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N-FRACTIONAL CALCULUS OF SOME PRODUCTS OF SOME FUNCTION
Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and its Applications Porto, Portugal, July 19-21, 2006
N-FRACTIONAL CALCULUS OF SOME PRODUCTS OF SOME FUNCTION Katsuyuki Nishimoto ∗ ∗
Institute for Applied Mathematics, Descartes Press Co. 2- 13- 10 Kaguike, Koriyama, 963 - 8833, JAPAN Fax : + 81 - 24 - 922 - 7596
Abstract: In this article, N- fractional calculus of products of power functions (z − c)α and (z − c)β , (z − c = 0) and that of products of logarithmic functions (log(z − c))−α , and (log(z − c))−β , (z − c = 0, 1) are discussed . Keywords: Fractional Calculus, N- Fractional Calculus Operator, N-Fractional Calculus of Products
O ≤ arg(ζ − z) ≤ 2π f or C+ ,
1. INTRODUCTION ( DEFINITION OF FRACTIONAL CALCULUS )
ζ = z, z ∈ C, ν ∈ R,
(I) Definition. (Nishimoto, 1984)
Γ; Gamma f unction,
Let D = {D− , D+ } , C = {C− , C+ }, where C− be a curve along the cut joining two points z and −∞+iIm(z), C+ be a curve along the cut joining two points z and ∞ + iIm(z), D− be a domain surrounded by C− , D+ be a domain surrounded by C+ ( Here D contains the points over the curve C ). Moreover, let f = f (z) be a regular function in D(z ∈ D) , fν = (f )ν = C (f )ν Γ(ν + 1) f (ζ)dζ = 2πi (ζ − z)ν+1
then (f )ν is the fractional differintegration of arbitrary order ν ( derivatives of order ν for ν > 0, and integrals of order −ν for ν < 0 ), with respect to z , of the function f , if |(f )ν | < ∞. (II) On the fractional calculus operator N ν (Nishimoto, 1991) Theorem A. Let fractional calculus operator ( Nishimoto’s Operator ) N ν be Nν = (
(ν ∈ / Z − ), (1)
ν→−m
(2)
C
dζ ) (ζ − z)ν+1
with N −m = lim N ν ν→−m
where −π ≤ arg(ζ − z) ≤ π f or C− ,
(ν ∈ / Z − ), (3)
(Ref er to (N ishimoto, 1984))
C
(f )−m = lim (f )ν (m ∈ Z + ),
Γ(ν + 1) 2πi
(m ∈ Z + ),
and define the binary operation ◦ as
(4)
N β ◦ N α f = N β N α f = N β (N α f )
(5)
(α, β ∈ R),
2. N- FRACTIONAL CALCULUS OF PRODUCTS OF SOME POWER FUNCTIONS In the following α, β, γ ∈ R , for our convenience.
then the set {N ν } = {N ν |ν ∈ R}
(6)
is an Abelian product group ( having continuous index ν ) which has the inverse transform operator (N ν )−1 = N −ν to the fractional calculus operator N ν , for the function f such that f ∈ F = {f ; 0 = |fν | < ∞, ν ∈ R}, where f = f (z) and z ∈ C. ( vis. −∞ < ν < ∞ ). β
Theorem 1. Let P = P (α, β, γ) :=
sinπα · sinπ(γ − α − β) (1) sinπ(α + β) · sinπ(γ − α)
(|P (α, β, γ)| = M < ∞) and Q = Q(α, β, γ) := P (β, α, γ)
( For our convenience, we call N ◦ N as product of N β and N α ) ν
Theorem B. ” F.O.G. {N }) ” is an ” Action product group which has continuous index ν ” for the set of F . ( F.O.G. ; Fractional calculus operator group )
(|P (β, α, γ)| = M < ∞) When α, β, γ ∈ / Z0+ , we have ; (i) ((z − c)α · (z − c)β )γ = e−iπγ P (α, β, γ)
Γ(γ − α − β) Γ(−α − β)
×(z − c)α+β−γ .
(3)
Theorem C. Let
(Re(α + β + 1) > 0, (1 + α − γ) ∈ / Z0− ),
S := {± N ν } ∪ {0} = {N ν } ∪ {− N ν } ∪ {0} . (7) (ν ∈ R) Then the set S is a commutative ring for the function f ∈ F , when the identity Nα + Nβ = Nγ
(N α , N β , N γ ∈ S)
(ii) ((z − c)β · (z − c)α )γ = e−iπγ Q(α, β, γ)
Γ(γ − α − β) Γ(−α − β)
×(z − c)α+β−γ .
(4)
(8)
holds. (Nishimoto, 2003) (III)Lemma. We have (Nishimoto, 1984) (i) ((z − c)β )α = e−iπα
(2)
α
Γ(α − β) (z − c)β−α Γ(−β) Γ(α − β) (| | ≤ ∞) Γ(−β)
(Re(α + β + 1) > 0, (1 + β − γ) ∈ / Z0− ), Γ(γ − α − β) (iii) ((z − c)α+β )γ = e−iπγ (z − c)α+β−γ .(5) Γ(−α − β) where Γ(γ − α − β) z − c = 0, | | < ∞. Γ(−α − β) Proof of ( i ). We have ((z − c)α · (z − c)β )γ =
∞ k=0
Γ(γ + 1) k!Γ(γ + 1 − k)
× ((z − c)α )γ−k ((z − c)β )k
(ii) (log(z − c))α = −eiπα Γ(α)(z − c)−α (|Γ(α)| ≤ ∞)
(6) by Lemma (iv). Next we have
(iii) ((z − c)−α )−α
1 = −eiπα log(z − c), Γ(α) (|Γ(α)| ≤ ∞)
where z − c = 0 in (i), and z − c = 0, 1 in (ii) and (iii) , (iv) (u · v)α :=
∞ k=0
((z − c)α )γ−k = e−iπ(γ−k)
(|
Γ(γ − k − α) | < ∞). Γ(−α)
and ((z − c)β )k = e−iπk
Γ(α + 1) uα−k vk k!Γ(α + 1 − k) (u = u(z), v = v(z))
Γ(γ − k − α) (z − c)α−γ+k (7) Γ(−α)
Γ(k − β) (z − c)β−k Γ(−β)
by Lemma (i), respectively. Substitute (7) and (8) into (6), we have then
(8)
((z − c)α · (z − c)β )γ ∞ Γ(γ + 1)Γ(γ − α − k)Γ(k − β) = e−iπγ k!Γ(γ + 1 − k)Γ(−α)Γ(−β) k=0 α+β−γ
×(z − c) . Γ(γ − α) = e−iπγ (z − c)α+β−γ Γ(−α) ∞ [−β]k [−γ]k × k![1 + α − γ]k
(9)
(10)
k=0
using the relationship Γ(λ + 1 − k) = (−1)−k
Γ(λ + 1)Γ(−λ) Γ(k − λ)
((z − c)α+β )γ = e−iπγ
×(z − c)α+β−γ Γ(γ − α − β) | < ∞) Γ(−α − β) directly by Lemma (i). (|
Note. (u · v)α , (which have · ) means the calculation by the use of generalized Leibniz rule, that is, the use of Lemma (iv).
(11)
[λ]k = λ(λ + 1) · · · (λ + k − 1) = Γ(λ + k)/Γ(λ),
3. N- FRACTIONAL CALCULUS OF PRODUCTS OF SOME LOGARITHMIC FUNCTIONS Theorem 2. When α, β, γ ∈ / Z0+ , we have ;
with [λ]0 = 1. (notation of P ochhammer).
(i)
Therefore, applying the following relationships
k=0
k![c]k
(((log(z − c))−α · (log(z − c))−β )γ = eiπ(α+β−γ) P (α, β, γ)T (α, β, γ)(z − c)α+β−γ
= 2 F1 (a, b; c; 1) =
(1)
Γ(c)Γ(c − a − b) Γ(c − a)Γ(c − b)
(Re(α + β + 1) > 0, (1 + α − γ) ∈ /
(12)
(((log(z − c))−β · (log(z − c))−α )γ
and
= eiπ(α+β−γ) Q(α, β, γ)T (α, β, γ)(z − c)α+β−γ
(λ ∈ / Z)
(13)
(2) (Re(α + β + 1) > 0, (1 + β − γ) ∈ /
to (10), we obtain
×(z − c)α+β−γ
(iii) (((log(z − c))−α · (log(z − c))−β )γ = eiπ(α+β−γ) T (α, β, γ)(z − c)α+β−γ (3) where z − c = 0, 1, T (α, β, γ) = B(−α, −β)Γ(γ − α − β) (|T (α, β, γ)| = M < ∞),
(15) Z0− )
and P (α, β, γ) and Q(α, β, γ) are the ones shown by (1) and (2) , respectively. ( B(* ,* ) ; beta function ). (16)
Proof of ( i ). We have (log(z − c))−α = −eiπα Γ(−α)(z − c)α
We have then (3) from (16) using the notation (1), under the conditions. Proof of ( ii ) . In the same way as the proof of (i) we obtain (4) using (2) instead of (1), under the conditions. ( Simply, changing α and β in (3) we obtain (4), using (2). Proof of ( iii ). We have
Z0− ),
and
((z − c)α · (z − c)β )γ Γ(γ − α) = e−iπγ (z − c)α+β−γ Γ(−α) ×2 F1 (−β, −γ; 1 + α − γ; 1) (14) Γ(γ − α)Γ(1 + α + β)Γ(1 + α − γ) = e−iπγ Γ(−α)Γ(1 + α)Γ(1 + α + β − γ) (Re(α + β + 1) > 0, (1 + α − γ) ∈ / sinπα · sinπ(γ − α − β) = e−iπγ sinπ(α + β) · sinπ(γ − α) Γ(γ − α − β) × (z − c)α+β−γ . Γ(−α − β)
Z0− ),
(ii)
(Re(c − a − b) > 0, c ∈ / Z0− ) π Γ(λ)Γ(1 − λ) = sinπλ
(17)
(uv)α (without · ) means the calculation without the use of generalized Leibniz rule.
where
∞ [a]k [b]k
Γ(γ − α − β) Γ(−α − β)
(4)
(α ∈ / Z0+ ) and (log(z − c))−β = −eiπβ Γ(−β)(z − c)β (β ∈ / Z0+ ) by Lemma (ii) , respectively, hence we obtain
(5)
(((log(z − c))−α · (log(z − c))−β )γ iπ(α+β)
=e
α
β
Γ(−α)Γ(−β)((z − c) · (z − c) )γ(6)
from (4) and (5). Therefore, applylng (3) of section [1] to (6) we obtain (i) clearly, under the conditions. Proof of ( ii ). In the same way described as above, we obtain (ii) under the conditions stated before. Or change the order (log(z − c))α , and (log(z − c))β in (6), then applying (4) in section [1] to it we obtain (ii) clearly, under the conditions. Proof of ( iii ). We have (((log(z − c))−α (log(z − c))−β )γ = eiπ(α+β) Γ(−α)Γ(−β)((z − c)α (z − c)β )γ(7) = eiπ(α+β) Γ(−α)Γ(−β)((z − c)α+β )γ
(8)
applying (4) and (5). Therefore, we obtain (iii) from (8) and (5) of section [1], under the conditions.
REFERENCES Hilfer, R.(Ed.) (2000). Applications of Fractional Calculus in Physics. World Scienfic,Singapor, New Jersey, London, Hong Kong. Kiryakova, V. (1994) Generalized fractional calculus and applications .Pitman Research Notes. No. 301. Longman. Lin,S.D., Shih- Tong Tu, Tsai- Ming Hsieh and H.M. Srivastava (2002). Some Finite and Infinite Sums Associated with the Digamma and Related Functions. J. Frac.Calc .Vol. 22, Nov., pp.103- 114. McBride, A.C. (1996). Fractional Calculus and Integral Transforms of Generalized Functions. Research Notes. No. 31. Pitman. Miller, K.S. and B. Ross (1993). An Introduction to The Fractional Calculus and Fractional Differential Equations. John Wiley & Sons. Nishimoto, K. (1984). Fractional Calculus, Vol. 1. Descartes Press, Koriyama, Japan. Nishimoto, K. (1987). Fractional Calculus, Vol. 2. Descartes Press, Koriyama, Japan. Nishimoto, K. (1989). Fractional Calculus, Vol. 3. Descartes Press, Koriyama, Japan. Nishimoto, K. (1991). Fractional Calculus, Vol. 4. Descartes Press, Koriyama, Japan. Nishimoto, K. (1996). Fractional Calculus, Vol. 5. Descartes Press, Koriyama, Japan. Nishmoto, K. (1991). An Essence of Nishimoto’s Fractional Calculus (Calculus of the 21st Century);Integrals and Differentiations of Arbitrary Order. Descartes Press, Koriyama, Japan.
Nishmoto, K. (1993). On Nishlmoto’s fractional calculus operatorN ν ( On an action group ). J.Frac. Calc.. Vol. 4, Nov.. pp. 1 - 11. Nishmoto, K. (1994). Unification of the integrals and derivatives ( A serendipity in fractional calculus). J. Frac. Calc.. Vol. 6, Nov.. pp. l - 14. Nishmoto, K. (2003). Ring and Field Produced from The Set of N- Fractional Calculus Operator, J. Frac Calc.. Vol. 24, Nov.. pp.29 36. Nishmoto, K. Ding- Kuo Chyan, Shy- Der Lin and Shih- Tong Tu (2001). On some infinite sums derived by N- fractional calculus. J. Frac. Calc..Vo1. 20. pp.91 - 97. Nishmoto, K. (2002). N- Fractional Calculus of the Power and Logarithmic Functions and Some Identities ( Continue ). J. Frac. Calc..Vo1. 22, Nov. pp.59 - 65. Oldham, K.B. and J. Spanier (1974). The Fractional Calculus. Academic Press. Podlubny, I. (1999). Fractional Differential Equations. Academic Press. Samko, S.G., Kilbas and O.I. Marichev (1987). Fractional Integrals and Derivatives, and Some Their Applications. Nauka, USSR. Wang, P.Y., Tsu- Chen Wu and Shih- Tong Tu (2002). Some Infinite Sums via N- fractional calculus. J. Frac. Calc. Vo1. 21, May. pp.71 -77.
N-FRACTIONAL CALCULUS OF SOME PRODUCTS OF SOME FUNCTION Katsuyuki Nishimoto ∗ ∗
Institute for Applied Mathematics, Descartes Press Co. 2- 13- 10 Kaguike, Koriyama, 963 - 8833, JAPAN Fax : + 81 - 24 - 922 - 7596
Abstract: In this article, N- fractional calculus of products of power functions (z − c)α and (z − c)β , (z − c = 0) and that of products of logarithmic functions (log(z − c))−α , and (log(z − c))−β , (z − c = 0, 1) are discussed . Keywords: Fractional Calculus, N- Fractional Calculus Operator, N-Fractional Calculus of Products
O ≤ arg(ζ − z) ≤ 2π f or C+ ,
1. INTRODUCTION ( DEFINITION OF FRACTIONAL CALCULUS )
ζ = z, z ∈ C, ν ∈ R,
(I) Definition. (Nishimoto, 1984)
Γ; Gamma f unction,
Let D = {D− , D+ } , C = {C− , C+ }, where C− be a curve along the cut joining two points z and −∞+iIm(z), C+ be a curve along the cut joining two points z and ∞ + iIm(z), D− be a domain surrounded by C− , D+ be a domain surrounded by C+ ( Here D contains the points over the curve C ). Moreover, let f = f (z) be a regular function in D(z ∈ D) , fν = (f )ν = C (f )ν Γ(ν + 1) f (ζ)dζ = 2πi (ζ − z)ν+1
then (f )ν is the fractional differintegration of arbitrary order ν ( derivatives of order ν for ν > 0, and integrals of order −ν for ν < 0 ), with respect to z , of the function f , if |(f )ν | < ∞. (II) On the fractional calculus operator N ν (Nishimoto, 1991) Theorem A. Let fractional calculus operator ( Nishimoto’s Operator ) N ν be Nν = (
(ν ∈ / Z − ), (1)
ν→−m
(2)
C
dζ ) (ζ − z)ν+1
with N −m = lim N ν ν→−m
where −π ≤ arg(ζ − z) ≤ π f or C− ,
(ν ∈ / Z − ), (3)
(Ref er to (N ishimoto, 1984))
C
(f )−m = lim (f )ν (m ∈ Z + ),
Γ(ν + 1) 2πi
(m ∈ Z + ),
and define the binary operation ◦ as
(4)
N β ◦ N α f = N β N α f = N β (N α f )
(5)
(α, β ∈ R),
2. N- FRACTIONAL CALCULUS OF PRODUCTS OF SOME POWER FUNCTIONS In the following α, β, γ ∈ R , for our convenience.
then the set {N ν } = {N ν |ν ∈ R}
(6)
is an Abelian product group ( having continuous index ν ) which has the inverse transform operator (N ν )−1 = N −ν to the fractional calculus operator N ν , for the function f such that f ∈ F = {f ; 0 = |fν | < ∞, ν ∈ R}, where f = f (z) and z ∈ C. ( vis. −∞ < ν < ∞ ). β
Theorem 1. Let P = P (α, β, γ) :=
sinπα · sinπ(γ − α − β) (1) sinπ(α + β) · sinπ(γ − α)
(|P (α, β, γ)| = M < ∞) and Q = Q(α, β, γ) := P (β, α, γ)
( For our convenience, we call N ◦ N as product of N β and N α ) ν
Theorem B. ” F.O.G. {N }) ” is an ” Action product group which has continuous index ν ” for the set of F . ( F.O.G. ; Fractional calculus operator group )
(|P (β, α, γ)| = M < ∞) When α, β, γ ∈ / Z0+ , we have ; (i) ((z − c)α · (z − c)β )γ = e−iπγ P (α, β, γ)
Γ(γ − α − β) Γ(−α − β)
×(z − c)α+β−γ .
(3)
Theorem C. Let
(Re(α + β + 1) > 0, (1 + α − γ) ∈ / Z0− ),
S := {± N ν } ∪ {0} = {N ν } ∪ {− N ν } ∪ {0} . (7) (ν ∈ R) Then the set S is a commutative ring for the function f ∈ F , when the identity Nα + Nβ = Nγ
(N α , N β , N γ ∈ S)
(ii) ((z − c)β · (z − c)α )γ = e−iπγ Q(α, β, γ)
Γ(γ − α − β) Γ(−α − β)
×(z − c)α+β−γ .
(4)
(8)
holds. (Nishimoto, 2003) (III)Lemma. We have (Nishimoto, 1984) (i) ((z − c)β )α = e−iπα
(2)
α
Γ(α − β) (z − c)β−α Γ(−β) Γ(α − β) (| | ≤ ∞) Γ(−β)
(Re(α + β + 1) > 0, (1 + β − γ) ∈ / Z0− ), Γ(γ − α − β) (iii) ((z − c)α+β )γ = e−iπγ (z − c)α+β−γ .(5) Γ(−α − β) where Γ(γ − α − β) z − c = 0, | | < ∞. Γ(−α − β) Proof of ( i ). We have ((z − c)α · (z − c)β )γ =
∞ k=0
Γ(γ + 1) k!Γ(γ + 1 − k)
× ((z − c)α )γ−k ((z − c)β )k
(ii) (log(z − c))α = −eiπα Γ(α)(z − c)−α (|Γ(α)| ≤ ∞)
(6) by Lemma (iv). Next we have
(iii) ((z − c)−α )−α
1 = −eiπα log(z − c), Γ(α) (|Γ(α)| ≤ ∞)
where z − c = 0 in (i), and z − c = 0, 1 in (ii) and (iii) , (iv) (u · v)α :=
∞ k=0
((z − c)α )γ−k = e−iπ(γ−k)
(|
Γ(γ − k − α) | < ∞). Γ(−α)
and ((z − c)β )k = e−iπk
Γ(α + 1) uα−k vk k!Γ(α + 1 − k) (u = u(z), v = v(z))
Γ(γ − k − α) (z − c)α−γ+k (7) Γ(−α)
Γ(k − β) (z − c)β−k Γ(−β)
by Lemma (i), respectively. Substitute (7) and (8) into (6), we have then
(8)
((z − c)α · (z − c)β )γ ∞ Γ(γ + 1)Γ(γ − α − k)Γ(k − β) = e−iπγ k!Γ(γ + 1 − k)Γ(−α)Γ(−β) k=0 α+β−γ
×(z − c) . Γ(γ − α) = e−iπγ (z − c)α+β−γ Γ(−α) ∞ [−β]k [−γ]k × k![1 + α − γ]k
(9)
(10)
k=0
using the relationship Γ(λ + 1 − k) = (−1)−k
Γ(λ + 1)Γ(−λ) Γ(k − λ)
((z − c)α+β )γ = e−iπγ
×(z − c)α+β−γ Γ(γ − α − β) | < ∞) Γ(−α − β) directly by Lemma (i). (|
Note. (u · v)α , (which have · ) means the calculation by the use of generalized Leibniz rule, that is, the use of Lemma (iv).
(11)
[λ]k = λ(λ + 1) · · · (λ + k − 1) = Γ(λ + k)/Γ(λ),
3. N- FRACTIONAL CALCULUS OF PRODUCTS OF SOME LOGARITHMIC FUNCTIONS Theorem 2. When α, β, γ ∈ / Z0+ , we have ;
with [λ]0 = 1. (notation of P ochhammer).
(i)
Therefore, applying the following relationships
k=0
k![c]k
(((log(z − c))−α · (log(z − c))−β )γ = eiπ(α+β−γ) P (α, β, γ)T (α, β, γ)(z − c)α+β−γ
= 2 F1 (a, b; c; 1) =
(1)
Γ(c)Γ(c − a − b) Γ(c − a)Γ(c − b)
(Re(α + β + 1) > 0, (1 + α − γ) ∈ /
(12)
(((log(z − c))−β · (log(z − c))−α )γ
and
= eiπ(α+β−γ) Q(α, β, γ)T (α, β, γ)(z − c)α+β−γ
(λ ∈ / Z)
(13)
(2) (Re(α + β + 1) > 0, (1 + β − γ) ∈ /
to (10), we obtain
×(z − c)α+β−γ
(iii) (((log(z − c))−α · (log(z − c))−β )γ = eiπ(α+β−γ) T (α, β, γ)(z − c)α+β−γ (3) where z − c = 0, 1, T (α, β, γ) = B(−α, −β)Γ(γ − α − β) (|T (α, β, γ)| = M < ∞),
(15) Z0− )
and P (α, β, γ) and Q(α, β, γ) are the ones shown by (1) and (2) , respectively. ( B(* ,* ) ; beta function ). (16)
Proof of ( i ). We have (log(z − c))−α = −eiπα Γ(−α)(z − c)α
We have then (3) from (16) using the notation (1), under the conditions. Proof of ( ii ) . In the same way as the proof of (i) we obtain (4) using (2) instead of (1), under the conditions. ( Simply, changing α and β in (3) we obtain (4), using (2). Proof of ( iii ). We have
Z0− ),
and
((z − c)α · (z − c)β )γ Γ(γ − α) = e−iπγ (z − c)α+β−γ Γ(−α) ×2 F1 (−β, −γ; 1 + α − γ; 1) (14) Γ(γ − α)Γ(1 + α + β)Γ(1 + α − γ) = e−iπγ Γ(−α)Γ(1 + α)Γ(1 + α + β − γ) (Re(α + β + 1) > 0, (1 + α − γ) ∈ / sinπα · sinπ(γ − α − β) = e−iπγ sinπ(α + β) · sinπ(γ − α) Γ(γ − α − β) × (z − c)α+β−γ . Γ(−α − β)
Z0− ),
(ii)
(Re(c − a − b) > 0, c ∈ / Z0− ) π Γ(λ)Γ(1 − λ) = sinπλ
(17)
(uv)α (without · ) means the calculation without the use of generalized Leibniz rule.
where
∞ [a]k [b]k
Γ(γ − α − β) Γ(−α − β)
(4)
(α ∈ / Z0+ ) and (log(z − c))−β = −eiπβ Γ(−β)(z − c)β (β ∈ / Z0+ ) by Lemma (ii) , respectively, hence we obtain
(5)
(((log(z − c))−α · (log(z − c))−β )γ iπ(α+β)
=e
α
β
Γ(−α)Γ(−β)((z − c) · (z − c) )γ(6)
from (4) and (5). Therefore, applylng (3) of section [1] to (6) we obtain (i) clearly, under the conditions. Proof of ( ii ). In the same way described as above, we obtain (ii) under the conditions stated before. Or change the order (log(z − c))α , and (log(z − c))β in (6), then applying (4) in section [1] to it we obtain (ii) clearly, under the conditions. Proof of ( iii ). We have (((log(z − c))−α (log(z − c))−β )γ = eiπ(α+β) Γ(−α)Γ(−β)((z − c)α (z − c)β )γ(7) = eiπ(α+β) Γ(−α)Γ(−β)((z − c)α+β )γ
(8)
applying (4) and (5). Therefore, we obtain (iii) from (8) and (5) of section [1], under the conditions.
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