General Mathematics Vol. 18, No. 1 (2010), 71–80
On a general class of modified gamma
approximating operators 1
Vasile Miheşan
Abstract
By using the generalized gamma distribution we shall define the
(a,b)
general modified gamma transform Γ α,β,γ , a, b ∈ R from which we
obtain as a special case both general modified gamma operators of the
first and second kind. We obtain generalization of a several positive
linear operator, as a special case of this general gamma oper ators.
2000 Mathematics Subject Classification: 41A36.
Key words and phrases: Euler’s gamma distribution, generalized
gamma distribution, linear positive operators, generalized modified
gamma transform, modified gamma operators of the first and second kind.
1
Received 06 August, 2009
Accepted for publication (in revised form) 19 September, 2009
71
72
1
V. Miheşan
Introduction
In this paper we continue our earlier investigations [5], [6], [7], [8], [9] concerning to use Euler’s gamma distribution for constructing linear positive
operators.
In probability theory and statistics, the gamma distribution (G) is a two
parameters family of continuous probability distribution. The probability
density function (p.d.f.) of the gamma distribution can be expressed in
terms of the gamma function parametrized in terms of a shape parameter
α and an inverse scale parameter β = 1/θ, called a rate parameter
(1)
G(t; α, β) =
β α α−1 −βt
t e
for t > 0 and α, β > 0.
Γ(α)
The Weibull distribution (named after Naloddi Weibull) is a continuous
probability distribution given by
(2)
γ
W (t; β, γ) = γβ γ tγ−1 e−(βt) for t > 0,
where γ > 0 is the shape parameter and β > 0 is a rate parameter.
The most general form of the gamma distribution is the generalized
gamma distribution (GG). It was introduced by Stacy and Mihran [12], [13],
in order to combine the power of two distribution: the gamma distribution
(1) and the Weibull distribution (2).
The generalized gamma distribution is a three-parameter distribution,
with the probability density function (p.d.f.) given by
(3)
γβ αγ γα−1 −(βt)γ
GG(t; α, β, γ) =
t
e
Γ(α)
On a general class of modified gamma approximating operators
73
for t > 0, where α > 0 and γ > 0 are shape parameter and β > 0 is rate
parameter.
The moments of (3) can be shown to be
EGG (tk ) = β k
(4)
Γ (α + k/γ)
.
Γ(α)
The generalized gamma distribution is a flexible distribution and it includes as special cases several distributions: the exponential distribution,
the gamma distribution, the half normal distribution, the Levy distribution, the Weibull distribution and the log-normal distribution in limit case
(α tends to infinity). For more details for the generalized gamma distribution see also [1], [2], [3].
The generalized beta distribution (GB) was introduced by J.B. McDonald and Y.J. Xu [4]. It is five-parameter distribution, with the probability
density function (p.d.f.) given by
q−1
tγp−1 (1 − (1 − c) (t/d)γ )
GB(t; γ, c, d, p, q) =
p+q
dγp B(p, q) (1 + c (t/d)γ )
(5)
for 0 < tγ < dγ /(1 − c) and zero otherwise, with 0 ≤ c ≤ 1 and γ, d, p, q,
positive, γ ∈ R∗ .
The moments of (5) can be shown to be [4]


k k
p
+
,
B (p + k/γ, q)
γ γ

(6)
EGB (tk ) = dk
; c
2 F1
k
B(p, q)
p+q+ γ
where
2 F1
denotes the hypergeometric series which converges for all k if
c < 1, or for kγ < q if c = 1. Substituting k = 0 into (6) verifies that (5)
integrates to one.
74
V. Miheşan
The generalized beta distribution (GB) includes the generalized beta of
the first kind (GB1) and the generalized beta of the second kind (GB2),
corresponding to c = 0 and c = 1, (see [4]).
The generalized gamma is a limiting case of GB, a.e.
1 γ1
(7)
GG(t; α, β, γ) = lim GB t; γ, c, d = q , α, q .
q→∞
β
Hence, the generalized beta includes generalized gamma as a limiting
case for all admissible values of c. We obtain by (6)
EGG (tk ) = lim EGB (tk )
(8)
q→∞
k
q γ β k B (α + k/γ, q)
Γ (α + k/γ)
= lim
= βk
.
q→∞
B(α, q)
Γ(α)
By using the generalized gamma distribution we shall define the general
(a,b)
modified gamma transform Γα,β,γ , a, b ∈ R from which we obtain as a special
case both general modified gamma operators of the first and second kind.
We obtain generalization of a several positive linear operator, as a special
case of this general gamma operators.
2
The general modified gamma transform
By using (3) we define the general modified gamma transform of a function
f
(9)
(a,b)
Γα,β,γ f
β αγ
=γ
Γ(α)
Z
∞
γ
γ
tαγ−1 e−(βt) f (cta e−bt )dt
0
(a,b)
where α, β, γ > 0; a, b ∈ R and f ∈ L1,loc (0, ∞) such that Γα,β,γ |f | < ∞.
On a general class of modified gamma approximating operators
75
(a,b)
We determine c ∈ R such that Γα,β,γ e1 = e1 , that is
c=
Γ(α)x
(β γ + b)α+a/γ
·
αγ
β
Γ (α + a/γ)
and we obtain from (9) the (a, b)-general modified gamma operators
(a,b)
(10)
(Γα,β,γ f )(x)
β αγ
=γ
Γ(α)
Z
∞
t
αγ−1 −(βt)γ
e
0
f
(β γ + b)α+a/γ
Γ(α)
a −btγ
·
t e
x dt.
β αγ
Γ (α + a/γ)
(a,b)
One observe that Γα,β,γ is a positive linear operator.
(a,b)
Theorem 1 The moment of order k of the operator Γα,β,γ has the following
value
(11)
(a,b)
(Γα,β,γ ek )(x) =
xk
(β γ + b)k(α+a/γ) Γ (α + ka/γ) Γk−1 (α)
·
·
.
(β γ + kb)α+ka/γ
Γk (α + a/γ)
β αγ(k−1)
Proof. We have
(a,b)
=
=
=
=
=
=
(Γα,β,γ ek )(x)
γ
k
Z ∞
β αγ
(β + b)α+a/γ Γ(α)
Γ(α)
γα−1 −(βt)γ
a −btγ
t
e
·
t e
x dt
γ
Γ(α) 0
β αγ
Γ(α + a/γ)
Z ∞
γ
k(α+a/γ)
β αγ
Γk (α)
γ (β + b)
γ
γ
tγα−1 e−(βt)
·
tka e−kbt xk dt
kαγ
k
Γ(α) 0
β
Γ (α + a/γ)
Z ∞
αγ
γ
k(α+a/γ)
k
k
β
(β + b)
Γ (α)x
γ
γ
γ
·
· k
tγα+ka−1 e−(βt) −kbt dt
kαγ
Γ(α)
β
Γ (α + a/γ) 0
Z ∞
αγ
γ
k(α+a/γ)
ka
(β + b)
Γk (α)xk
β
γ
1/γ
γ
γ
·
· k
tγ (α+ γ )−1 e−((β +(kb) )t) dt
kαγ
Γ(α)
β
Γ (α + a/γ) 0
γ
k(α+a/γ)
k−1
(β + b)
Γ (α)xk
Γ (α + ka/γ) 1
·
γ
·
·
β αγ(k−1)
Γk (α + a/γ) (β γ + kb)α+ ka
γ
γ
(β γ + b)α+a/γ
(β γ + kb)
α+ ka
γ
·
Γ (α + ka/γ) Γk−1 (α)
xk
·
.
Γk (α + a/γ)
β αγ(k−1)
76
V. Miheşan
Consequently, we obtain
(a,b)
(12)
(Γα,β,γ e2 )(x) =
(β γ + b)2(α+a/γ) Γ (α + 2a/γ) Γ(α)xk
· 2
·
2a
Γ (α + a/γ)
β αγ
(β γ + 2b)α+ γ
and
(a,b)
Γα,β,γ ((t − x)2 ; x)
(13)
2a
(β γ + b)2(α+a/γ) Γ(α)Γ (α + 2a/γ) − β αγ (β γ + 2b)α+ γ Γ2 (α + a/γ) 2
=
·x .
β αγ (β γ + 2b)α + 2a/γΓ2 (α + a/γ)
3
The modified gamma first kind operators
If we put in (10) b = 0 we obtain the general modified gamma first kind
operators
(14)
(a)
(Γα,β,γ f )(x)
β αγ
=γ
Γ(α)
Z
γ
=
Γ(α)
Z
∞
t
αγ−1 −(βt)γ
e
f
0
Γ(α)(βt)a
x dt
Γ (α + a/γ)
or equivalent
(15)
(a)
(Γα,β,γ f )(x)
∞
u
αγ−1 −uγ
0
e
f
Γ(α)ua x
Γ (α + a/γ)
du
(a)
where f ∈ L1,loc (0, ∞) such that Γα,β,γ |f | < ∞.
(a)
We observe that Γα,β,γ does not depend on β and we may consider β = 1.
(a)
Corollary 1 The moment of order k of the operator Γα,β,γ has the following
value
(16)
(a)
(Γα,β,γ ek )(x) =
Γk−1 (α)Γ (α + ka/γ) k
x .
Γk (α + a/γ)
On a general class of modified gamma approximating operators
77
Proof. The result follows from (11) for b = 0.
Consequently, we obtain
Γ(α)Γ (α + 2a/γ) 2
x .
Γ2 (α + a/γ)
2
Γ(a)
α,γ ((t − x) ) =
(17)
For γ = 1 we obtain the modified gamma first kind operators (see [9])
and for α = 1 we obtain the modified Weibull first kind operators (see [11]).
Special cases
Case 1. If we consider a = 1 in (15) we obtain the modified gamma first
kind operator
(18)
γ
(Γα,γ f )(x) =
Γ(α)
Z
∞
t
αγ−1 −tγ
e
f
0
Γ(α)tx
Γ (α + 1/γ)
dt.
Corollary 2 The moment of order k of the operator Γα,γ has the following
value
(19)
(Γα,γ ek )(x) =
Γk−1 (α)Γ (α + k/γ) k
x .
Γk (α + 1/γ)
Proof. The result follows from (16) for a = 1.
We deduce
(20)
(Γα,γ e2 )(x) =
Γα,γ ((t − x)2 ; x) =
Γ(α)Γ (α + 2/γ) 2
x ,
Γ2 (α + 1/γ)
Γ(α)Γ (α + 2/γ) − Γ2 (α + 1/γ) 2
x .
Γ2 (α + 1/γ)
If we choose α = n, n ∈ N in (18) then we obtain the generalization of
the Post-Wider positive linear operator defined for f ∈ L1,loc (0, ∞) by (see
[9])
(21)
γ
(Pn,γ f )(x) =
Γ(n)
Z
∞
t
0
γn−1 −tγ
e
f
Γ(n)
tx dt.
Γ (n + 1/γ)
78
V. Miheşan
If we replace α = nx, n ∈ N in (18) then we obtain the generalization of
the Rathore positive linear operator defined for f ∈ L1,loc (0, ∞) by (see [9])
(22)
γ
(Rn,γ f )(x) =
Γ(nx)
Z
∞
t
γnx−1 −tγ
e
f
0
Γ(nx)tx
dt
Γ (nx + 1/γ)
Case 2. If we replace a = −1 in (15) we obtain the modified gamma
operator
(23)
e α,γ f )(x) = γ
(Γ
Γ(α)
Z
∞
t
αγ−1 −tγ
e
f
0
Γ(α)
x
·
Γ (α − 1/γ) t
dt.
e α,γ has the following
Corollary 3 The moment of order k of the operator Γ
value
(24)
Γk−1 (α)Γ (α − k/γ) k
e
x ,
(Γα,γ ek )(x) =
Γk (α − 1/γ)
0 ≤ k < αγ.
Proof. The result follows from (16) for a = −1.
We obtain
e α,γ e2 )(x) =
(Γ
Γ(α)Γ (α − 2/γ) 2
x
Γ2 (α − 1/γ)
2
e α,γ ((t − x)2 ; x) = Γ(α)Γ (α − 2/γ) − Γ (α − 1/γ) x2 .
Γ
Γ2 (α − 1/γ)
For α = n + 1, n ∈ N we obtain the generalization of the operator
introduced and studied by A. Lupaş and M. Müller
γ
(Gn,γ f )(x) =
Γ(n + 1)
Z
∞
t
0
(n+1)γ−1 −tγ
e
f
Γ(n + 1)
x
·
Γ (n + 1 − 1/γ) t
dt.
79
On a general class of modified gamma approximating operators
4
The modified gamma second kind
operators
If we choose in (10) a = 0 then we obtain the modified gamma second kind
operators
(25)
(b)
(Γα,β,γ f )(x)
β αγ
=γ
Γ(α)
Z
∞
t
αγ−1 −(βt)γ
e
f
0
βγ + b
βγ
α
e
−btγ
x dt
(b)
where f ∈ L1,loc (0, ∞) such that Γα,β,γ |f | < ∞.
(b)
Corollary 4 The moment of order k of the operator Γα,β,γ has the following
value
(b)
(26)
(Γα,β,γ ek )(x) =
xk
(β γ + b)kα
·
.
(β γ + kb)α β αγ(k−1)
Proof. The result follows from (11) for a = 0.
Consequently, we obtain
(b)
(27)
(28)
(Γα,β,γ (e2 )(x) =
(b)
Γα,β,γ ((t − x)2 ; x) =
(β γ + b)2α x2
·
(β γ + 2b)α β αγ
(β γ + b)2α − β αγ (β γ + 2b)α 2
·x .
β αγ (β γ + 2b)α
For γ = 1 we obtain the modified gamma second kind operators (see [9])
and for α = 1 we obtain the modified Weibull second kind operators (see
[11]).
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V. Miheşan
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Vasile Miheşan
Technical University of Cluj-Napoca
Department of Mathematics
400020 Cluj-Napoca, Romania
e-mail: Vasile.Mihesan@math.utcluj.ro