CERTAIN RESULTS INVOLVING A CLASS OF FUNCTIONS ASSOCIATED WITH THE
HURWITZ ZETA FUNCTION
R. K. RAINA and P. K. CHHAJED
Abstract. The purpose of this paper is to consider a new generalization of the Hurwitz zeta function. Generating
functions, Mellin transform, and a series identity are obtained for this generalized class of functions. Some of the results
are used to provide a further generalization of the Lambert transform. Relevance with various known results are depicted
invariably. Multivariable extensions are also pointed out briefly.
1.
Introduction and preliminaries
The generalized (Hurwitz’s) zeta function is defined by ([1], [2])
(1.1)
ζ(s, a) =
∞
X
(a + n)−s
(a 6= 0, −1, −2, . . . ; <(s) > 1),
n=0
so that when a = 1, we have
(1.2)
ζ(s, 1) =
∞
X
n−s = ζ(s),
n=0
2000 Mathematics Subject Classification. Primary 11M06, 11M35; Secondary 33C20.
Key words and phrases. Hurwitz zeta function, generating functions, Mellin transform, Gamma function.
The work of the first author was supported by Council for Scientific and Industrial Research, Government of India.
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where ζ(s) is the Riemann zeta function. The function Φ(x, s, a) extends (1.1) further, and this generalized
Hurwitz-Lerch zeta function [1, p. 316], is defined by
(1.3)
Φ(x, s, a)
=
∞
X
(a + n)−s xn
n=0
(a 6= 0, −1, −2, . . . , |x| < 1; <(s) > 1, when |x| = 1).
Evidently, we have
(1.4)
(1.5)
Φ(1, s, 1)
Φ(1, s, a)
=
=
ζ(s),
ζ(s, a).
The function Φ(x, s, a) has the integral representation
Z ∞
1
(1.6)
Φ(x, s, a) =
ts−1 e−at (1 − xe−t )−1 dt.
Γ(s) 0
(<(a) > 0; either |x| ≤ 1, x 6= 1, and <(s) > 0, or x = 1, <(s) > 1).
In the present paper we introduce a class of functions Θλb (x, α, a, b) which is defined by
Z ∞
−λ
1
(1.7)
Θλµ (x, α, a, b) =
tα−1 e−at−bt (1 − xe−t )−µ dt
Γ(α) 0
(λ > 0, µ ≥ 1, <(a) > 0, <(b) > 0 ; when <(b) = 0,
then either |x| ≤ 1 (x 6= 1), <(α) > 0, or x = 1, <(α))
The various results obtained in this paper include series representation, Mellin transform, and generating
functions involving the above class of functions (1.7). Some of the results are used to consider a new generalization
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of the Lambert transform. Relevance with several new and known results are pointed out. Multivariable extensions
are also briefly indicated in the concluding section.
Special Cases of (1.7)
(i) When λ = µ = 1, x = 1, we have
Θ11 (1, α, a, b)
(1.8)
= ζb (α, a)
Z ∞ α−1 −at−b/t
1
t
e
=
dt,
Γ(α) 0
(1 − e−t )
(0 ≤ a ≤ 1, <(b) > 0; b = 0, <(α) > 1)
where ζb (α,a) is the extended Hurwitz zeta function defined by [1, p. 308].
(ii) When λ = µ = 1, b = 0, we have
Θ11 (x, α, a, 0)
(1.9)
= Φ(x, α, a)
Z ∞ α−1 −at
t
e
1
dt,
=
Γ(α) 0 (1 − xe−t )
(<(a) > 0; either |x| ≤ 1 (x 6= 1), <(α) > 0 or x = 1, <(α) > 1)
where Φ(x, α, a) is the generalized zeta function defined by (1.6).
(iii) When λ = µ = 1, x = −1, a = 1, then
Θ11 (−1, α, 1, b) =
(1.10)
=
(1 − 21−α )ζb∗ (α, a)
Z ∞ α−1 −at−b/t
1
t
e
dt,
Γ(α) 0
(1 + e−t )
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where ζb∗ (α, a) is the extended Hurwitz zeta function defined by [1, p. 309].
(iv) When b = 0, then
Θλµ (x, α, a, 0) = φ∗µ (x, α, a)
Z ∞ α−1 −at
1
t
e
=
dt,
Γ(α) 0 (1 − xe−t )µ
(1.11)
(µ ≥ 1, <(a) > 0 ; either |x| ≤ 1 (x 6= 1), <(α) > 0, or x = 1, <(α) > µ)
which was studied recently by Goyal and Laddha [3].
The series representation of (1.11) is given by
Φ∗µ (x, α, a) =
(1.12)
∞
X
(µ)n xn
(a + n)α n!
n=0
(µ ≥ 1, <(a) > 0, <(α) > 0, |x| ≤ 1).
2.
Series repretentation and Mellin transform
In this section we first find series representation and the Mellin transform of the class of functions Θλb (x, α, a, b)
(defined by (1.7)).
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Making use of (1.7), changing the order of integration and summation (under the prescribed conditions stated
with (1.7)), we have
Z ∞
−λ
1
λ
Θµ (x, α, a, b) =
tα−1 e−at−bt (1 − xe−t )−µ dt
Γ(α) 0
!
Z ∞
∞
X
1
(−b)m −λm
α−1 −at
−t −µ
=
t
e (1 − xe )
t
dt
Γ(α) 0
m!
m=0
Z
∞
1 X (−b)m ∞ α−λm−1 −at
t
e (1 − xe−t )−µ dt.
=
Γ(α) m=0 m!
0
This gives the series representation as
(2.1)
Θλµ (x, α, a, b) =
∞
1 X Γ(α − λm)(−b)m ∗
Φµ (x, α − λm, a)
Γ(α) m=0
m!
(λ > 0, µ ≥ 1, <(a) > 0, <(b) ≥ 0, <(α) 6= νλ (ν ∈ N), |x| ≤ 1).
To find the Mellin transform of the function Θλµ (x, α, a, b), we have from the definition of the Mellin transform
[1, p. 10]:
ms {Θλµ (x, α, a, b)}
Z∞
=
bs−1 Θλµ (x, α, a, b)db
0
=
1
Γ(α)
Z∞
0

bs−1 
Z∞
t
α−1 −at−bt−λ

e
dt db
(1 − xe−t )µ
0
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=
1
Γ(α)
Z∞
0
=
Γ(s)
Γ(α)
Z∞
∞

Z
−λ
tα−1 e−at 
bs−1 e−bt db dt,
(1 − xe−t )µ
0
tα+λs−1 e−at
dt.
(1 − xe−t )µ
0
Thus
(2.2)
Γ(s)Γ(α + λs) ∗
ms Θλµ (x, α, a, b) =
Φµ (x, α + λs, a).
Γ(α)
(λ >0, µ ≥ 1, <(s) > 0, <(a) > 0; either |x| ≤ 1 (x 6= 1),
<(α) > −λ<(s), or x = 1, <(α) > 1 − λ<(s))
For s = 1 in (2.2), we at once have
Z∞
Γ(α + λ) ∗
(2.3)
Φµ (x, α + λ, a).
Θλµ (x, α, a, b)db =
Γ(α)
0
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3.
Generating relations
Using (2.1) and (1.12), we have
∞ X
α+k−1
=
=
=
=
=
k=0
∞ X
k
Θλµ (x, α + k, a, b)tk
∞
X
−bm
α+k−1 k
1
Γ(α + k − λm)Φ∗µ (x, α + k − λm, a)
t
k
Γ(α + k) m=0 m!
k=0
∞
∞ X
(−b)m X α + k − 1 Γ(α + k − λm) ∗
Φµ (x, α + k − λm, a)tk
m!
k
Γ(α
+
k)
m=0
k=0
∞
∞ ∞
m X
X
(−b)
α + k − 1 Γ(α + k − λm) k X
(µ)n xn
t
m!
k
Γ(α + k)
(a + n)α−λm+k n!
m=0
n=0
k=0
∞
∞
∞ X
X
(−b)m X
(µ)n xn
α + k − 1 Γ(α + k − λm)tk
m! n=0 (a + n)α−λm n!
k
Γ(α + k)(a + n)k
m=0
k=0
−(α−λm)
∞
∞
1 X Γ(α − λm)(−b)m X
(µ)n xn
t
1−
Γ(α) m=0
m!
(a + n)α−λm n!
a+n
n=0
=
∞
∞
1 X Γ(α − λm)(−b)m X
(µ)n xn
Γ(α) m=0
m!
(a + n − t)α−λm n!
n=0
=
∞
1 X Γ(α − λm)(−b)m ∗
Φµ (x, α − λm, a − t)
Γ(α) m=0
m!
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Hence
(3.1)
∞ X
α+k−1
k
k=0
=
Θλµ (x, α + k, a, b)tk
∞
m
1 X Γ(α − λm) ( - b)
Φ∗µ (x, α − λm, a − t)
Γ(α) m=0
m!
(λ > 0, µ ≥ 1, <(a) > 0, <(b) > 0, <(α) 6= ν<(λ) (ν ∈ N), |x| ≤ 1, |t| < |a|).
If we put λ = µ = 1, x = 1 in (3.1), then in view of (1.7), we have
∞ X
α+k−1
(3.2)
ζb (α + k, a)tk
k
k=0
∞
X
Γ(α − m)(−b)m
=
ζ(α − m, a − t),
m!
m=0
where ζb (x, a) is the extended Hurwitz zeta function defined by (1.8). The result [1, p. 321, Eqn. 7.220] corresponds to (3.2), when b = 0.
On the other hand, when λ = µ = 1, b = 0 in (3.1), then in view of (1.9) we receive
∞ X
α+k−1
(3.3)
Φ(x, α + k, a)tk = Φ(x, α, a − t)
k
k=0
(α 6= 1, |t| < |a|).
This result was given by Raina and Srivastava [6, p. 302].
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For λ = µ = 1, and x = −1 (3.1) in view of (1.10) and (1.11) gives
∞ X
α+k−1
k=0
k
ζb∗ (α + k, a) =
∞
X
1
bm
Φ(−1, α − m, a − t).
1 − 21−α m=0 (1 − α)m m!
Next, using (2.1) and (1.12), we have
∞ X
α + 2k − 1
k=0
=
2k
∞
X
k=0
=
Θλµ (x, α + 2k, a, b)t2k
∞
α + 2k − 1 2k X Γ(α + 2k − λm)(−b)m ∗
t
Φµ (x, α + 2k − λm, a)
2k
Γ(α + 2k)m!
m=0
∞
∞
∞
1 X (−b)m X Γ(α + 2k − λm) 2k X
(µ)i xi
t
Γ(α) m=0 m!
(2k)!
(a + i)α+2k−λm i!
i=0
k=0
=
=
∞
∞
∞
X
1 X Γ(α − λm)(−b)m X
(µ)i xi
(α − λm)2k t2k
Γ(α) m=0
m!
(a + i)α−λm i!
(a + i)2k (2k)!
i=0
k=0
"
−(α−λm) −(α−λm) #
∞
∞
1 X Γ(α − λm)(−b)m X
(µ)i xi
t
t
·
1+
+ 1−
Γ(α) m=0
m!
2(a + i)α−λm i!
a+i
a+i
i=0
=
∞
∞
i
1 X Γ(α − λm)(−b)m X (µ)i xi h
−(α−λm)
−(α−λm)
· (a + i + t)
+ (1 + i − t)
2Γ(α) m=0
m!
i!
i=0
=
∞
1 X Γ(α − λm)(−b)m ∗
Φµ (x, α − λm, a − t) + Φ∗µ (x, α − λm, a + t) .
2Γ(α) m=0
m!
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Thus
∞ X
α + 2k − 1
(3.4)
2k
k=0
=
Θλµ (x, α + 2k, a, b)t2k
∞
1 X Γ(α − λm)(−b)m ∗
Φµ (x, α − λm, a − t) + Φ∗µ (x, α − λm, a + t) ,
2Γ(α) m=0
m!
under the same conditions as stated with (3.1).
Setting λ = µ = 1, x = 1 in (3.4), then in view of (1.8) and (1.12), we have
∞ X
α + 2k − 1
(3.5)
2k
k=0
=
ζb (α + 2k, a)t2k
∞
1 X Γ(α − m)(−b)m
[ζ(α − m, a − t) + ζ(α − m, a + t)] .
2Γ(α) m=0
m!
The result [1, p. 32, Eqn. (7.223)] corresponds to (3.5) when a = 1, b = 0.
If we put λ = µ = 1 and x = −1 in (3.4), then in view of (1.10) and (1.11), we have
(3.6)
∞ X
α + 2k − 1
k=0
=
2k
ζb∗ (α + 2k, a)t2k
∞
X
1
Γ(α − m)(−b)m (1 − 21−α+m )
· [ζ0∗ (α − m, a − t) + ζ0∗ (α − m, a + t)] .
1−α
2Γ(α)(1 − 2
) m=0
m!
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Similarly we can establish the relation:
∞ X
α + 2k
(3.7)
k=0
=
2k + 1
Θλµ (x, α + 2k + 1, a, b)t2k+1
∞
1 X Γ(α − λm)(−b)m ∗
· Φµ (x, α − λm, a − t) − Φ∗µ (x, α − λm, a + t) ,
2Γ(α) m=0
m!
under the same condition as stated with (3.1).
Now we turn again to (2.1), and find a generating function which involves additional parameters. We have
∞
X
(θ)n (ϕ)n λ
tn
Θµ (x, θ + ϕ − υ + n, a, b)
(υ)n
n!
n=0
=
=
∞
∞
X
(θ)n (ϕ)n tn X Γ(θ + ϕ − υ − λm)(−b)m ∗
φµ (x, θ + ϕ − υ + n − λm, a)
(υ)n n! m=0
Γ(θ + ϕ − υ + n)m!
n=0
∞
X
∞
∞
X
Γ(θ + ϕ − υ + n − λm)(−b)m X
(µ)k xk
(θ)n (ϕ)n tn
·
(υ)n Γ(θ + ϕ − υ + n)n! m=0
m!
k!(a + k)θ+ϕ−υ+n−λm
n=0
k=0
=
=
∞
∞
X
(µ)k xk X
k=0
∞
X
m,k=0
k!
m
Γ(θ + ϕ − υ − λm)(−b)
Γ(θ
+ ϕ − υ)m!(a + k)θ+ϕ−υ−λm
m=0
∞
X
(θ)n (ϕ)n (θ + ϕ − υ − λm)n tn
·
(υ)n (θ + ϕ − υ)n (a + k)n n!
n=0
Γ(θ + ϕ − υ − λm)(µ)k (−b)m xk
θ,ϕ,θ+ϕ−υ−λm;
F
t/a
+
k
.
3
2
υ,θ+ϕ−υ
;
Γ(θ + ϕ − υ)(a + k)θ+ϕ−υ−λm m!k!
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Hence
(3.8)
∞
X
(θ)n (ϕ)n λ
tn
Θµ (x, θ + ϕ − υ + n, a, b)
(υ)n
n!
n=0
=
∞
X
1
Γ(θ + ϕ − υ)
m,k=0
Γ(θ + ϕ − υ − λm)(µ)k (−b)m xk
θ,ϕ,θ+ϕ−υ−λm;
·
F
t/a
+
k
3 2 υ,θ+ϕ−υ
;
(a + k)θ+ϕ−υ−λm m!k!
(<(θ + ϕ) > <(υ) > 0, <(θ + ϕ − υ) 6= rλ(r ∈ N), |t/a| < 1).
Substituting λ = µ = 1, x = 1 in (3.8), then in view of (1.9) we have
∞
∞
X
X
(θ)n (ϕ)n
tn
ςb (θ + ϕ − υ + n, a, ) =
(υ)n
n!
n=0
m,k=0
(θ + ϕ − υ)m (−b)m
θ,ϕ,θ+ϕ−υ−m;
F
t/a
+
k
3
2
υ,θ+ϕ−υ
;
(a + k)θ+ϕ−υ−m m!
(<(θ + ϕ) > <(υ) > 0, |t/a| < 1).
For λ = ν = 1, b = 0, (3.8) in view of (1.10) yields
∞
∞
X
X
(θ)n (ϕ)n
tn
xk
θ,ϕ;
Φ(x, θ + ϕ − υ + n, a) =
F
t/a
+
k
,
2
1
υ ;
(υ)n
n!
(a + k)θ+ϕ−υ
n=0
k=0
which is due to Raina and Srivastava [6, p. 302].
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Again using (2.1), we can express
Θλµ (x, α, a, b)
=
∞
1 X Γ(α − λm)(−b)m ∗
Φµ (x, α − λm, a)
Γ(α) m=0
m!
=
∞
∞
(µ)n−1 xn−1
1 X Γ(α − λm)(−b)m X
.
Γ(α) m=0
m!
(a + n − 1)α−λm (n − 1)!
n=1
Appealing to the series identity of Srivastva [5] (see also [1, p. 316, Eqn. 7.176])
∞
X
k=1
f (k) =
q X
∞
X
f (qk + j) (q ∈ N),
j=1 k=0
then (3.9) gives
Θλµ (x, α, a, b)
=
q
∞
∞
1 X Γ(α − λm)(−b)m X X (µ)nq+j−1
xnq+j−1
Γ(α) m=0
m!
Γ(nq + j) (a + nq + j − 1)α−λm
j=1 n=0
=
q
∞
∞
1 X Γ(α − λm)(−b)m q λm−α X X
Γ(α) m=0
m!
j=1 n=0
=
q
∞
1 X X Γ(α − λm)(−b)m q λm−α ∗
a+j−1
q
Φµ x , α − λm,
xj−1 .
Γ(α) m=0 j=1
m!
q
(µ)nq+j−1 xnq+j−1
α−λm ,
Γ(nq + j) a + nq+j−1
q
which yields
Θλµ (x, α, a, b)
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For b = 0, (3.9) reduces to the identity
(3.9)
Θλµ (x, α, a, 0) = q −α
q
X
a+j−1
Φ∗µ xq , α,
xj−1 ,
q
j=1
which when x = µ = 1 reduces to the following known identity ([5], see also [1, p. 317])
q
X
a+j−1
λ
−α
(3.10)
Θµ (1, α, a, 0) = ζ(α, a) = q
.
ζ α,
q
j=1
4.
Generalized Lambert Transform
Suppose f (t) (∀t ∈ [0, ∞)) is a continuous function such that
(4.1)
f (t) = O(ekt )
then, the Lambert transform of f (t) is defined by
Z ∞
(4.2)
F (s) = LM{f (t)} =
0
(t → ∞),
st
f (t)dt
− 1)
(est
(<(s) > 0).
In the papers [3], [4] and [6], various generalizations of the transform (4.2) were given.
We consider here a new generalization of (4.2) in the following form:
Z ∞
λ
k
(st) e−b(st)
(4.3)
G ∗ {f (t)} = GLMb {f (t)} =
f (t)dt,
(est − x)µ
0
(λ > 0, µ ≥ 1, <(s) > 0, f (t) ∈ A, <(b) > 0; when b = 0,
either |x| ≤ 1 (x 6= 1), <(k + γ) > −1, or x = 1, <(k + γ) > µ − 1)
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where A denotes the class of functions f (t) which are continuous for t > 0, and satisfy the order estimates:
(
O(tγ ), if t → 0+ ,
(4.4)
f (t) =
O(tδ ), if t → ∞.
Obviously for b = 0 and µ = 1, (4.3) becomes
Z
(4.5)
GLM{f (t)} =
0
∞
(st)k
f (t)dt,
(est − x)µ
(µ ≥ 1, <(s) > 0, f (t) ∈ A; either |x| ≤ 1 (x 6= 1),
<(γ + k) > −1 orx = 1, <(γ + k) > µ − 1)
which was studied in [6].
Put f (t) = tα−1 e−ast in (4.3), using (1.7), we have
Z ∞
−λ
∗ α−1 −ast
k
(4.6)
G {t
e
} = s
tα+k−1 e−ast−b(st) (1 − xe−st )−µ dt
0
=
Γ(α + k) λ
Θ (x, α + k, a + µ, b)
sα+k−1 µ
(λ > 0, µ ≥ 1, <(a) > 0, <(s) > 0, <(b) > 0; when b = 0,
then either |x| ≤ 1 (x 6= 1), <(α + k + γ) > −1, or x = 1, <(α + k + γ) > µ − 1).
Inversion formula for the transform (4.3)
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On applying the Mellin transform [7, p. 46], (4.3) gives
Z ∞
Z ∞
−λ
k
φ(m) =
s−m−1
(st) e−b(st) (est − x)−µ f (t)dt ds
0
Z 0∞
Z ∞
−λ
=
tk f (t)
sk−m−1 e−µst−b(st) (1 − xe−st )−µ ds dt.
0
0
In view of (1.7), this gives
(4.7)
φ(m) = Γ(k − m)Θλµ (x, k − m, µ, b)
Z
∞
tk+m−1 f (t)dt,
0
provided that <(m) < k, and the existence and convergence conditions stated with (4.3) hold true.
By the Mellin inversion [7, p. 46] theorem, we obtain the following inversion formula for the integral transform
(7.3):
Z σ+i∞
−1 −m
1
1
[f (t + 0) + f (t − 0)] =
Γ(k − m)Θλµ (x, k − m, µ, b)
t φ(m)dm,
2
2πi σ−i∞
provided that y m f (y) ∈ L(0, ∞), f (y) is of bounded variation in the neighbourhood of the point y = t, σ > 1/2,
<(k − m) > 0, and φ(m) is given by (4.7).
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5.
Concluding Remarks
In this concluding section we find it worthwhile to mention briefly here a multivariable extension of the class of
(λ,p )
functions Θλµ (x, α, a, b). This multivariable function Θ(µi )i (x1 , . . . , xn ; α; a, b) can be defined by
∞
(λ,p )
Θ(µi )i
(x1 , ..., xn ; α; a, b)
=
1 X Γ(α − λk) (−b)k
Γ(α)
k!
k=0
∞
X
(a + Ω)
−(α−λk)
m1, ...,mn =0
n i
Y
(µi )mi xm
i
,
mi !
i=1
(λ > 0, µi ≥ 1(i = 1, . . . , n), <(a) > 0, <(b) ≥ 0,
n
X
<(α) 6= νλ (ν ∈ N), max (|xi |) ≤ 1, Ω =
pi mi ).
1≤i≤n
i=1
(λ,p )
Equivalently, the integral representation of Θ(µi )i (x1 , ..., xn ; α; a, b) is given by
(λ,µ )
Θ(pi ) i (x1 , . . . , xn ; α; a, b)
=
1
Γ(α)
Z∞
−λ
tα−1 e−at−bt
n
Y
(1 − xi e−pi t )µi dt
i=1
0
(λ > 0, µi ≥ 1, <(pi ) > 0 (i = 1, . . . , n), <(a) > 0, <(b) > 0;
when b = 0, then either, max (|xi |) < 1 (xi 6= 1), <(α) > 0,
1≤i≤n
or xi = 1 (i = 1, . . . , n), <(α) > max (µi ))
1≤i≤n
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Corresponding to the above multivariable extension (5.1), we may also define a generalized Lambert transform
in the form
Z ∞
λ
(st)k e−b(st)
∗
(5.1)
H {f (t)} =
f (t)dt,
n
Q
0
(epi st − xi )µi
i=1
(λ > 0, µi ≥ 1, <(pi ) > 0 (i = 1, . . . , n), <(s) > 0, f (t) ∈ A, <(b) > 0;
when b = 0, either max (|xi |) ≤ 1 (xi 6= 1) (i = 1, . . . , n), <(k + γ) > −1,
1≤i≤n
or xi = 1 (i = 1, . . . , n), <(α) > max (µi − 1))
1≤i≤n
These obvious extensions can be manipulated in several ways and various results can be obtained by following
the same procedures as mentioned above (see also [4]). We do not pursue further, and skip further details in this
regard.
1. Chaudhry M. Aslam and Zubair Syed M., On a Class of Incomplete Gamma Functions with Applications, Chapman & Hall/CRC
(Boca Raton/London/NewYork/Washington D.C.) 2000.
2. Erdélyi A. and Magnus W., Oberhettinger F. and Tricomi F. G., Higher Transcendental Functions, Vol. 1, McGraw–Hill,
NewYork, Toronto and London, 1953.
3. Goyal S. P. and Laddha R. K., On the generalized Riemann zeta functions and the generalized Lambert transform, Ganita
Sandesh, 11(1997), 99–108.
4. Raina R. K and Nahar T. S., A note on certain class of functions related to Hurwitz zeta function and Lambert transform,
Tamkang J. Math. 31 (2000), 49–55.
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5. Srivastava H. M., Some formula for the Bernoulli and Euler polynomials at rational arguments, Math, Proc. Comb. Phil. Soc.129
(2000), 77–84.
6. Raina R. K. and Srivastava H. M., Certain results associated with the generalized Riemann zeta functions, Rev. Tec. Ing. Univ.
Zulia, 18 (3)(1995), 301–304.
7. Titchmarsh E. C., Introduction to the Theory of Fourier Integrals, Clarendon Press, Oxford, 1948.
R. K. Raina, Department of Mathematics, College of Technology and Engineering M. P. University of Agri. and Technology, Udaipur313001, Rajasthan, India
P. K. Chhajed, Department of Mathematics, College of Science, M. L. Sukhadia University, Udaipur-313001, Rajasthan, India
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