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Integral representations for products of two parabolic cylinder functions with different arguments and orders

This paper derives new integral representations for products of two parabolic cylinder functions. In particular, expressions are obtained for D_{nu}(x)D_{mu}(y), with x>0 and y>0, that allow for different orders and arguments in the two parabolic cylinder functions. Also, two integral representations are obtained for D_{nu}(-x)D_{mu}(y) by employing the connection between the parabolic cylinder function and the Kummer confluent hypergeometric function. The integral representations are specialized for products of two complementary error functions and of two modified Bessel functions of the second kind of order 1/4, as well as for the product of a parabolic cylinder function and a modified Bessel function of the first kind of order 1/4.


Introduction
The study of Nicholson-type integrals for the product of two parabolic cylinder functions recently gained renewed interest. An integral representation for the product of two parabolic cylinder functions with the same order and identical or opposite arguments, D ν (x)D ν (±x), was presented in [1]. Recently, [2] obtained an integral representation for the case of identical orders but unrelated arguments, D ν (x)D ν (y). Integral representations, on the other hand, for unrelated orders but identical or opposite arguments, D ν (±x)D ν+µ−1 (x), were documented in [3]. The approach in [4] offered a first, yet incomplete attempt at allowing both for differing arguments as well as orders given that the orders in the various expressions still had to be linearly related. In particular, [4] used the results of [5] to first obtain integral representations for D ν (x)D ν (y) and D ν (x)D ν−1 (y) that subsequently can be extended via the recurrence relation for the parabolic cylinder function into separate integral representations for other, linearly related orders such as D ν (x)D ν+1 (y) and D ν (x)D ν−2 (y). This paper generalizes existing results by deriving an integral representation for D ν (x)D µ (y) in which both the arguments as well as the orders are (linearly) unrelated.
The integral representation for D ν (x) D µ (y), with x 0 and y 0, is derived by applying the convolution theorem to two inverse Laplace transforms for single parabolic cylinder functions that are documented in [6]. The latter expressions are selected upon their ability to ultimately The results of this paper can be specialized in various directions. For instance, integral representations for products of two complementary error functions and of two modified Bessel functions of the second kind of order 1/4 can immediately be obtained from the properties of the parabolic cylinder function. It is interesting to note that the integrand then simplifies into the arcsine function and the complete elliptic integral of the first kind, respectively. Also, a representation is given for the product of a parabolic cylinder function and a modified Bessel function of the first kind of order 1/4.
The remainder of this paper is organized as follows. Section 2 lists the definitions and the properties that are used throughout the paper. The required inverse Laplace transforms are derived in Section 3. Section 4 specifies the integral representations for D ν (x) D µ (y) and obtains two expressions for D ν (−x) D µ (y). Section 5 presents some specializations of the main results.

Definitions and useful properties
Extensive detail on the parabolic cylinder function, the Laplace transform and the hypergeometric function can be found in, for instance, [7], [8] and [9], respectively, such that this section merely lists the definitions and properties that will be used in the paper.
The parabolic cylinder function of order ν and argument z is denoted by D ν (z). Equation (4) on (2.3b) The following two identities for the parabolic cylinder function will be used within the below specializations of results see p. 326 in [11] and (9.254.1) in [12], where K ν (z) denotes the modified Bessel function of the second kind with order ν and erfc(z) is the complementary error function. We also use the following relation between Φ (a; b; z) and I ν (z), i.e. the modified Bessel function of the first kind Φ (a; 2a; z) = Γ (a + 1/2) (z/4) 1/2−a exp (z/2) I a−1/2 (z/2) , (2.5) see (5) on p. 579 in [13].
The convolution theorem will be used to obtain inverse Laplace transforms for products of two parabolic cylinder functions. As the inverse Laplace transforms that will be used for Φ (a; b; z) are defined over a finite interval rather than for 0 t < ∞, the convolution theorem becomes somewhat more involved. In particular, [14] defines the Laplace transform of the original functions f 1 (t) and f 2 (t) on p. 578 as where Re p > 0. The convolution theorem then is specified in [14] as is the convolution of f 1 (t) and f 2 (t) that is obtained from The inverse Laplace transforms for single parabolic cylinder functions and Kummer's confluent hypergeometric function that will be used within the convolution theorem (2.7) are chosen in terms of their ability to ultimately produce integrands that contain the Gaussian hypergeometric function F (a, b; c; z). In particular, the latter function will be obtained in two steps. First, the convolution integral will be transformed into the integral representation for the Appell function for Re c > Re a > 0, see (8.2.5) in [15]. Second, the Appell function F 1 (a, b 1 , b 2 ; c; x, y) simplifies into the Gaussian hypergeometric function for [15].
The paper also intensively uses the following (linear transformation) formulas for F (a, b; c; z) and the quadratic transformation formula  [9].
Also, the following special cases of F (a, b; c; z) are used respectively. The associated Legendre function of the first kind is denoted by P a b (x), B (z; a, b) is the incomplete beta function and K (z) is the complete elliptic integral of the first kind.

Three inverse Laplace transforms
The integral representations for D ν (x) D µ (y) and D ν (−x) D µ (y) in the products (2.3a) and (2.3b) will be obtained from three inverse Laplace transforms.

An inverse Laplace transform that contains two parabolic cylinder functions
We start from the two inverse Laplace transforms for single parabolic cylinder functions that are specified in (5) and (6) on p. 290 in [6] These inverse transforms, in our notation, can be written as The original functions f 1 (t) and f 2 (t) in the convolution theorem (2.6) and (2.7) are taken from the inverse transforms (3.1) and (3.2), respectively The substitution τ = tu gives The definition (2.8) allows to rewrite this result as which via (2.10c) is rewritten into t (x + y + t) (x + t) (y + t) .

Integral representations for products of parabolic cylinder functions
Setting p = 1 in the inverse Laplace transform (3.3) and rescaling 2 1/2 x 1/2 and 2 1/2 y 1/2 to x and y, respectively, gives the following integral representation for D ν (x) D µ (y) Note that the linear transformation formulas (2.10b)-(2.10d) and the quadratic transformation formula (2.11) generate alternative expressions for D ν (x) D µ (y) in which the integrands differ but still evolve around the Gaussian hypergeometric function. The identity (2.12a), on the contrary, allows to rephrase the integrand of (4.1) in terms of the associated Legendre function of the first The first integral representation for D ν (−x) D µ (y) is obtained by setting p = 1 and rescaling 2 1/2 y 1/2 → y in the inverse Laplace transform (3.7) and plugging the outcome together with (4.1) into the product (2.3a) Setting p = 1 and rescaling 2 1/2 y 1/2 → y in (3.9) and subsequently inserting the result together with (4.1) into the product (2.3b) gives a second integral representation for D ν (−x) D µ (y) into F (1/4, 1/4; 1; z) = (2/π) K 1/2 1 − √ 1 − z as can be seen from (2.11) and (2.12e) Setting ν = µ = −1 in (4.1) and using (2.4b) yields an integral representation for the product of two complementary error functions for which the integrand contains the arcsine function in view of identity (2.12f) The next example illustrates an integral representation for the product of a parabolic cylinder function with a special function that follows from choosing a special value for Φ (a; b; z) in the inverse Laplace transform (3.7). First, the latter inverse transform gives the following integral x 2 y − t x 2 − 4y + 2t x 2 y dt × t x 2 − 4y + 2t −1/4 P The final example focuses on a limiting case for D ν (−x) D µ (y) for which identity (2.12d) can be specialized in terms of the incomplete beta function. Setting µ at −1 in the integral representation (4.3) gives an integral representation for D ν (−x)erfc(y). The first two Gaussian hypergeometric functions reduce into the incomplete beta function via the identities (2.10a), (2.10d) and (2.12c).
The third Gaussian hypergeometric function can also be simplified into the incomplete beta function given that (2.12d) can be written as This simplification is possible in view of the fact that the degree and order in the associated Legendre functions in (2.12d) both are at 1/2 − b. The associated Legendre function of the first kind is defined as 1 − x 2 for − 1 < x < 1, see (6) on p. 143 of [17]. For identical degree and order at, for instance, ν this gives ; −ν, −ν for − 1 < x < 1, on account of the identities (2.10a) and (2.12c).
Using the latter result then implies that all integrands in this integral representation for