Extended incomplete Riemann-Liouville fractional integral operators and related special functions

: In this study, we introduce the extended incomplete versions of the Riemann-Liouville (R-L) fractional integral operators and investigate their analytical properties rigorously. More precisely, we investigate their transformation properties in L 1 and L ∞ spaces, and we observe that the extended incomplete fractional calculus operators can be used in the analysis of a wider class of functions than the extended fractional calculus operator. Moreover, by considering the concept of analytical continuation, deﬁnitions for extended incomplete R-L fractional derivatives are given and therefore the full fractional calculus model has been completed for each complex order. Then the extended incomplete τ -Gauss, conﬂuent and Appell’s hypergeometric functions are introduced by means of the extended incomplete beta functions and some of their properties such as integral representations and their relations with the extended R-L fractional calculus has been given. As a particular advantage of the new fractional integral operators, some generating relations of linear and bilinear type for extended incomplete τ -hypergeometric functions have been derived.


Introduction
There is a close relationship between the special functions and fractional calculus, which is a branch gaining popularity, especially in the last decades, because of its potential usefulness in real world applications (see [1][2][3][4][5][6][7][8][9][10]). Different definitions of fractional calculus have been introduced in the literature, each of which has their own advantages or disadvantages. One direction of research has been to add 1 y f (uz)(1 − u) −µ−1 du, Re(µ) < 0. (1. 2) The incomplete fractional integrals and derivatives have been subjected to an in-depth analysis in the papers [15][16][17][18].
Very recently, in order to introduce a different variant of incomplete Gauss hypergeometric functions which is more suitable for the fractional calculus results as well, the authors introduced the incomplete Pochhammer ratios as follows [15] Several properties of these functions were obtained, such as integral representations, derivative formulae, transformation formulae, and recurrence relations. Also, the incomplete Appell's functions were defined and expressed using integral representations. It should be mentioned that, in a recent paper, the incomplete Gauss hypergeometric function was used in the derivation of some new estimates for the generalized Simpson's quadrature rule [31].
One of the generalisation of Gauss hypergeometric function was defined by Chaudhry [13,14] is the extended beta function. On the other hand, the incomplete version of the extended beta function has been defined as and investigated in [14]. Some applications, where these functions are used, can be found in [14]. We saw from the above discussion that both structures of generalisations by adding new parameters or incomplitifications are interesting topics of study with potential real world applications. By combining both forms of generalisations, we are able to construct new functions and operators which have the advantages of both the incomplete versions and the parametric versions. Therefore in this paper, in Sections 2 and 3 we combine these generalisations and investigate them thoroughly. More precisely in Section 2, we introduce the extended incomplete versions of the Riemann-Liouville (R-L) fractional integral operator and investigate their transformation properties in L 1 and L ∞ spaces. We observe that the extended incomplete fractional calculus operators can be used in the analysis of a wider class of functions than the extended fractional calculus operator. Moreover, by considering the concept of analytical continuation, definitions for extended incomplete R-L fractional derivatives are given and therefore the full fractional calculus model has been completed for each complex order. In Section 3, similar treatment has been considered to introduce extended incomplete τ-Gauss, confluent and Appell's hypergeometric functions. Some of their properties such as integral representations and their relations with the extended R-L fractional calculus has been given. In the last section, one particular advantage of the new fractional integral operators has been exhibited by deriving some generating relations of linear and bilinear type for extended incomplete τ-hypergeometric functions.
Integrating both sides of this inequality over z ∈ [0, A] , we get Integrating both sides of this inequality over z ∈ [0, A] , we get Whence the result.
Therefore, using the above two Theorems, we can give the following definitions.
Then for all f ∈ L 1 0, yA the µ th order extended incomplete lower fractional integral is defined by Definition 2.4. Let A > 0, 0 < y < 1 and Re(µ) > 1. Then for all f ∈ L 1 [0, A] the µ th order extended incomplete upper fractional integral is defined by There is a gap in the definition of the extended incomplete upper fractional integral for the case 0 < Re(µ) ≤ 1. In order to fill this gap, in the following theorem, we consider the operators in the space L ∞ .
Theorem 2.5. Let A > 0, 0 < y < 1 and Re(µ) > 0. Then we have Proof. For any z ∈ [0, A], since Re(p) > 0, we have Taking essup over all z ∈ [0, A] on both sides of the inequality, we complete the proof of the first statement.
Since Re(p) > 0, we have for any x ∈ [0, A] that Taking essup over all z ∈ [0, A] on both sides of the inequality, we complete the proof of the second statement.
Using the above Theorem, in the following we give definition of the µ th order extended incomplete upper fractional integrals for the case 0 < Re(µ) ≤ 1.
Remark 2.7. It should be remarked that the transformation properties of has not been investigated and it can be easily proved by majorizing the exponential term that D −µ,p z (·) : Proof. Firstly, using the usual technique of differentiation under the integral sign, we have Secondly, taking derivative from the definition of the upper incomplete integral operator, we get Using the above theorem, in the following definitions, we extend the domain of analyticity of both D µ,p z f (z); y and D µ,p z { f (z); y} to the right half-plane and hence we call them as the µth order extended upper and lower R-L derivative operators.
Definition 2.9. The µth order extended upper R-L derivative operator is defined by Definition 2.10. The µth order extended lower R-L derivative operator is defined by Remark 2.11. It is important to mention that the definition of D µ,p z f (z); y given in (10) does not require the condition Re(µ) < 0 since the interval of integration in this definition is 0, y with 0 < y < 1. Therefore the formula (10) is valid for all µ ∈ C.
Example 2.12. Let Re(λ) > −1, Re(µ) < 0 and Re(p) > 0. Then In the next theorem, we present useful representations of the extended upper and lower R-L derivatives of an analytic function.
is an analytic function on the disk |z| < R and has a power series expansion f (z) = ∞ n=0 c n z n , then for Re(λ) > 0 and Re(p) > 0 we have
Proof. Since the function is analytic in the given disc, its series expansion is uniformly convergent. Using the relation in the above Example, ∞ n=0 a n B y (λ + n, −µ; p) z n . Similarly, ∞ n=0 a n B 1−y (−µ, λ + n; p) z n , using uniform convergence of the series and absolute convergence of the integral under the given conditions. Now we consider the Mellin transform of the extended incomplete beta function. For Re(s) > 0, we have Therefore, from the inverse Mellin transform, we have Using this result, we have where we have interchanged the contour integral and the series by considering that the function is analytic in the given disc and Re(s) > 0, Re(p) > 0, Re(λ) > 0. Now we consider the case p → 0, which gives the incomplete lower R-L fractional integral operator. Recalling the incomplete beta ratio the following complex contour representation of this function was given in [32] for p < 1, p + q > 0, 0 < d < 1; The condition p < 1, which is important for the evaluation of the contour integral, can be cancelled by using the analytic continuation principle. Using the above integral representation, we have for λ−µ > 0 that For instance, let's choose f (z) = p F q (a 1 , · · · , a p ; b 1 , · · · , b q ; z) with p ≤ q, which is an entire function on the whole domain. From the above result, we can immediately write an elegant contour integral representation: The heading levels should not be more than 4 levels. The font of heading and subheadings should be 12 point normal Times New Roman. The first letter of headings and subheadings should be capitalized.

Extended incomplete τ-Gauss and confluent hypergeometric functions
The main aim of this section is to initiate the study of the extended incomplete τ-hypergeometric type function and the extended incomplete τ-Appell functions, where, as mentioned in the introduction, the investigation of the usual cases was a concern of the recent years. The second aim of the section is to make the preparation for the next section, where we obtain their generating relations. We should note here that the results obtained in Sections 3 and 4 are reduced to the incomplete versions in the case p → 0, where the reduced results will be new for τ−incomplete special functions discussed in these sections.
We shall introduce the extended incomplete τ-Gauss and confluent hypergeometric functions in terms of the extended incomplete beta function B y (x, z; p) , as follows : Also, it should be mentioned that in the special case of (3.1) and (3.2) when p → 0, we arrive at the new definitions which can be called as the incomplete τ-Gauss and confluent hypergeometric functions as follows: In the following propositions, we obtain integral representations and derivative formulas for incomplete τ-Gauss and confluent hypergeometric functions.
Proposition 3.2. The extended incomplete τ-Gauss hypergeometric function can be represented by an integral as follows: Proof. Replacing the extended incomplete beta function in the definition (3.5) by its integral representation given by (1.8), we have From the uniform convergence, summation and integration can be interchanged. Then, we have Corollary 3.3. If p is set to 0 in the above proposition, we reach the result corresponding to the incomplete τ-Gauss hypergeometric function which is given as follows: Proposition 3.4. For the extended incomplete τ-confluent hypergeometric function, we have the following integral representation: Corollary 3.5. If we set p = 0 in the above theorem, we can give the corresponding result for the incomplete τ-confluent hypergeometric function as follows: Proposition 3.6. The equation shown below holds true for the incomplete τ-Gauss hypergeometric function: Proof. Using (3.3), differentiating on both sides with respect to z, we get which is (3.6) for n = 1. Recursive application of this procedure yields the general result.
In a similar manner, we have the following. In the following theorem, we give expressions for the Mellin transforms of the extended incomplete τ -Gauss hypergeometric function an expression which involves incomplete τ-Gauss hypergeometric function.
Theorem 3.8. The extended incomplete τ-hypergeometric function has a Mellin transform which can be written as follows : Proof. To get the Mellin transform, multiplying (3.5) by p s−1 and integrate over the interval [0, ∞) with respect to p to get Thus we get Remark 3.9. Setting s = 1 in (3.7), we get By means of the extended incomplete beta function B y (x, z; p) stated by (1.8) , we introduce the extended incomplete τ−Appell's functions as follows: where max{|x| , |z|} < 1, and x m m! z n n! . (3.12) where |x| + |z| < 1.
Remark 3.10. The special case of the definitions (3.9-3.12) when τ = 1 and p → 0 are easily seen to reduce to the incomplete Appell's functions [15]: Also, in the case of p → 0, we can give the incomplete versions of the τ−Appell's functions as follows: x m m! z n n! . (3.16) and we call these functions as the incomplete τ-Appell's functions.
We can rewrite the series for incomplete τ-Appell's functions in terms of the incomplete τ-Gauss hypergeometric functions, so that In the following proposition, integral representations of the extended incomplete τ-Appell's functions are given.
If p is set to 0 in the above proposition, we have the following corollary: Corollary 3.12. The incomplete τ-Appell's functions can be represented by an integral as follows: Now we evaluate the following fractional derivative formulas, which we shall need them in the derivation of the generating functions in Section 4.
Interchanging the order, we get To get the desired result, we use Propositions 6 and 7.