# Contribution for the Summation Formulas for the Generalized Hypergeometric SeriesF65−1

Authors: Prathima Jayarama; Dongkyu Lim; Arjun Kumar Rathie
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, our aim is to provide as many as one hundred and ninety-two summation formulas for the generalized hypergeometric seriesF65−1 in terms of gamma functions. For this, we have established eight theorems containing general results. This is achieved by means of applying generalized Watson’s and Dixon’s summation formulas obtained earlier by Lavoie et al. into a known transformation formula available in the literature. Results obtained earlier by Zhao follows special cases of our main findings.


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## Body

## 1. Introduction and Results Required

The well-known generalized hypergeometric functionFpq which is a natural generalization of the Gauss’s hypergeometric function F21 is defined as follows [1]: (1)Fpqα1,α2,…,αpβ1,β2,…,βq;z=1+∑n=1∞∏i=1pαin∏j=1qβjn.znn!,where αn is the well-known Pochhammer symbol defined by(2)αn=αα+1,…,α+n−1,n∈ℕ,1,n=0.In terms of well-known gamma function, we have(3)αn=Γα+nΓα.Here as usual,p and q are nonnegative integers and the parameters αj1≤j≤p and βj1≤j≤q can have arbitrary complex values with zero or negative integer values of βj excluded. It is clear that the generalized hypergeometric function Fpqz converges for z<∞p≤q, z<1p=q+1, and z=1p=q+1andℜδ>0 where δ is the parametric excess defined as(4)δ=∑j=1qβj−∑j=1pαj.It is not out of place to mention here that the generalized hypergeometric function occurs in many theoretical and practical applications such as mathematics, theoretical physics, engineering, and statistics.For more details about this function, we refer [1–6].It should be remarked here that whenever a generalized hypergeometric function reduces to the gamma function, the results are very important from the applications point of view. Thus, the classical summation theorems such as those of Gauss, Gauss second, Kummer and Bailey for the seriesF21, Watson, Dixon, Whipple, and Saalschu¨tz for the series F32 and others play an important role. Recently, a good deal of progress has been made in the direction of generalizing and extending the abovementioned classical summation theorems. For this, we refer here the research papers by Chu [7], Kim et al. [8], Lavoie et al. [9–11], and Rakha and Rathie [12].However, in our present investigation, we are interested in the following classical Watson and Dixon summation theorems. These are(5)F32α,β,γα+β+1/2,2γ;1=Γ1/2Γγ+1/2Γα/2+β/2+1/2Γγ−α/2−β/2+1/2Γα/2+1/2Γβ/2+1/2Γγ−α/2+1/2Γγ−β/2+1/2,provided ℜ2γ−α−β>−1, and(6)F32α,β,γ1+α−β,1+α−γ;1=Γ1+α−βΓ1+α−γΓ1+α/2Γ1+α/2−β−γΓ1+αΓ1+α/2−βΓ1+α/2−γΓ1+α−β−γ,provided ℜα−2β−2γ>−2.During 1992–1994, in a series of three research papers, Lavoie et al. [9, 10] generalized the abovementioned classical Watson and Dixon summation theorems and obtained twenty-five results closely related to Watson theorem and twenty-three results closely related to Dixon theorem in the form of two general formulas given as follows:Generalization of Watson summation theorem [9](7)F32a,b,c;112a+b+i+1,2c+j=Aij2a+b+i−2Γ12a+12b+12i+12×Γc+j/2+1/2Γc−1/2a−1/2b−i+j/2−j/2−1/2Γ1/2ΓaΓb×BijΓ1/2a+1/41−−1iΓ1/2bΓc−1/2a+1/2+j/2−−1j/41−−1iΓc−1/2b+1/2+j/2+CijΓ1/2a+1/41−−1iΓ1/2b+1/2Γc−1/2a+j+1/2+−1j/41−−1iΓc−1/2b+j+1/2,for i,j=0,±1,±2.Also, herex denotes the greatest integer less than or equal to x and its modulus is denoted by x. The coefficients Aij,Bij, and Cij are given in Tables 1–3 at the end of this paper.Table 1
Table for the coefficientsAij.
i/j−2−10122A2−2A2−1A20A21A21A1−21/a−b1/a−b1/2a−bA1201/2c−11111/2c+1−11/c−11222/c+1−21/2c−11122/c+1Table 2
Table for the coefficientsBij.
i/j−2−10122B2−2B2−1B202ca+b−1B221c−a−111B112cc+10B0−2111B02−1B−1−22c−a+b−211c−b+1−2B−2−2B−2−1B−20a+b−1B−22Table 3
Table for the coefficientsCi,j.
i/j−2−10122−4−4c−a−b−3−8C21C221−c−a−1−1−1−2c+a−bC220410−1−4−1C−1−22c+a−b−211c−b+1−2C−2−2C−2−184c−a−b+14Clearly, fori,j=0, the result (7) reduces to the Watson theorem (5).Generalization of Dixon summation theorem [10](8)F32a,b,c;11+a−b+i,1+a−c+i+j=2−2c+i+jΓ1+a−b+iΓ1+a−c+i+j×Γb−i/2−j/2Γc−1/2i+j+i+jΓa−2c+i+j+1Γa−b−c+i+j+1ΓbΓc×DijΓ1/2a−c+1/2+i+j+1/2Γ1/2a−b−c+i+1+j+1/2Γ1/2a+1/2Γ1/2a−b+1+i/2+EijΓ1/2a−c+1+i+j/2Γ1/2a−b−c+3/2+i+j/2Γ1/2aΓ1/2a−b+1/2+i+1/2,provided that Rea−2b−2c>−2−2i−j, for i=−3,−2,−1,0,1,2 and j=0,1,2,3.Also, as usual, herex denotes the greatest integer less than or equal to x and its modulus is denoted by x.The coefficientsDij and Eij appear in Tables 4 and 5 at the end of this paper.Table 4
Table for the coefficientsDij.
i/j01233D30−−−2D20D21D22−11−a−c+1D12−01−1D02D03−111b+c−1D−13−2D−20a−b−1D−22D−23−3D−30D−31D−32D−33Table 5
Table for the coefficientsEij.
i/j01233E30−−−2−2E21E22−1−1a−2b−c+3E12−001−2E03−111−b−c+1E−13−22a−b−2c−12E−23−3E−30E−31E−32E−33Clearly, fori,j=0, the result (8) reduces to the Dixon theorem (6).By applying classical summation theorem (5) and Dixon summation theorem (6) into the following well-known and useful transformation formula available in the literature [2], viz.,(9)F65a,a2+1,b,c,d,e;−1a2,1+a−b,1+a−c,1+a−d,1+a−e=Γ1+a−dΓ1+a−eΓa+1Γ1+a−d−eF321+a−b−c,d,e;11+a−b,1+a−c,very recently Zhao [13] established among several results the following five interesting summation formulas for the series F65−1. These are(10)P11=F65a,a2+1,b,c,1−b,1−c;−1a2,1+a−b,1+a−c,a+b,a+c=π21−2aΓa−b+1Γa+bΓa−c+1Γa+cΓaΓa+1Γa/2−b/2−c/2+1Γa/2+b/2−c/2+1/2Γa/2+c/2−b/2+1/2Γa/2+b/2+c/2,(11)P12=F65a,a2+1,2b,c,1+a−2c,1+2a−2b−2c;−1a2,1+a−2b,2c,1+a−c,−a+2b+2c=πΓc+1/2Γa−2b+1Γa−c+1Γ−a+2b+2cΓa+1Γb+1/2Γ−b+c+1/2Γa−b−c+1Γ−a+b+2c,(12)P13=F65a,a2+1,2b,2c,a−b−c+12,−a+2b+2c;−1a2,1+a−2b,1+a−2c,2a−2b−2c+1,b+c+12=πΓb+c+1/2Γa−2b+1Γa−2c+1Γa−b−c+1Γa+1Γb+1/2Γc+1/2Γa−b−2c+1Γa−2b−c+1,(13)P14=F6512,a,a+34,2a−12,2a−b,b;−1a−14,2a,a+12,2a−b+12,b+12=πΓa+1/22Γb+1/2Γ2a−b+1/2Γ2a+1/2Γb/2+1/22Γa−b/2+1/22,(14)P15=F65a,a2+1,a−2b+1,a−b+12,a−b+12,b;−1a2,a−b+1,2b,b+12,b+12=πΓb+1/23Γa−b+1Γa+1Γa/2−b/2+3/42Γ−a/2+3b/2+1/42.Our aim of this paper is to provide as many as one-hundred and ninety-two summation formulas for the generalized hypergeometric seriesF65−1. This is achieved by means of applying generalized Watson’s and Dixon’s summation formulas (7) and (8) into the transformation formula (9). Results (10) to (14) obtained earlier by Zhao follow special cases of our main findings.


## 2. A New Class of Summation Formulas for the Generalized Hypergeometric SeriesF65−1

In this section, we shall establish as many as one-hundred and ninety-two summation formulas in the form of eight general summation formulas asserted in the following theorems.(15)F65a,a2+1,2b,c−j,1+a−2c+j,1+2a−2b−2c−i+j;−1a2,1+a−2b,2c−j,1+a−c+j,−a+2b+2c+i−j=A1ij22a−4b−1Γ1+a−2bΓ1+a−c+jΓc+1/2−j+j/2Γ1/2Γa+1Γ2c−2b−j×Γ−a+2b+2c+i−jΓ−a+2b+c+i/2−j/2−i+j/2Γ−a+2b+c+iΓ1+2a−2b−2c−i+jB1ijΓc−b−j/2+1−−1i/4Γa−b−c+1/2−i/2+j/2Γb+1/2−j/2+j/2−−1j/41−−1iΓ−a+b+2c+i/2−3j/2+j/2+C1ijΓc−b−j/2+1+−1i/4Γ1+a−b−c−i/2+j/2Γb−j/2+j+1/2+−1j/41−−1iΓ−a+b+2c−1/2+i/2−3j/2+j+1/2.Theorem 1.
Fori,j=0,±1,±2 and ℜ2a−b+c+i>0, the following general summation formula holds true:The coefficientsA1ij, B1ij, and C1ij can be obtained from the tables of Aij, Bij, and Cij by simply changing a⟶2c−2b−j, b⟶2a−2b−2c+1−i+j, and c⟶c−j, respectively.(16)F65a,a2+1,2b−i,2c,12+a−b−c,−a+2b+2c+j;−1a2,12+b+c,1+a−2c,1+a−2b+i,1+2a−2b−2c−j=A2ij4b+c−1Γ1/2+b+cΓ1+a−2cΓ1+a−2b+iΓ1/2Γa+1Γ2b−i×Γ1+a−b−c−j+j/2Γ1+a−2b−2c+i/2−j/2−i+j/2Γ2cΓ1+a−2b−2c+iB2ijΓcΓb−i/2+1−−1i/4Γ1+a−b−2c−j+j/2Γ1+a−2b−c+i/2−j+j/2−−1j/41−−1i+C2ijΓc+1/2Γb−i/2+1+−1i/4Γ1/2+a−b−2c−j+j+1/2Γ1/2+a−2b−c+i/2−j+j+1/2+−1j/41−−1i.Theorem 2.
Fori,j=0,±1,±2 and ℜa−2b−2c+i>−1, the following general summation formula holds true:The coefficientsA2ij, B2ij, and C2ij can be obtained from the tables of Aij, Bij, and Cij by simply changing a⟶2b−i, b⟶2c, and c⟶1/2+a−b−c−j, respectively.(17)F652a−12,a+34,b,a−j,12+j,2a−b−i+j;−1a−14,12+2a−b,2a−j,12+a+j12+b+i−j=A3ij42a−b−1Γ1/2+2a−bΓ1/2+a+jΓ1/2+a−j+j/2Γ1/2Γ2a+1/2Γ2a−b−j×Γ1/2+b+i−jΓ−a+b+1/2+i/2−j/2−i+j/2Γ1/2−a+b+iΓ2a−b−i+jB3ijΓa−b/2−j/2+1−−1i/4Γa−b/2−i/2+j/2Γb/2+1/2+i/2−3j/2+j/2Γb/2+1/2−j/2+j/2−−1j/41−−1i+C3ijΓa−b/2+1/2−i/2+j/2Γa−b/2−j/2+1+−1i/4Γa/2+b/2+i/2−3j/2+j+1/2Γa/2+b/2−j/2+j+1/2+−1j/41−−1i.Theorem 3.
Fori,j=0,±1,±2 and ℜb−a+i>−1/2, the following general summation formula holds true:The coefficientsA3ij, B3ij, and C3ij can be obtained from the tables of Aij, Bij, and Cij by simply changing a⟶2a−b−j, b⟶2a−b−i+j, and c⟶a−j, respectively.(18)F65a2+1,a,b,a−b+12,a−b+12−i,1+a−2b+j;−1a2,b+12,1+a−b,b+12+i,2b−j=A4ij22a−2b−1Γ1/2+bΓ1+a−bΓ1/2+b+iΓ1/2Γa+1Γa−b+1/2×Γ1/2+b−j+j/2Γ−a+2b+i/2−j/2−i+j/2Γ1/2+a−b−iΓ−a+2b+iB4ijΓa/2−b/2+1/4Γa/2−b/2+1/4−i/2+1−−1i/4Γ−a/2+3b/2+1/4−j+j/2Γ−a/2+3b/2+i/2−j+j/2−−1j/41−−1i+C4ijΓa/2−b/2+3/4Γa/2−b/2+1/4−i/2+1+−1i/4Γ−a/2+3b/2−1/4−i/2−j+j+1/2Γ−a/2+3b/2−1/4+i/2−j+j+1/2+−1j/41−−1i.Theorem 4.
Fori,j=0,±1,±2 and ℜ2b−a+i>0, the following general summation formula holds true:The coefficientsA4ij, B4ij, and C4ij can be obtained from the tables of Aij, Bij, and Cij by simply changing a⟶1/2+a−b−i, b⟶1/2 +a−b, and c⟶b−j, respectively.(19)F65a,a2+1,b,c,1−b+i+j,1−c+i;−1a2,1+a−b,1+a−c,a+b−i−j,a+c−i=22b−2−i−jΓ1+a−bΓ1+a−cΓa+b−i−jΓa+c−iΓa+1Γa−iΓ1−b+i+jΓ1−c+i×Γ1−c+i/2−i/2Γ1−b+i/2+j/2−i+j/2Γa−b+c−i−jΓa+b+c−1−2i−jD1ijΓa/2+b/2−c/2−i−j+i+j+1/2Γa/2+b/2+c/2−1/2−i−j+j+1/2Γa/2−b/2−c/2+1Γa/2−b/2+c/2+1/2−i+i/2+E1ijΓa/2+b/2−c/2+1/2−i−j+i+j/2Γa/2+b/2+c/2−i−j+j/2Γa/2−b/2−c/2+1/2Γa/2−b/2+c/2−i+i+1/2.Theorem 5.
Fori=−3,−2,−1,0,1,2 and j=0,1,2,3, the following general summation formula holds true:The coefficientsD1ij and E1ij can be obtained from the tables of Dij and Eij by simply changing a⟶1+a−b−c, b⟶1−c+i, and c⟶1−b+i+j, respectively.(20)F65a,a2+1,2b,c,1+a−2c−j,1+2a−2b−2c−i−j;−1a2,1+a−2b,1+a−c,2c+j,−a+2b+2c+i+j=2−2a+4b+2c−2+i+jΓ1+a−2bΓ1+a−cΓ2c+jΓa+1Γ2bΓ1+a−2b−c×Γ−a+2b+2c+i+jΓ1+a−2c+i/2−j+i/2Γ1+a−2b−c−i/2−j/2−i+j/2Γc+jΓ1+a−2c−jΓ−2a+2b+4c−1+i+2jD2ijΓb−i/2−j/2+i+j+1/2Γ−a+b+2c−1/2+i/2+j/2+j+1/2Γ1+a−b−c−i/2−j/2Γ−b+c+1/2−i/2+j/2+i/2+E2ijΓb+1/2−i/2−j/2+i+j/2Γ−a+b+2c+i/2+j/2+j/2Γa−b−c+1/2−i/2−j/2Γ−b+c−i/2+j/2+i+j/2.Theorem 6.
Fori=−3,−2,−1,0,1,2 and j=0,1,2,3, the following general summation formula holds true:The coefficientsD2ij and E2ij can be obtained from the tables of Dij and Eij by simply changing a⟶1+2a−2b−2c−i−j, b⟶1+a−2c−j, and c⟶1+a−2b−c, respectively.(21)F65a,a+34,b,2a−12,2a−b−i,12+j;−1a−14,a+12,12+2a−b,12+b+i,2a−j=2i−j−1Γa+1/2Γ1/2+2a−bΓ2a−jΓ1/2+b+iΓaΓ2a+1/2Γ1/2+a−b×Γ1/2+a−b−i/2−i/2Γ1/2−i/2+j/2−i+j/2Γ1/2+jΓb+i−jΓ2a−b−jD3ijΓa−b/2−i/2−j+i+j+1/2Γb/2+i/2−j+j+1/2Γa−b/2+1/2−i/2Γb/2+1/2−i/2+i/2+E3ijΓa−b/2+1/2−i/2−j+i+j/2Γb/2+1/2+i/2−j+j/2Γa−b/2−i/2Γb/2−i/2+i+1/2.Theorem 7.
Fori=−3,−2,−1,0,1,2 and j=0,1,2,3, the following general summation formula holds true:The coefficientsD3ij and E3ij can be obtained from the tables of Dij and Eij by simply changing a⟶2a−b−i, b⟶1/2+a−b, and c⟶1/2+j, respectively.(22)F65a,b,a2+1,a−b+12,a−b+12−i,1+a−2b+j;−1a2,b+12,1+a−b,b+12+i,2b−j=2i+j−1Γb+1/2−iΓb+1/2−jΓ2b−jΓa−b+jΓ1−i−j−i+j/2πΓa+1ΓbΓ1+a−2b+jΓa−b+1/2+jΓ−a+3b−1/2−i−jD4ijΓ−a/2+3b/2−1/4+i/2−j+j+1/2Γa/2−b/2+1/4−i/2+i+j+1/2Γa/2−b/2+3/4−i/2Γ−a/2+3b/2+1/4−i/2−j+i/2+E4ijΓa/2−b/2+3/4−i/2+i+j/2Γ−a/2+3b/2+1/4+i/2−j+j/2Γa/2−b/2+1/4−i/2Γ−a/2+3b/2−1/4−i/2−j+i+1/2.Theorem 8.
Fori=−3,−2,−1,0,1,2 and j=0,1,2,3, the following general summation formula holds true:The coefficientsD4ij and E4ij can be obtained from the tables of Dij and Eij by simply changing a⟶a−b+1/2−i, b⟶1/2, and c⟶a−2b+1+j, respectively.Proof.
In order to establish (15) asserted in Theorem 1, we proceed as follows. In the transformation formula (9), if we take d=1+a−2c+j and e=1+2a−2b−2c−i+j, then for i,j=−2,−1,0,1,2, it takes the following form:(23)F65a,a2+1,2b,c−j,1+a−2c+j,1+2a−2b−2c−i+j,;−1a2,1+a−2b,2c−j,1+a−c+j,−a+2b+2c+i−j=Γ1+a−c+jΓ−a+2b+2c+i−jΓ1+aΓ−a+2b+c+j×F322c−2b−j,2a−2b−2c+1−i+j,c−j;11+a−2b,,2c−j.
We now observed that theF32 appearing on the right hand side of (20) can be evaluated with the help of generalized Watson’s summation theorem (7) by taking a⟶2c−2b−j, b⟶2a−2b−2c+1−i+j, and c⟶c−j and after some algebras, we easily arrive at the desired result (15).In exactly the same manner, the generalized summation formulas (16) to (22) asserted in Theorems 2 to 8 can be established. We, however, prefer to omit the details.


## 3. Corollaries

In this section, we shall mention some of the known as well as unknown results of our main findings.(24)F65a,a2+1,2b,c,1+a−2c,1+2a−2b−2c;−1a2,1+a−2b,2c,1+a−c,−a+2b+2c=22a−4b−1Γ1+a−2bΓ1+a−cΓc+1/2Γ1/2Γa+1Γ2c−2bΓ−a+2b+2cΓ1+2a−2b−2cΓc−bΓa−b−c+1/2Γb+1/2Γ2c−a+b,(25)F65a,a2+1,2b,c−1,2+a−2c,2+2a−2b−2c;−1a2,1+a−2b,2c−1,2+a−c,−a+2b+2c−1=22a−4b−1Γ1+a−2bΓ2+a−cΓc−1/2Γ1/2Γa+1Γ2c−2b−1Γ−a+2b+2c−1Γ−a+2b+c−1Γ−a+2b+cΓ2+2a−2b−2cΓc−b−1/2Γa−b−c+1ΓbΓ2c−a+b−3/2−Γc−bΓ3/2+a−b−cΓb+1/2Γ−a+b+2c−1,(26)F65a,a2+1,2b,c,1+a−2c,2a−2b−2c;−1a2,1+a−2b,2c,1+a−c,1−a+2b+2c=4a−2b−112c−aΓ1+a−2bΓ1+a−cΓc+1/2Γ−a+2b+cΓ1/2Γa+1Γ2c−2bΓ2a−2b−2cΓc−b+1/2Γa−b−cΓbΓ−a+b+2c+1/2−Γc−bΓa−b−c+1/2Γb−1/2Γ−a+b+2c.(27)F65a,a2+1,2b,c−1,2+a−2c,1+2a−2b−2c;−1a2,1+a−2b,2c−1,2+a−c,−a+2b+2c=4a−2b−112c−a−1Γ1+a−2bΓ2+a−cΓc−1/2Γ1/2Γa+1Γ2c−2b−1×Γ−a+2b+2cΓ−a+2b+c−1Γ−a+2b+c+1Γ1+2a−2b−2ca−cΓc−bΓa−b−c+1/2Γb+1/2Γ−a+b+2c−1−2c−a−2Γc−b−1/2Γ1+a−b−cΓbΓ−a+b+2c−1/2.Corollary 1.
In Theorem1, if we take ii=0,j=0,iii=0,j=1.\\iiii=1,j=0, and ivi=1,j=1, we get the following interesting results:The result (24) is the known result (11) due to Zhao [13] written in a different form.Similarly, other results can be obtained.(28)F65a,a2+1,2b,2c,a−b−c+12,−a+2b+2c;−1a2,1+a−2b,1+a−2c,2a−2b−2c+1,b+c+12=4b+c−1ΓbΓcΓ1+a−2b−cΓ1+a−b−2c×Γ1/2+b+cΓ1+a−2bΓ1+a−2cΓ1+a−b−c+Γ1/2Γa+1Γ2bΓ2c.(29)F65a,a2+1,2b,2c,12+a−b−c,1−a+2b+2c;−1a2,12+b+c,1+a−2b,1+a−2c,2a−2b−2c=4b+c−1Γ1/2+b+cΓ1+a−2bΓ1+a−2cΓa−b−cΓa−2b−2cΓ1/2Γa+1Γ2bΓ2cΓ1+a−2b−2cΓbΓcΓa−2b−cΓa−b−2c−Γb+1/2Γc+1/2Γ1/2+a−2b−cΓ1/2+a−b−2c.(30)F65a,a2+1,2b−1,2c,12+a−b−c,−a+2b+2c;−1a2,12+b+c,2+a−2b,1+a−2c,1+2a−2b−2c=4b+c−1Γ1/2+b+cΓ2+a−2bΓ1+a−2cΓ1+a−b−cΓ1+a−2b−2c2b−2c−1Γ1/2Γa+1Γ2b−1Γ2cΓ2+a−2b−2cΓbΓcΓ1+a−2b−cΓ1+a−b−2c−Γb−1/2Γc+1/2Γ1/2+a−b−2cΓ3/2+a−2b−c,(31)F65a,a2+1,2b−1,2c,12+a−b−c,1−a+2b+2c;−1a2,12+b+c,2+a−2b,1+a−2c,2a−2b−2c=4b+c−1Γ1/2+b+cΓ2+a−2bΓ1+a−2cΓa−b−cΓa−2b−2c2b−2c−1Γ1/2Γa+1Γ2b−1Γ2cΓ2+a−2b−2ca−2bΓbΓcΓ1+a−2b−cΓa−b−2c−a−2c−1Γb−1/2Γc+1/2Γ1/2+a−b−2cΓ1/2+a−2b−c.Corollary 2.
In Theorem2, if we take ii=0,j=0,iii=0,j=1.\\,iiii=1,j=0, and ivi=1,j=1, we get the following interesting results:The result (28) is the known result (12) due to Zhao [13] written in a different form.Similarly, other results can be obtained.(32)F6512,a,a+34,2a−12,2a−b,b;−1a−14,2a,a+12,2a−b+12,b+12=42a−b−1Γ1/2+a2Γ1/2+bΓa−b/22Γ1/2+2a−bΓ1/2Γ2a+1/2Γb/2+1/22Γ2a−b2,(33)F652a−12,a+34,a−1,b,32,1+2a−b;−1a−14,32+a,2a−1b−12,12+2a−b=42a−b−1Γa−1/2Γ3/2+aΓ1/2+2a−bΓ1/2Γ2a+1/2Γ2a−b−1Γb−1/2Γ−a+b−1/2Γ1/2−a+bΓ1+2a−bΓa−b/2−1/2Γa−b/2+1/2Γb/2Γb/2−1−Γa−b/2Γa−b/2−1Γa/2+b/2+1/2Γa/2+b/2−1/2,(34)F652a−12,a+34,a,b,12,2a−b−1;−1a−14,12+a,2ab+32,12+2a−b=42a−b−1Γa+1/22Γ3/2+bΓ1/2+2a−bΓ−a+b+1/2Γ1/2Γ2a+1/2Γ2a−bΓ3/2−a+bΓ2a−b−1Γa−b/2−1/2Γa−b/2+1/2Γb/2Γb/2+1−Γa−b/22Γa/2+b/2+1/22,(35)F652a−12,a+34,a−1,b,32,2a−b;−1a−14,32+a,2a−1b+12,12+2a−b=24a−2b−3Γa−1/2Γ3/2+aΓb+1/2Γ−a+b−1/2Γ1/2+2a−bΓ1/2Γ2a+1/2Γ1/2−a+bΓ1+2a−bΓ2a−b−12a−3Γa−b/2−1/2Γa−b/2+1/2Γa/2+b/22−2a−1Γa−b/22Γb/2+1/2Γb/2−1/2.Corollary 3.
In Theorem3, if we take ii=0,j=0,iii=0,j=1\\,iiii=1,j=0, and ivi=1,j=1, we get the following interesting results:The result (32) is the known result (13) due to Zhao [13] written in a different form.Similarly, other results can be obtained.(36)F65a,a2+1,a−2b+1,a−b+12,a−b+12,b;−1a2,a−b+1,2b,b+12,b+12=22a−2b−1Γ1/2+b3Γ1+a−bΓa/2−b/2+1/42Γ1/2Γa+1Γ−a/2+3b/2Γa−b+1/22Γ−a/2+3b/2+1/4,(37)F65a2+1,a,b,a−b+12,a−b+12,2+a−2b;−1a2,2b−1,b+12,b+12,1+a−b=22a−2b−1Γb+1/22Γb−1/2Γ1+a−bΓ−a+2b−1Γ1/2Γa+1Γ−a+2bΓ1/2+a−b2Γa/2−b/2+1/42Γ−a/2+3b/2−1−Γa/2−b/2+3/42Γ−a/2+b/2−1/42,(38)F65a2+1,a,b,a−b−12,a−b+12,1+a−2b;−1a2,2b,b+12,b+32,1+a−b=22a−2b−1Γb+1/22Γb+3/2Γ−a+2bΓ1+a−bΓ1/2Γa+1Γa−b+1/2Γa−b−1/2Γ1−a+2bΓa/2−b/2−1/4Γa/2−b/2+3/4Γ−a/2+3b/2−3/4Γ−a/2+3b/2+3/4−Γa/2−b/2+1/42Γ−a/2+3b/2Γ−a/2+3b/2+1/4,(39)F65a2+1,a,b,a−b−12,a−b+12,2+a−2b;−1a2,2b−1,b+12,b+32,1+a−b=4a−b−1Γb−1/2Γb+1/2Γb+3/2Γ−a+2b−1Γ1+a−bΓ1/2Γa+1Γa−b+1/2Γa−b−1/2Γ1−a+2bΓa/2−b/2−1/4Γa/2−b/2+3/4Γ−a/2+3b/2−1/4Γ−a/2+3b/2+1/4−Γa/2−b/2+1/42Γ−a/2+3b/2Γ−a/2+3b/2−1/4.Corollary 4.
In Theorem4, if we take ii=0,j=0,iii=0,j=1\\,iiii=1,j=0, and ivi=1,j=1, we get the following interesting results:The result (36) is the known result (14) due to Zhao [13] written in a different form.Similarly, other results can be obtained.(40)F65a,a2+1,b,c,1−b,1−c;−1a2,1+a−b,1+a−c,a+b,a+c=4b−1Γa+bΓa+cΓ1+a−bΓ1+a−cΓa+b−c/2Γa+b+c−1/2ΓaΓa+1Γa−b+cΓa+b+cΓa−b+c+1/2Γa−b−c/2+1,(41)F65a,a2+1,b,c,2−b,1−c;−1a2,1+a−b,a+b−1,1+a−c,a+c=2−2c−1Γ1+a−bΓa+b−1Γa+cΓ1+a−c1−bΓaΓa+1Γa−b+c−1Γa+b+c−2Γa+b−c−1/2Γa+b+c/2−1Γa−b−c+1/2Γa−b+c/2−Γa+b−c/2Γa+b+c−1/2Γa−b−c/2+1Γa−b+c+1/2,(42)F65a,a2+1,b,c,2−b,2−c;−1a2,1+a−b,a+b−1,1+a−c,a+c−1=22b−3Γ1+a−bΓa+b−1Γ1+a−cΓa+c−11−b1−cΓa−1Γa+1Γa−b+c−1Γa+b+c−3Γa+b−c/2Γa+b+c−3/2Γa−b−c/2+1Γa−b+c−1/2−Γa+b−c−1/2Γa+b+c/2−1Γa−b−c+1/2Γa−b+c/2−1.(43)F65a,a2+1,b,c,3−b,2−c;−1a2,1+a−b,a+b−2,1+a−c,a+c−1=4b−2Γ1+a−bΓa+b−2Γ1+a−cΓa+c−11−b2−b1−cΓa−1Γa+1Γa−b+c−2Γa+b+c−4a+c−3Γa+b−c−1/2Γa+b+c/2−2Γa−b−c+1/2Γa−b+c/2−a−c−1Γa+b−c/2−1Γa+b+c−3/2Γa−b−c/2+1Γa−b+c−1/2.Corollary 5.
In Theorem5, if we take ii=0,j=0,iii=0,j=1\\,iiii=1,j=0, and ivi=1,j=1, we get the following interesting results:The result (40) is the known result (10) due to Zhao [13] written in a different form.Similarly, other results can be obtained.(44)F65a,a2+1,2b,c,1+a−2c,1+2a−2b−2c;−1a2,1+a−2b,2c,1+a−c,−a+2b+2c=4−a+2b+c−1ΓbΓ2cΓ1+a−2bΓ1+a−cΓa+1Γ2bΓcΓ−b+c+1/2Γ1+a−b−cΓ−a+2b+2cΓ−a+b+2c−1/2Γ1+a−2b−cΓ−a+2b+4c−1,(45)F65a,a2+1,2b,c,a−2c,2a−2b−2c;−1a2,1+a−2b,2c+1,1+a−c,1−a+2b+2c=2−2a+4b+2c−1Γ2c+1Γ1+a−2bΓ1+a−cΓa+1Γ2bΓc+1×Γa−2b−cΓ1−a+2b+2cΓ1+a−2b−cΓ2−2a+2b+4cΓbΓ1/2−a+b+2cΓ1/2−b+cΓa−b−c−Γb+1/2Γ−a+b+2c+1Γa−b−c−1/2Γ1−b+c,(46)F65a,a2+1,2b,c,1+a−2c,2a−2b−2c;−1a2,1+a−2b,2c,1+a−c,1−a+2b+2c=2−2a+4b+2c−1Γ2cΓ1+a−2bΓ1+a−cΓ1/2+a−2cΓa+1Γ2bΓcΓ1+a−2c×Γa−2b−cΓ1−a+2b+2cΓ1+a−2b−cΓ−2a+2b+4cΓb+1/2Γ−a+b+2cΓ−b+cΓ1/2+a−b−c−ΓbΓ1/2−a+b+2cΓa−b−cΓ1/2−b+c,(47)F65a,a2+1,2b,c,a−2c,2a−2b−2c;−1a2,1+a−2b,2c+1,1+a−c,1−a+2b+2c=4a+2b+cΓ2c+1Γ1+a−2bΓ1+a−cΓ1/2+a−2cΓa+1Γ2bΓc+1Γa−2c×Γa−2b−c−1Γ2−a+2b+2cΓ1+a−2b−cΓ2−2a+2b+4c1−a+cΓbΓ3/2−a+b+2cΓ1/2−b+cΓa−b−c−1−a+3cΓb+1/2Γ1−a+b+2cΓa−b−c−1/2Γ1−b+c.Corollary 6.
In Theorem6, if we take ii=0,j=0,iii=0,j=1\\,iiii=1,j=0,andivi=1,j=1, we get the following interesting results:The result (44) is the known result (11) due to Zhao [13] written in a different form.Similarly, other results can be obtained.(48)F6512,a,a+34,2a−12,2a−b,b;−1a−14,2a,a+12,2a−b+12,b+12=Γ2aΓa+1/2Γb/2Γb+1/2Γa−b/2Γ1/2+2a−b2ΓaΓ2a+1/2ΓbΓb/2+1/2Γ2a−bΓ1/2+a−b/2,(49)F65a,a+34,2a−12,b,32,2a−b;−12a−1,a−14,a+12,b+12,12+2a−b=Γ2a−1Γa+1/2Γb+1/2Γ1/2+2a−b2ΓaΓ2a+1/2Γb−1Γ2a−b−1Γb/2−1/2Γa−b/2−1/2Γb/2Γa−b/2−Γb/2Γa−b/2Γb/2+1/2Γa−b/2+1/2,(50)F65a,a+34,2a−12,b,12,2a−b−1;−12a,a−14,a+12,b+32,12+2a−b=2Γ2aΓa+1/2Γb+3/2Γa−bΓ1/2+2a−bΓaΓ2a+1/2Γb+1Γ2a−bΓ1/2+a−bΓb/2+1Γa−b/2Γb/2+1/2Γa−b/2−1/2−Γb/2+1/2Γa−b/2+1/2Γb/2Γa−b/2,(51)F65a,a+34,2a−12,b,32,2a−b−1;−12a−1,a−14,a+12,b+32,12+2a−b=2Γ2a−1Γa+1/2Γb+3/2Γa−bΓ1/2+2a−bΓaΓ2a+1/2ΓbΓ2a−b−1Γ1/2+a−b2a−b−32Γb/2+1/2Γa−b/2−1/2Γb/2Γa−b/2−b−12Γb/2Γa−b/2Γb/2+1/2Γa−b/2−1/2.Corollary 7.
In Theorem7, if we take ii=0,j=0,iii=0,j=1,iiii=1,j=0, and ivi=1,j=1, we get the following interesting results:The result (48) is the known result (13) due to Zhao [13] written in a different form.Similarly, other results can be obtained.(52)F65a,a2+1,a−2b+1,a−b+12,a−b+12,b;−1a2,a−b+1,2b,b+12,b+12=Γ2bΓb+1/22Γa−b2Γa+1ΓbΓ1+a−2bΓa−b+1/2Γa/2−b/2+1/4Γ−a/2+3b/2−1/4Γ−a+3b−1/2Γa/2−b/2+3/4Γ−a/2+3b/2+1/4,(53)F65a,a2+1,b,a−b+12,a−b+12,a−2b+2;−1a2,2b−1,b+12,b+12,1+a−b=2Γb−1/2Γb+1/2Γ2b−1Γ1+a−bΓa+1ΓbΓa−2b+2Γa−b+3/2Γ−a+3b−3/2Γa/2−b/2+5/4Γ−a/2+3b/2−1/4Γa/2−b/2+3/4Γ−a/2+3b/2−3/4−Γa/2−b/2+3/4Γ−a/2+3b/2−3/4Γa/2−b/2+1/4Γ−a/2−3b/2−5/4.(54)F65a,a2+1,b,a−b+12,a−b−12,a−2b+1;−1a2,2b,b+12,b+32,1+a−b=2Γb−1/2Γb+1/2Γ2bΓa−bΓa+1ΓbΓa−2b+1Γa−b+1/2Γ−a+3b−3/2,Γ−a/2+b/2+1/4Γ−a/2+3b/2+3/4Γ−a/2−b/2−1/4Γ−a/2+3b/2+1/4−Γa/2−b/2+3/4Γ−a/2+3b/2+1/4Γa/2−b/2+1/4Γ−a/2+3b/2−1/4,(55)F65a,a2+1,b,a−b+12,a−b−12,a−2b+2;−1a2,2b−1,b+12,b+32,1+a−b=4Γb−1/22Γ2b−1Γ1+a−b3Γa+1ΓbΓa−2b+2Γa−b+3/2Γ−a+3b−5/2b−aΓa/2−b/2+3/4Γ−a/2+3b/2+1/4Γa/2−b/2+1/4Γ−a/2+3b/2−5/4−a−3b+2Γa/2−b/2+5/4Γ−a/2+3b/2−1/4Γa/2−b/2+1/4Γ−a/2+3b/2−3/4.Corollary 8.
In Theorem8, if we take ii=0,j=0,iii=0,j=1,iiii=1,j=0, and ivi=1,j=1, we get the following interesting results:The result (52) is the known result (14) due to Zhao [13] written in a different form.Similarly, other results can be obtained.


## 4. Concluding Remark

In this paper, we have provided as many as one-hundred and ninety-two summation formulas for the generalized hypergeometric seriesF65−1 in terms of gamma functions. This is achieved by means of applying generalized Watson’s and Dixon’s summation formulas obtained earlier by Lavoie et al. into a known transformation formula available in the literature. We believe that the results established in this paper have not been appeared in the literature before and represent a definite contribution to the theory of generalized hypergeometric function. Since the results established are in terms of gamma function, it is hoped that the results could be of potential use in the area of mathematics, statistics, and mathematical physics.Here,(56)A2−2=12c−1a−b−1a−b+1,A2−1=12a−b−1a−b+1,A20=14a−b−1a−b+1,A21=14a−b−1a−b+1,A22=18c+1a−b−1a−b+1,A1−2=1c−1a−b,A12=12c+1a−b.Here,(57)B2−2=ca+b−1−a+1b+1+2,B2−1=a+b−1+b2c−b−2c+1,B20=a2c−a−a−b2+1,B22=2cc+12c+1a+b−1−aa−1−bb−13cm−a−b−1a−b+1c+12c−a−b+1+ab,B11=2c−a+b−a−bc−b+1,B0−2=c−a−1c−b−1+c−1c−2,B02=c−a+1c−b+1+cc+1,B−1−2=2c−1c−2−a−bc−b−1,B−2−2=2c−1c−22c−1a+b−1−aa+1−bb+1+2,−a−b−1a−b+1c−12c−a−b−3+abB−2−1=2c−1a+b−1−a−b2+1,B−20=a2c−a+b2c−b−2c+1,B−22=ca+b−1−a−1b−1.Here,(58)C21=−8c2−2ca+b−1−a−b2+1,C22=−42c+a−b+12c−a+b+1−42c+a−b+12c−a+b+1,C12=−2cc+1+a−bc−a+1,C−1−2=2c−1c−2+a−bc−b−1,C−2−2=42c−a+b−32c+a−b−3,C−2−1=8c2−2c−1a+b+7−a−b2−7.Here,(59)D30=5a−b2+a+12−2a−b+1b+c,D20=12a−1a−4−b2−5a+1−a−b+1b+c,D21=b−1b−2−a−b+1a−b−c+3,D22=12a−c+2a−2b−c+5a−c+2a−2b+2−ac−3−b−1b−2c−2c−3,D12=aa−1+b+c−3c−2a−1,D02=12a−b−c+12+c−1c−3−b2+aD03=3ab+ca−b−c+4−a+1a+2−a−1b−1,D−13=c−1c−2−ba−c+1,D−20=12a−1a−2b−2−ca−b−1,D−22=12a−1a−2b−2c+bb+c,D−23=a−b−1c−1−bb+1,D−30=a−1a−2b−2c−4+bc,D−31=a−b−2a−c−1−ac,D−32=a−b−1a−b−2c−2−bc,D−33=bb+1+a−1a−b−c2a−b−2.Here,(60)E30=−a+3b2−a+32+2a−3b+5b+c,E21=−b−1b−2+a−b−2c+5a−b−c+3,E22=−2a−c+2a−2b−c+5,E12=−a−b−c+2a−b−c+3+b−1b−c+1,E03=a+2a+4−b2a+5−3ca−b−c+4+3,E−13=c−1c−2+ba−2b−c+1,E−23=ba−2c+2−b−c+1a−b−2c+1,E−30=a−2a−2b−2c−3+3bc,E−31=a−b−2a−2b−2c−3+bc,E−32=a−b−2a−b−2c−1+bc,E−33=a−1a−2−3ba−b−2−c2a−3b−4.



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*Source: 2788208-2022-04-25.xml*