Some Fourier transforms involving confluent hypergeometric functions

ABSTRACT In this paper, we derive some Fourier transforms of confluent hypergeometric functions. We give generalizations of several well-known results involving Fourier transforms of gamma functions. In particular, the generalizations include some Ramanujan's remarkable formulas.


Introduction
The confluent hypergeometric function 1 F 1 (a; b; z) is defined by The function 1 F 1 (a; b; z) is entire in z and a, and is a meromorphic function of b with simple poles at the points b = 0, −1, . . .Thus, therein).Because of these applications to physics and technical sciences, it is of an ever increasing significance in recent years.
In the paper, we derive some Fourier transforms of the confluent hypergeometric functions.The transforms obtained here generalize several well-known results, involving Fourier transforms of gamma functions, including the following Ramanujan's remarkable formulas (see [5,Chapter 13]) , a > 0. (1. 2) It should be mentioned that these results, as well as other integrals involving gamma functions, have been derived by several authors using different approaches (see, for instance, [4,[6][7][8][9][10], and references therein).Note that in [11], Ramanujan derived a Fourier transform for products of Bessel functions of the first kind, which in terms of the confluent hypergeometric functions can be written as where Re(α + β) > −1 and ξ ∈ R\{±π }.For further Fourier transforms involving confluent hypergeometric functions, as well as for their particular cases, see [12,13].

Main results
First, we derive a new Fourier transform for the hypergeometric function 2 F 2 , which is defined by the series as well as for its particular case -confluent hypergeometric function 1 F 1 .

Theorem 2.1:
(2.1) where Re(a ) > 0 and Re(c) > 0. Thus, substituting x = 1 − e −ξ and a = b + it in (2.3), we obtain where F[φ](t) is the Fourier transform of φ given by and 1 A is the indicator function of a subset A ⊂ R. Since it follows that there exists the inverse Fourier transform of φ such that This completes the proof of the theorem.
Among the various particular cases which may be derived from Theorem 2.1, the following are of interest.

Corollary 2.4:
(2.7) . (2.8) and the Fourier transform of the confluent hypergeometric function (2.2) we have ), and we applied the following relations Substituting u = 1 − e −ξ in the integral on the right hand side of above equation, then taking into account Equation (2.3) for a = c = α and a + c = β, i.e.
It should be mentioned that formula (2.8) can be written as an evaluation of Meijer's G function G 11  22 (1), i.e. .
Proof: (i) The proof is based on the partial fraction expansion for the product of confluent hypergeometric functions (see [20]) where z, a, α, β ∈ C and Re(b) > 0. Putting b = a, α = c + it and β = c − it in (2.12), yields We multiply this equation by e iξ t , ξ ∈ R, then integrate it with respect to t over R to get where we used the integral (see [21, p.323 On the other hand, using the integral representation for 1 F 1 given by (2.9), the series in the right hand side of equation (2.13) can be written as follows where Re(a) > 0 and Re(c) > 0.
Now, using the addition theorem (see [4,Section 13.13]) A combination of (2.13) and (2.14) yields where where Re(a) > 1 2 and Re(c) > 0. (ii) Setting α = t and β = −t (t ∈ R) in Equation (2.12), we get Using the identity Multiply the last equation by e iξ t , ξ ∈ R, then integrate it with respect to t over R, taking into account Ramanujan's integral (see [11]) where Re(α + β) > 1.This yields Now, let us consider some particular cases of Theorem 2.5.
In particular,

Corollary 2.8: It is evident that Equation (2.11) can be written as
i.e. the summand and integrand are of exactly the same form.
There have been a number of studies of this kind of sum-integral equality for the confluent hypergeometric functions, for example, [2,25, Section 6.15.3], and for its special cases see [26][27][28], and references therein.Finally we remark that, from Theorem 2.5(i), we may easily deduce the following Mellin transform of 1 1 0 x c−1 (1 − x) a−1 e z(1−x) cos t ln 1 x 1 a, 1 − a; a + 2c; For t = 0, we get For further Mellin transforms involving hypergeometric functions of one and several variables, we suggest the reader consults [14,19].