Generalized Summation Formulas for the Kampé de Fériet Function

: By employing two well-known Euler’s transformations for the hypergeometric function Wang established numerous general transformation and reduction formulas for the Kampé de Fériet function and deduced many new summation formulas for the Kampé de Fériet function with the aid of classical summation theorems for the 2 F 1 due to Kummer, Gauss and Bailey. Here, by making a fundamental use of the above-mentioned reduction formulas, we aim to establish 32 general summation formulas for the Kampé de Fériet function with the help of generalizations of the above-referred summation formulas for the 2 F 1 due to Kummer, Gauss and Bailey. Relevant connections of some particular cases of our main identities, among numerous ones, with those known formulas are explicitly indicated.

In Wolfram's MATHEMATICA, the function p F q is implemented as HypergeometricPFQ and is suitable for both symbolic and numerical calculation. For p = q + 1, it has a branch cut discontinuity in the complex z-plane running from 1 to ∞. If p ≤ q the series (1) converges for each z ∈ C. For some recent results on this subject, especially on transformations, summations and some applications, see [4].
Here and elsewhere, let C, Z and N be the sets of complex numbers, integers and positive integers, respectively, and let N 0 := N ∪ {0} and Z − 0 := Z \ N.
It is worthy of note that whenever the generalized hypergeometirc function p F q (z) (including 2 F 1 (z)) with its specified argument z (for example, z = 1 or z = 1/2) can be summed to be expressed in terms of the Gamma functions, the result may be very important from both theoretical and applicable points of view. Here, the classical summation theorems for the generalized hypergeometric series such as those of Gauss and Gauss second, Kummer, and Bailey for the series 2 F 1 ; Watson, Dixon, Whipple and Saalschütz summation theorems for the series 3 F 2 and others play important roles in theory and application. During 1992During -1996, in a series of works, Lavoie et al. [10][11][12] have generalized the above-mentioned classical summation theorems for 3 F 2 of Watson, Dixon, and Whipple and presented a large number of special and limiting cases of their results, which have been further generalized and extended by Rakha and Rathie [13] and Kim et al. [14]. Those results have also been obtained and verified with the help of computer programs in MATHEMATICA and MAPLE.
The vast popularity and immense usefulness of the hypergeometric function and the generalized hypergeometric functions of one variable have inspired and stimulated a large number of researchers to introduce and investigate hypergeometric functions of two or more variables. A serious, significant and systematic study of the hypergeometric functions of two variables was initiated by Appell [15] who presented the so-called Appell functions F 1 , F 2 , F 3 and F 4 , which are generalizations of the Gauss' hypergeometric function. Here, we recall the Appell function F 3 (see, e.g., ([8] p. 23, Equation (4))) The confluent forms of the Appell functions were studied by Humbert [16]. A complete list of these functions can be seen in the standard literature, see, e.g., [5]. Later, the four Appell functions and their confluent forms were further generalized by Kampé de Fériet [17], who introduced more general hypergeometric functions of two variables. The notation defined and introduced by Kampé de Fériet for his double hypergeometric functions of superior order was subsequently abbreviated by Burchnall and Chaudndy [18,19]. We recall here the definition of a more general double hypergeometric function (than one defined by Kampé de Fériet) in a slightly modified notation given by Srivastava and Panda ([20] p. 423, Equation (26)). The convenient generalization of the Kampé de Fériet function is defined as follows: where (h H ) denotes the sequence of parameters (h 1 , h 2 , . . . , h H ) and ((h H )) n is defined by the following product of Pochhammer symbols: where the product when n = 0 is to be accepted as unity. For more details about the function (2) including its convergence, the reader may be referred (for example) to ([8] pp. [26][27][28][29][30][31][32][33]. When some extensively generalized special functions like (2) were appeared, it has been an interesting and natural research subject to consider certain reducibilities of the functions. In this regard, many researchers have investigated the reducibility and transformation formulas of the Kampé de Fériet function. In fact, there are numerous reduction formulas and transformation formulas of the Kampé de Fériet function in the literature, see, e.g., . In the above-cited references, most of the reduction formulae were related to both cases H + A = 3 and G + C = 2. In 2010, by using Euler's transformation formula for the 2 F 1 , Cvijović and Miller [26] established a reduction formula for the case H + A = 2 and G + C = 1. Motivated essentially by the work [26], recently, Liu and Wang [43] used Euler's first and second transformation formulas for the 2 F 1 and the above-mentioned classical summation theorems for p F q to present a number of very interesting reduction formulas and then deduced summation formulas for the Kampé de Fériet function. Indeed, only a few summation formulas for the Kampé de Fériet function are available in the literature.
In this paper, by choosing to make a basic use of 7 reduction formulas due to Liu and Wang [43], we aim to establish 32 general summation formulas for the Kampé de Fériet function, which are provided in 16 theorems, each one containing two formulas, with the help of generalizations of Kummer summation theorem, Gauss second summation theorem and Bailey summation theorem due to Rakha and Rathie [13]. The 32 general formulas afforded here are explicitly indicated to reduce to correspond with some special cases of the main results in Liu and Wang [43] and contain all of the main identities in Choi and Rathie [44].

General Summation Formulas for the Kampé de Fériet Function
In this section, we establish 32 general summation formulas for the Kampé de Fériet function, which are stated in Theorems 1-16. Each theorem includes two summation formulas. Additionally, some particular cases of the general summation formulas here are explicitly pointed out to correspond to those known identities in Remarks 1-16.
and F 1:1;2 (3), we get Now, the 2 F 1 in the right side of (18) can be evaluated with the help of the result (12) by taking a = β − ε and b = 1 − 2α − ε + β + i. After some simplification, we get the result (16).

Remark 2.
The particular case i = 0 in (19) or (20) and F 1:1;2 Proof. A similar process of the proof of Theorem 1 can establish the results here. Setting (i) (4) with the help of (10) and (11) offers, respectively, (21) and (22). The details are omitted.
Proof. A similar process of the proof of Theorem 1 can establish the results here. Setting (i) (5) with the help of (10) and (11) yields, respectively, (26) and (27). The details are omitted.
and F 1:1;3 Proof. A similar process of the proof of Theorem 1 can establish the results here. Setting (i) (5) with the help of (12) and (13) produces, respectively, (28) and (29). We omit the details.
and F 1:1;3 Proof. A similar process of the proof of Theorem 1 can establish the results here. Setting (i) (5) with the help of (14) and (15) affords, respectively, (30) and (31). The details are omitted.

Proof.
A similar process of the proof of Theorem 1 can establish the results here. Setting (i) (6) with the help of (10) and (11) gives, respectively, (32) and (33). We omit the details.

Proof.
A similar process of the proof of Theorem 1 can establish the results here. Setting (6) with the help of (12) and (13) yields, respectively, (34) and (35). The details are omitted.

Proof.
A similar process of the proof of Theorem 1 can establish the results here. Setting (6) with the help of (14) and (15) offers, respectively, (36) and (37). The details are omitted. ; and F 2:0;1 Proof. A similar process of the proof of Theorem 1 can establish the results here. Setting (7) with the help of (12) and (13) affords, respectively, (38) and (39). We omit the details. ; and F 2:0;1 1:0;1 α, γ : Proof. A similar process of the proof of Theorem 1 can establish the results here. Setting (7) with the help of (14) and (15) presents, respectively, (40) and (41). The details are omitted.

Concluding Remarks
We chose to make an essential use of 7 reduction formulas for the Kampé de Fériet function due to Liu and Wang [43], with the help of generalizations of Kummer summation theorem, Gauss second summation theorem and Bailey summation theorem due to Rakha and Rathie [13], to present 32 general summation formulas for the Kampé de Fériet function.
With the aid of other general summation formulas for 2 F 1 (if any), a similar method used in this paper is available to provide the corresponding summation formulas for the Kampé de Fériet function, which remains for future investigation.
As commented in the beginning of Section 2, constraints of each formula in this paper are omitted. The restrictions of Formula (3) is demonstrated to provide as follows: x, α, β, γ, ε ∈ C such that |x| < 1, β ∈ C \ Z − 0 , γ + β ∈ C \ Z − 0 . Funding: The work of the second author was supported in part by the Serbian Academy of Sciences and Arts (Φ-96).