A New Generalization of Pochhammer Symbol and Its Applications

In this paper, we introduce a new generalization of the Pochhammer symbol by means of the generalization of extended gamma function (4). Using the generalization of Pochhammer symbol, we give a generalization of the extended hypergeometric functions one or several variables. Also, we obtain various integral representations, derivative formulas and certain properties of these functions.


A New Generalization of the Pochhammer Symbol
In this section, we denote a new generalization of Pochhammer symbol (5). Also, we give some useful properties.
Theorem 1. For the generalization of the Pochhammer symbol (5) following integral representation holds true: Proof. Using the equality (4) in the definition of the (5), we get the desired result (6).
Theorem 4. The following integral representation holds true: Proof. Using the integral representation given by (6) in the definition (9), we led to desired result (12).
Theorem 7. The following derivative formula holds true: Proof. Differentiating (9) with respect to z and then replacing n → n + 1 in the right-hand side term, we obtain repeating the same procedure n-times gives the formula (19). Choosing r = s = 1 and r = 2, s = 1 in (19), we have the derivative formulas for the (10) and (11), respectively.
Corollary 9. Each of the following integral representations hold true: and on condition that the integrals involves are convergent.
Corollary 10. The following integral representation holds true: on condition that the integrals involves are convergent.

A New Generalization of the extended Appell hypergeometric functions
In this section, we introduce extended Appell hypergeometric series and some extended multivariable hypergeometric functions.
Let us introduce the extensions of the Appell's functions and extended Lauricella's hypergeometric function and other functions defined by  Theorem 11. The following integral representations for (30) hold true: Proof. Using the generalization of the extended Pochhammer symbol (a 1 ; p, q; κ, µ) in the definition (30) by its integral representation given by (6), we led to desired result (39). Similar way, we can prove the (40).
Proof. Using the generalization of the extended Pochhammer symbol (a 1 ; p, q; κ, µ) in the definition (31) by its integral representation given by (6), we led to desired result (41). Similar way, we can prove the (42).
Theorem 13. The following integral representation for (32) holds true: Proof. Using the generalization of the extended Pochhammer symbol (a 1 ; p, q; κ, µ) in the definition (32) by its integral representation given by (3), we led to desired result (43).
Theorem 14. The following integral representation for (33) holds true: Proof. Using the generalization of the extended Pochhammer symbol (a 1 ; p, q; κ, µ) in the definition (33) by its integral representation given by (6), we led to desired result (44).
Theorem 15. The following integral representations for (34) hold true: Proof. Using the generalization of the extended Pochhammer symbol (a 1 ; p, q; κ, µ) in the definition (34) by its integral representation given by (6), we led to desired result (45). Similar way, we can prove the (46). and [a + n, b, c + n; d + n; x, y]. (54) Proof. Multiplying the (30) with y c+n−1 and taking the derivative n-times with respect to y, we have Proof. The proof of theorem would be parallel to those of the Theorem 17.

Recursion Formulas for Extended Appell Hypergeometric Functions
In this section, we present some recursion formulas for Appell hypergeometric functions. Let's we start following theorem.
Theorem 21. The following recursion formulas for (30) hold true: Calculating the function p,q F (κ,µ) 1 (.) with the parameter b+n by equation (66) for n times, we obtain the required result (63). Setting the b = b − n in the equation (66) and making same calculation as above equation, we can be yield the desired result (64). The proof of (65) is omitted to readers because it is similar to the proof of (63).
Proof. The proof of the Theorem 24 is same as the proof of Theorem 21.
Remark 2. Taking p = 1 and q = κ = µ = 0 in the relation Theorem 21-Theorem 24, it is easily seen that the special case of recursion formulas of Appell hypergeometric functions [32].

Conclusions
We may also give point to that results obtained in this work are of general character and can appropriate to give a new generalization of the Pochhammer symbol by means of the generalization of extended gamma function (4) [31]. Using the generalization of Pochhammer symbol, we give a generalization of the extended hypergeometric functions one or several variables. Also, we obtain various integral representations, derivative formulas and certain properties of these functions.