A New Identity for Generalized Hypergeometric Functions and Applications

We establish a new identity for generalized hypergeometric functions and apply it for firstand second-kind Gauss summation formulas to obtain some new summation formulas. The presented identity indeed extends some results of the recent published paper (Some summation theorems for generalized hypergeometric functions, Axioms, 7 (2018), Article 38).


Introduction
Let R and C denote the sets of real and complex numbers and z be a complex variable.For real or complex parameters a and b, the generalized binomial coefficient denotes the well-known gamma function for Re(z) > 0, can be reduced to the particular case where (a) b denotes the Pochhammer symbol [1] given by ( By referring to the symbol (1), the generalized hypergeometric functions [2] p F q a 1 , ... .
There are two important cases of the series (2) arising in many physics problems [5,6].The first case (convergent in |z| ≤ 1) is the Gauss hypergeometric function Replacing z = 1 in (3) directly leads to the well-known Gauss identity The second case, which converges everywhere, is the Kummer confluent hypergeometric function In this paper, we explicitly obtain the simplified form of the hypergeometric series p F q a 1 , ... , a p−1 , m + 1 b 1 , ... , b q−1 , n + 1 z , when m, n are two natural numbers and m < n.

A New Identity for Generalized Hypergeometric Functions
Let m, n be two natural numbers so that m < n.By noting (1), since Hence, substituting (5) into a special case of (2) yields In [7], two particular cases of ( 6) for m = 0 and m = 1 were considered and other cases have been left as open problems.In this section, we wish to consider those open problems and solve them for any arbitrary value of m.For this purpose, since is simplified as It is clear in (7) that To evaluate (.), we can directly use Chu-Vandermonde identity, which is a special case of Gauss identity (4), i.e., Now if in (9), p = j − n + 1 and q = −n + 1, we have Hence, replacing (10) in S * 1 gives It is important to note in the second equality of (11) that (−j) k = 0 for any j = 0, 1, 2, . . ., k − 1.Therefore, the lower index is starting from j = k instead of j = 0. Now since is simplified as On the other hand, the well-known identity To compute the finite sum 8), we can directly use the identity Finally, by noting the identity the main result of this paper is obtained as follows.
Main Theorem.If m, n are two natural numbers so that m < n, then p F q a 1 , ...
Note that the case m > n in (14) leads to a particular case of Karlsson-Minton identity, see e.g., [8,9].

Some Special Cases of the Main Theorem
Essentially whenever a generalized hypergeometric series can be summed in terms of gamma functions, the result will be important as only a few such summation theorems are available in the literature.In this sense, the classical summation theorems such as Kummer and Gauss for 2 F 1 , Dixon, Watson, Whipple and Pfaff-Saalschutz for 3 F 2 , Whipple for 4 F 3 , Dougall for 5 F 4 and Dougall for 7 F 6 are well known [1,10].In this section, we consider some special cases of the above main theorem to obtain new hypergeometric summation formulas.

Special case 1.
Note that if m = 0, the first equality of (13) reads as Hence, the main theorem is simplified as which is a known result in the literature [10] (p.439).
As a very particular case, replacing p = 3 and q = 2 in the above relation yields Special case 3.For p = q = 1, the main theorem is simplified as For instance, by referring to the special case 1, we have [7,10] 1 Special case 4. For p = 2 and q = 1, the main theorem is simplified as in which we have used the relation 1 F 0 a − z = (1 − z) −a .For instance, by referring to the special case 1, we have [7,10] 2 Special case 5.For p = 3 and q = 2, the main theorem is simplified as As a particular case and by noting the first kind of Gauss formula (4), if z = 1 is replaced in (15) then we get As a numerical example for the result (16), we have (1 , where b 1 = (a 1 + a 2 + 1)/2.