A New Generalization of Confluent Hypergeometric Function and Whittaker Function

In this article, we introduce a further generalizations of the confluent hypergeometric function and Whittaker function by introducing an extra parameter in the extended con uent hypergeometric function dened by Parmar [15]. We also investigate some integral representations, some integral transforms, differential formulas and recurrence relations of these new generalizations


By using B
(α,β;m) σ (x, y), Parmar [15] defined a new generalization of extended Gauss hypergeometric function and confluent hypergeometric function as follows: and gave their Euler's type integral representation On substituting t = 1−u in (16), Parmar [15] obtained the following new extension of Kummer's relation for the generalized extended confluent hypergeometric function of the first kind: For σ = 0, (17) reduces to the Kummer's first formula for the classical confluent hypergeometric function [4].
Afterwards, Srivastava et al. [9] introduced a new generalized Gauss hypergeometric functions as follows: where the generalized beta function B (α,β;m,n) σ (x, y) is defined by On substituting m = n in (18) and (19), respectively, we get the extended Gauss hypergeometric function and extended beta function defined by Parmar [15].

Generalized extended confluent hypergeometric function
In this section, we give the definition of the generalized extended confluent hypergeometric function.An integral representation and a Kummer type relation of this function are also indicated.
Definition 2.1.The generalized extended confluent hypergeometric function (for and is defined as follows: Remark 2.2.On setting m = n, (20) reduces to the generalized extended confluent hypergeometric function defined by Parmar [15], which further for n = 1 gives the known extension of the confluent hypergeometric function given by Özergin et al. [7].Further, if we set α = β and m = n in (20) then we get the generalized confluent hypergeometric function defined by Lee et al. [3] and if we put α = β and m = n = 1 in (20) then we obtain the extended confluent hypergeometric function defined by Chaudhry et al. [13].
Integral representation: The integral representation of the generalized extended confluent hypergeometric function can be obtained by using the definition of generalized beta function defined by (19).
Theorem 2.3.For the generalized extended confluent hypergeometric function, we have the following integral representation: Proof.We have This completes the proof.
On substituting t = 1 − u in (21), we obtain the following new extension of Kummer's relation for the generalized extended confluent hypergeometric function of the first kind: For σ = 0, equation ( 22) reduces to the Kummer's first formula for the classical confluent hypergeometric function [4].
Proof.Taking the derivative of Φ (α,β;m,n) σ (b; c; z) with respect to z, we obtain Replacing r by r + 1, we get In a similar procedure, by induction, we can obtain the desired result.

Mellin transforms and transformation Formula
Certain interesting Mellin transforms and a transformation formula for the generalized extended confluent hypergeometric function are given in the following theorems: Theorem 4.1.For the generalized extended confluent hypergeometric function, we have the following Mellin transform representation: Proof.To obtain Mellin transform, multiply both sides of (21) by σ s−1 , and integrating with respect to σ over the interval [0, ∞), and changing the order of integral, we get where Γ (α,β) (s) is the generalized Gamma function [7].
Thus we have This completes the proof.
Proof.Taking Mellin inversion of Theorem (4.1), we get the required result.
Theorem 4.3.For the generalized extended confluent hypergeometric function, we have the following transformation formula: Proof.Using the definition of the generalized extended confluent hypergeometric function, we have Replacing t by t − 1, we get the result.

Generalized Extended Whittaker Function
In this section, we give the definition of the generalized extended Whittaker function in terms of generalized extended confluent hypergeometric function.Some integral representations and a relation of this function are also derived. where is the generalized extended confluent hypergeometric function of the first kind defined by (21).
On setting m = n = 1 in (28), we obtain the following (presumably) a new representation of the extended Whittaker function: where Remark 5.2.On setting m = n in (28), we obtain the generalized extended Whittaker function defined by Choi et al. [10].Further, on setting α = β, m = n in (28), we get the extended Whittaker function given by Khan and Ghayasuddin [14], which further for n = 1 gives the extended Whittaker function due to Nagar et al. [2].For σ = 0, (28) reduces to the classical Whittaker function defined by (1).
Remark 5.5.On using (21) in the equation (31), we get Thus it is seen that the generalized extended Whittaker function can also be expressed by (35).
Theorem 5.6.The following relation holds true: Proof.Replacing z by −z in (28), we get Now using ( 22) in (37) and then writing the resulting expression by using (28), we get the desired result.

Integral transforms of M
Certain interesting integral transforms of the generalized extended Whittaker function are given as follows: Theorem 6.1.The following Mellin transformation holds true: given by (30) and changing the order of integration, we get where Γ (α,β) σ (s) is the generalized gamma function defined by Özergin et al. [7].
So that, we have On using the integral representation of confluent hypergeometric function 1 F 1 or Φ in the above equation, we get the required result.
Remark 6.2.The case n = m of (38) on arranging the resulting expression in terms of classical Whittaker function, is seen to yield the Mellin transform of the extended Whittaker function given by Choi et al. [ [10], p.6535, eq.( 27)].
Proof.On using the integral representation of M (α,β;m,n) σ,k,µ (z) on the left hand side of (39) given by (30), and changing the order of integration, and integrating with respect to z by using the definition of gamma function, we arrive at (40) On using the integral representation of F (α,β;m,n) σ (a, b; c; z) (which can be easily obtained by using the integral representation of extended beta function given by ( 19) in ( 18)) in (40), yield the desired result.Corollary 6.4.If we put b = a = 1 in (39), we obtain following special case.

Remark 2 . 4 .
On taking m = n, (21) reduces to the integral representation of the generalized extended confluent hypergeometric function defined by Parmar[15], which further for n = 1 gives the integral representation of the extended confluent hypergeometric function given by Özergin et al.[7].Further, if we put α = β and m = n in (21) then we get the integral representation of generalized confluent hypergeometric function defined by Lee et al.[3] and if we set α = β and m = n = 1 in (21) then we obtain the integral representation of extended confluent hypergeometric function defined by Chaudhry et al.[13].3.Derivative of Φ(α,β;m,n) σ (b; c; z) The derivative of generalized extended confluent hypergeometric function Φ (α,β;m,n) σ (b; c; z) with respect to the variable z in terms of a shift operator is obtained by using the following formulas: B(b, c − b) = b c B(b + 1, c − b) and (a) n+1 = a(a + 1) n .(23) Theorem 3.1.For the generalized extended confluent hypergeometric function Φ (α,β;m,n) σ (b; c; z), the following differentiation formula holds true: