function as, p Q

≡

i=1 q Q

α Qi

γk (ai )

α n kαi n (αi ) i βj n βj n k (β ) j γk (bj )

×

j=1

Γ(ϑs )

R1

r=1 βj Q

tµr −n−1 (1 − t)ϑs −µr −1 dt,

(19)

Γ(ϑs −µr )Γ(µr ) 0

s=1

From (16) and (19), we obtain p Q   βj Γk (ai ) Q αi Q R1 µ −1 (a , kα ) i i 1,p k 1 z = i=1 × t r (1 − t)ϑs −µr −1 dt q p Ψq Q B(µr ,ϑs −µr ) (bj , kβj )1,q Γk (bj ) r=1 s=1 0 j=1

This gives equation (15). Theorem 5.2. For k ∈ R; z ∈ C, ai , bj ∈ C; αi , βj ∈ R (αi , βj 6= 0; i = 1, 2, ..., p; j = 1, 2, ..., q) and (ai + αi n), (bj + βj n) ∈ C\kZ− , then " # p p p   P P P a a a ∞ ( ki )−p R ( ki )−p ( pi ) (ai , kαi )1,p k k −t ¯ i=1 z = k i=1 e ti=1 z dt(20) p Ψq 0 Ψq (bj , βj )1,q (kt) (bj , kβj )1,q 0 Proof: Using the definition of generalized K-Wright function (10), we have

k p Ψq

(z) =

k p Ψq





(ai , αi )1,p z = (bj , βj )1,q

∞ X n=0

p Q

Γk (ai + αi n)z n

i=1 q Q

, Γk (bj + βi n)n!

j=1

using equation (4), we get

k p Ψq

(z) =

∞ X

p Q

k

ai +αi n −1 k

i=1

n=0

q Q

in Γ( ai +α )z n k

, Γk (bj + βj n)n!

j=1

this can be written as in the form

k p Ψq

(z) =

∞ X n=0

p Q i=1 q Q

k

ai +αi n −1 k

z n Z∞

Γk (bj + βj n)n!

j=1

0

e−t t

ai +αi n −1 k

dt,

ON GENERALIZATION OF K-WRIGHT FUNCTIONS AND ITS PROPERTIES

7

above equation reduces to,

k p Ψq

(z) =

Z∞ Y p

(kt)

0 i=1

ai k

−1 −t

e

p Q

∞ X n=0

(kt)

nαi k

i=1 q Q

Γk (bj + βj n)

zn dt, n!

j=1

Finally, we arrived at k p Ψq



" # p p p  P P P a a a ∞ ( ki )−p R ( pi ) ( ki )−p (ai , kαi )1,p k −t ¯ i=1 z = k i=1 z dt e ti=1 0 Ψq (bj , βj )1,q (kt) (bj , kβj )1,q 0

Particular Cases: For suitable values of parameters the results obtained in references Dorrego and Cerutti [2] , Gehlot [3] are special cases of this paper.

Acknowledgement: Authors are grateful to reviewers for their valuable suggestions and comments from the betterment of paper and also thankful to Mr. Krunal Kachhia (Lecturer, CHARUSAT)for LaTeX typing of paper.

References [1] Diaz, R. and Pariguan, E. On hypergeometric functions and Pochhammer k-symbol. Divulgaciones Mathematicas, Vol. 15 No. 2 (2007) 179-192. [2] Dorrego, G.A. and Cerutti, R.A. The K-Mittag-Leffler Function. Int. J. Contemp. Math. Sciences, Vol. 7 (2012) No. 15, 705-716. [3] Gehlot K.S. The generalized K- Mittag-Leffler function, Int. J. Contemp. Math. Sciences, Vol. 7 (2012) No. 45, 2213-2219. [4] Kilbas, A.A. Fractional Calculus of the Generalized Wright function, FCAA, Vol.(8), No 2 (2005), 113-126. [5] Mittag-Leffler, G.. Sur la nouvellefonction . C.RAcad. Sci. Paris 137 (1903) 554-558. [6] Prabhakar, T. R. A singular integral equation with a generalized Mittag- Leffler function in the kernel. Yokohama Math. J. 19 (1971), 7-15. [7] Samko, S. G., Kilbas, A.A. and Marichev,O.I. ,Fractional Integral and Derivatives, Theory and Applications. Gordon Breach, Yverdon et al. (1993). [8] Shukla, A. K. and Prajapati, J.C.. On the generalization of Mittag-Leffler function and its properties. Journal of Mathematical Analysis and Applications, 336 (2007) 797-811. [9] Wiman, A. Uber den fundamental Satz in der Theories der Funktionen . Acta Math. 29 (1905) 191-201. [10] Wright, E.M. The asymptotic expansion of the generalized hypergeometric function. J. London Math. Soc. 10 (1935), 287-293.

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KULDEEP SINGH GEHLOT AND JYOTINDRA C. PRAJAPATI

Kuldeep Singh Gehlot Department of Mathematics, Government Bangur College, Pali-306401, Rajasthan, India. E-mail address: [email protected] Jyotindra C. Prajapati Charotar University of Science and Technology, Changa, Anand-388421, Gujarat, India. E-mail address: [email protected]