A Quadruple Integral Involving Product of the Struve HHH v ( β t ) and Parabolic Cylinder D u ( α x ) Functions

: The objective of the present paper is to obtain a quadruple inﬁnite integral. This integral involves the product of the Struve and parabolic cylinder functions and expresses it in terms of the Hurwitz–Lerch Zeta function. Almost all Hurwitz-Lerch Zeta functions have an asymmetrical zero distributionSpecial cases in terms fundamental constants and other special functions are produced. All the results in the work are new.


Introduction
In this paper we derive the quadruple definite integral given by: where the parameters k, a, α, β, u, v, and m are general complex numbers and Re(u) < Re(m) < 1/2, Re(m) < Re(v). This definite integral will be used to derive special cases in terms of special functions and fundamental constants. The derivations follow the method used by us in [3]. This method involves using a form of the generalized Cauchy's integral formula given by: y k Γ(k + 1) = 1 2πi C e wy w k+1 dw. (2) where C is, in general, an open contour in the complex plane where the bilinear concomitant has the same value at the end points of the contour. We then multiply both sides by a function of x, y, z, and t, then take a definite quadruple integral of both sides. This yields a definite integral in terms of a contour integral. Then, we multiply both sides of Equation (2) by another function of x, y, z, and t and take the infinite sums of both sides such that the contour integral of both equations are the same.

Definite Integral of the Contour Integral
We used the method in [3]. The variable of integration in the contour integral is r = w + m. The cut and contour are in the first quadrant of the complex r-plane. The cut approaches the origin from the interior of the first quadrant; the contour goes round the origin with zero radius and is on opposite sides of the cut. Using a generalization of Cauchy's integral formula, we form the quadruple integral by replacing y with log , then taking the definite integral with respect to x ∈ (0, ∞), y ∈ (0, ∞), z ∈ (0, ∞), and t ∈ (0, ∞) to obtain from Equation (3.326.2) in [4], equations (3.9.1.3) and (3.15.1) in [5] where Re(β) > 0, 0 < Re(w + m) < 3/2, |Re(w + m + β)| < 1, | arg α| < π/4, and using the reflection formula (8.334.3) in [4] for the Gamma function. We are able to switch the order of integration over x, y, z, and t using Fubini's theorem for multiple integrals see (9.112) in [6], as the integrand is of bounded measure over the space

The Hurwitz-Lerch Zeta Function and Infinite Sum of the Contour Integral
In this section we use Equation (2) to derive the contour integral representations for the Hurwitz-Lerch Zeta function.

Infinite Sum of the Contour Integral
Using Equation (2) and replacing y with log(a) + log(α) 2 (2)  4 , then multiplying both sides by taking the infinite sum over y ∈ [0, ∞), and simplifying in terms of the Hurwitz-Lerch Zeta function we obtain (6) from Equation (1.232.2) in [4] where Im( π 2 (m + w)) > 0 in order for the sum to converge.

Definite Integral in Terms of the Hurwitz-Lerch Zeta Function
Proof. The right-hand sides of relations (3) and (6) are identical; hence, the left-hand sides of the same are identical too. Simplifying with the Gamma function yields the desired conclusion.

The Invariance of Indices u and v Relative to the Hurwitz-Lerch Zeta Function
In this section, we evaluate Equation (7) such that the indices of the Struve H H H v (βt) and parabolic cylinder D u (αx) functions are independent of the right-hand side. These types of integrals could involve properties related to orthogonal functions. This invariant property occurs as a result of how the gamma function is chosen for the definite integral of the contour integral to reduce to a trigonometric function. The derivation of this invariant property is not dependent on all the parameters involved.

Plots of a Special Case Involving ζ(k)
In this section, we evaluate the right-hand side of Equation (15) for complex values of the parameter k. The Figures 1-3 are below:

Conclusions and Observation
In this paper, we have presented a novel method for deriving a new integral transform involving the product of the Struve and parabolic cylinder functions along with some interesting definite integrals using contour integration. We also derived an invariant index form of the quadruple integral. We observed that the single integral of the product of the Struve and parabolic cylinder functions is in terms of the gamma function, while the higher dimensional integrals of the product of these functions can be constant. We will be investigating this property in future work. The Figures 1-3, and results presented were numerically verified for real, imaginary, and complex values of the parameters in the integrals using Mathematica by Wolfram.
Author Contributions: Conceptualization, R.R.; methodology, R.R.; writing-original draft preparation, R.R.; writing-review and editing, R.R. and A.S.; and funding acquisition, A.S. All authors have read and agreed to the published version of the manuscript.

Funding: This research is supported by NSERC Canada under grant 504070.
Institutional Review Board Statement: Not applicable.