As a consequence of definitions (1.1)-(1.9) the following results hold:
Theorem 2.1.
If , Re(α) > 0, Re(β) > 0, Re(γ) > 0, Re(δ) > 0, Re(μ) > 0, Re(ν) > 0, Re(ρ) > 0, Re(σ) > 0 and p, q > 0 a n d q ≤ Re(α) + p, then
(2.1.1)
(2.1.2)
(2.1.3)
In particular,
(2.1.4)
(2.1.5)
(2.1.6)
Proof.
which is (2.1.1).
The proof of (2.1.2) can easily be followed from the definition (1.9). Now
which proves (2.1.3). □
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Substituting μ = ν, ρ = σ and p = 1 in (2.1.1) immediately leads to (2.1.4).
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Substituting μ = ν, ρ = σ and p = 1 in (2.1.2) immediately leads to (2.1.5).
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Putting μ = ν, ρ = σ and p = 1 in (2.1.3) immediately leads to (2.1.6).
Theorem 2.2.
If , Re(α) > 0, Re(β) > 0, Re(γ) > 0, Re(δ) > 0, Re(μ) > 0, Re(ν) > 0, Re(ρ) > 0, Re(σ) > 0, Re(w) > 0; a n d q ≤ Re(α) + p then for
(2.2.1)
(2.2.2)
In particular,
(2.2.3)
(2.2.4)
Proof.
From (1.9),
which is the proof of (2.2.1).
Again using (1.9) and term by term differentiation under the sign summation(which is possible in accordance with the uniform convergence of the series (1.9) in any compact set ), we have
which is the proof of (2.2.2). □
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Setting μ = ρ, ν = σ, in (2.2.1), we get (2.2.3).
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Setting μ = ρ, ν = σ, in (2.2.2), we get (2.2.4).
Theorem 2.3.
If , with relatively prime; and q < Re(α + p), then
(2.3.1)
(2.3.2)
Proof.
which proves (2.3.1). □
Corollary 2.3.
For μ = ν, ρ = σ, δ = p = 1, (2.3.1) reduces to the known result of Shukla and Prajapati Shukla and Prajapati (2007) (2.3.1).
Remark 2.3.
Setting μ = ν, ρ = σ and p = 1 in (2.3.1), we get (2.3.2).
Special Properties: Setting putting μ = ν, ρ = σ and p = q = δ = 1 in (2.3.1), we have
(2.3.3)
For β = γ = δ = q = 1 in (2.3.2), we have
(2.3.4)
Theorem 2.4.
If then
(2.4.1)
(2.4.2)
(2.4.3)
In particular,
(2.4.4)
(2.4.5)
(2.4.6)
(2.4.7)
(2.4.8)
Proof.
which proves (2.4.1).
Now change the variable from s to Then the L.H.S. of (2.4.2) becomes
which proves (2.4.2).
Now
which proves (2.4.3).
Putting q = δ = 1 and γ = q = δ = 1 in (2.4.1) and (2.4.3) yields (2.4.4) and (2.4.5) respectively. □