89
Acta Math. Univ. Comenianae
Vol. LXXIII, 1(2004), pp. 89–100
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
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).
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.
90
R. K. RAINA and P. K. CHHAJED
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 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
(1.8)
Θ11 (1, α, a, b) = 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
1
t
e
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 )
where ζb∗ (α, a) is the extended Hurwitz zeta function defined by [1, p. 309].
FUNCTIONS ASSOCIATED WITH THE HURWITZ ZETA FUNCTION
91
(iv) When b = 0, then
Θλµ (x, α, a, 0)
(1.11)
= φ∗µ (x, α, a)
Z ∞ α−1 −at
1
t
e
=
dt,
Γ(α) 0 (1 − xe−t )µ
(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)).
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
tα−1 e−at−bt (1 − xe−t )−µ dt
Θµ (x, α, a, b) =
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
∞

Z α−1 −at−bt−λ
t
e
bs−1 
dt db
(1 − xe−t )µ
0
92
R. K. RAINA and P. K. CHHAJED
=
1
Γ(α)
Z∞
∞

Z
tα−1 e−at 
s−1 −bt−λ 
b e
db dt,
(1 − xe−t )µ
0
=
Γ(s)
Γ(α)
Z∞
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∞
(2.3)
Θλµ (x, α, a, b)db =
Γ(α + λ) ∗
Φµ (x, α + λ, a).
Γ(α)
0
3. Generating relations
Using (2.1) and (1.12), we have
∞ X
α+k−1
Θλµ (x, α + k, a, b)tk
k
k=0
∞
∞ X
X
α+k−1 k
1
−bm
=
t
Γ(α + k − λm)Φ∗µ (x, α + k − λm, a)
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
∞
∞ ∞
X
(−b)m X 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!
FUNCTIONS ASSOCIATED WITH THE HURWITZ ZETA FUNCTION
93
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].
For λ = µ = 1, and x = −1 (3.1) in view of (1.10) and (1.11) gives
∞ X
α+k−1 ∗
(3.4)
ζb (α + k, a)
k
k=0
=
∞
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
α + 2k − 1
Θλµ (x, α + 2k, a, b)t2k
2k
k=0
∞ ∞
X
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
∞ X
k=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
=
∞
∞
∞
X
1 X Γ(α − λm)(−b)m X
(µ)i xi
(α − λm)2k t2k
α−λm
Γ(α) m=0
m!
(a + i)
i!
(a + i)2k (2k)!
i=0
k=0
k=0
94
R. K. RAINA and P. K. CHHAJED
=
=
=
∞
∞
1 X Γ(α − λm)(−b)m X
(µ)i xi
Γ(α) m=0
m!
2(a + i)α−λm i!
i=0
"
−(α−λm) −(α−λm) #
t
t
·
1+
+ 1−
a+i
a+i
∞
∞
1 X Γ(α − λm)(−b)m X (µ)i xi
2Γ(α) m=0
m!
i!
i=0
h
i
−(α−λm)
−(α−λm)
· (a + i + t)
+ (1 + i − t)
∞
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!
Thus
(3.5)
∞ X
α + 2k − 1
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 (3.5), then in view of (1.8) and (1.12), we have
(3.6)
∞ X
α + 2k − 1
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 (3.6) when a = 1, b = 0.
If we put λ = µ = 1 and x = −1 in (3.5), then in view of (1.10) and (1.11), we
have
(3.7)
∞ X
α + 2k − 1
k=0
=
2k
ζb∗ (α + 2k, a)t2k
∞
X
1
Γ(α − m)(−b)m (1 − 21−α+m )
2Γ(α)(1 − 21−α ) m=0
m!
· [ζ0∗ (α − m, a − t) + ζ0∗ (α − m, a + t)] .
FUNCTIONS ASSOCIATED WITH THE HURWITZ ZETA FUNCTION
95
Similarly we can establish the relation:
∞ X
α + 2k
(3.8)
Θλµ (x, α + 2k + 1, a, b)t2k+1
2k + 1
k=0
=
∞
1 X Γ(α − λm)(−b)m
2Γ(α) m=0
m!
· Φ∗µ (x, α − λm, a − t) − Φ∗µ (x, α − λm, a + t) ,
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
(θ)n (ϕ)n tn
(υ)n Γ(θ + ϕ − υ + n)n! m=0
m!
n=0
·
∞
X
(µ)k xk
k!(a + k)θ+ϕ−υ+n−λm
k=0
=
∞
∞
X
(µ)k xk X
k=0
∞
X
·
=
k!
Γ(θ + ϕ − υ − λm)(−b)m
Γ(θ + ϕ − υ)m!(a + k)θ+ϕ−υ−λm
m=0
(θ)n (ϕ)n (θ + ϕ − υ − λm)n tn
(υ)n (θ + ϕ − υ)n (a + k)n n!
n=0
∞
X
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
;
Γ(θ + ϕ − υ)(a + k)θ+ϕ−υ−λm m!k!
Hence
(3.9)
∞
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 − λm)(µ)k (−b)m xk
Γ(θ + ϕ − υ)
(a + k)θ+ϕ−υ−λm m!k!
m,k=0
θ,ϕ,θ+ϕ−υ−λm;
·3 F2 υ,θ+ϕ−υ
t/a
+
k
;
(<(θ + ϕ) > <(υ) > 0, <(θ + ϕ − υ) 6= rλ(r ∈ N), |t/a| < 1).
96
R. K. RAINA and P. K. CHHAJED
Substituting λ = µ = 1, x = 1 in (3.9), then in view of (1.9) we have
(3.10)
∞
X
(θ)n (ϕ)n
tn
ςb (θ + ϕ − υ + n, a, )
(υ)n
n!
n=0
∞
X
=
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, (3.9) in view of (1.10) yields
(3.11)
∞
X
(θ)n (ϕ)n
tn
Φ(x, θ + ϕ − υ + n, a)
(υ)n
n!
n=0
=
∞
X
k=0
xk
θ,ϕ;
F
t/a
+
k
,
2
1
υ ;
(a + k)θ+ϕ−υ
which is due to Raina and Srivastava [6, p. 302].
Again using (2.1), we can express
(3.12)
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 (3.12) 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
(µ)nq+j−1 xnq+j−1
α−λm ,
Γ(nq + j) a + nq+j−1
q
which yields
(3.13) Θλµ (x, α, a, b)
=
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
FUNCTIONS ASSOCIATED WITH THE HURWITZ ZETA FUNCTION
97
For b = 0, (3.13) reduces to the identity
(3.14)
Θλµ (x, α, a, 0)
=q
−α
q
X
j=1
Φ∗µ
a+j−1
x , α,
q
q
xj−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
(3.15)
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
f (t) = O(ekt )
(4.1)
(t → ∞),
then, the Lambert transform of f (t) is defined by
Z ∞
st
(4.2)
f (t)dt
F (s) = LM{f (t)} =
st − 1)
(e
0
(<(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)
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 ∞
(st)k
GLM{f (t)} =
(4.5)
f (t)dt,
(est − x)µ
0
(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].
98
R. K. RAINA and P. K. CHHAJED
Put f (t) = tα−1 e−ast in (4.3), using (1.7), we have
Z ∞
−λ
(4.6)
G ∗ {tα−1 e−ast } = sk
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)
On applying the Mellin transform [7, p. 46], (4.3) gives
Z ∞
Z ∞
k −b(st)−λ st
−m−1
−µ
s
(st) e
(e − x) f (t)dt ds
φ(m) =
0
Z 0∞
Z ∞
k
k−m−1 −µst−b(st)−λ
−st −µ
=
t f (t)
s
e
(1 − xe ) ds dt.
0
0
In view of (1.7), this gives
φ(m) = Γ(k − m)Θλµ (x, k − m, µ, b)
(4.7)
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):
1
[f (t + 0) + f (t − 0)]
2
Z σ+i∞
−1 −m
1
(4.8)
=
Γ(k − m)Θλµ (x, k − m, µ, b)
t φ(m)dm,
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).
5. Concluding Remarks
In this concluding section we find it worthwhile to mention briefly here a multivariable extension of the class of functions Θλµ (x, α, a, b). This multivariable function
(λ,p )
Θ(µi )i (x1 , . . . , xn ; α; a, b) can be defined by
(λ,p )
Θ(µi )i (x1 , ..., xn ; α; a, b)
∞
(5.1) =
1 X Γ(α − λk) (−b)k
Γ(α)
k!
k=0
∞
X
m1, ...,mn =0
(a + Ω)
−(α−λk)
n i
Y
(µi )mi xm
i
,
mi !
i=1
FUNCTIONS ASSOCIATED WITH THE HURWITZ ZETA FUNCTION
99
(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
Equivalently, the integral representation of
(λ,p )
Θ(µi )i
(x1 , ..., xn ; α; a, b) is given by
(λ,µ )
Θ(pi ) i (x1 , . . . , xn ; α; a, b)
(5.2)
=
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
Corresponding to the above multivariable extension (5.2), we may also define a generalized Lambert transform in the form
Z ∞
λ
(st)k e−b(st)
∗
H {f (t)} =
(5.3)
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.
References
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.
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.
100
R. K. RAINA and P. K. CHHAJED
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, Udaipur-313001, Rajasthan, India
P. K. Chhajed, Department of Mathematics, College of Science, M. L. Sukhadia University,
Udaipur-313001, Rajasthan, India