On a new class of double integrals involving hypergeometric function

Available online at www.ispacs.com/jiasc Volume 2012, Year 2012 Article ID jiasc-00004, 20 pages

doi:10.5899/2012/jiasc-00004 Research Article

On a new class of double integrals involving

hypergeometric function

Medhat A. Rakha∗

Department of Mathematics and Statistics, College of Science

Sultan Qaboos University - Muscat - OMAN

Copyright 2012 c⃝Medhat A. Rakha. This is an open access article distributed under the Creative Com-mons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The aim of this research paper is to provide a large number of double integrals in the form of four master formulas. A large number of very interesting integrals follow special cases of our main findings. These results are obtained with the help of contiguous extensions of classical summation theorems such as of Watson, Dixon and Whipple on the sum of a3F2.

The double integrals established in this paper are simple, interesting, easily established and may be useful.

Keywords : q+1Fq hypergeometric functions, general hypergeometric functions, integral represen-tations.

1

Introduction

We recall an interesting double integral recorded in Edwards [1]

1 ∫

0 1 ∫

0

yα(1−x)α−1(1−y)β−1(1−xy)1−α−βdxdy = Γ(α)Γ(β)

Γ(α+β) (1.1)

provided Re(α)>0 and Re(β)>0.

In 1992, Lavoie et. al., [2] obtained the following contiguous extensions of the well known classical Watson’s theorem on the sum of a3F2 as follows:

∗_{Corresponding author. Permanent Address: Department of Mathematics - Faculty of Science, Suez}

3F2 



a, b, c

; 1

1

2(a+b+i+ 1), 2c+j





=Aij 2a+b+i−2Γ (

1 2a+

1 2b+

1 2i+

1 2

)

×

Γ(c+[j₂]+1₂)Γ(c−₂1a− 1₂b− |i+j₂ |−₂j −1₂)

Γ(₁ 2 )

Γ (a) Γ (b)

×

 



Bij

Γ(₁

2a+14(1−(−1)ij) )

Γ(₁ 2b

)

Γ(c−1₂a+1₂ +[j₂]− (−₄1)j(1−(−1)ij₎)_Γ(_c−1

2b+12 + [

j 2

])

+Cij

Γ(₁

2a+14(1 + (−1)ij) )

Γ(₁ 2b+12

)

Γ(c−1₂a+[j+1₂ ]+(−₄1)j(1−(−1)ij₎)_Γ(_c−1 2b+

[ j+1 2 ])    (1.2)

for i, j = 0,±1,±2 and [x] denotes the greatest integer less than or equal to x. The coefficients Aij, Bij and Cij are given in Tables (1, 2 and 3). The case (i= 0 =j) of

(1.2) reduces immediately to the well know Watson’s theorem. In the same paper [2], the following fifty special cases of (1.2) are given

3F2 



−2n, a+ 2n, c

; 1

1

2(a+i+ 1), 2c+j





=Dij (₁ 2 ) n ( 1

2a−c+34 −14(−1)i− [

j

2 +14(1−(−1)i) ])

n (

c+1₂ +[j₂])

n (₁

2a+14(1 + (−1)i) )

n

(1.3)

and

3F2 



−2n−1, a+ 2n+ 1, c

; 1

1

2(a+i+ 1), 2c+j





=Eij (₃ 2 ) n ( 1

2a−c+54 +14(−1)i− [

j

2 +14(1 + (−1)i) ])

n (

c+1₂ +[j+1₂ ])

n (₁

2a+ 14(3−(−1)i) )

n

(1.4)

fori, j= 0,±1,±2, and the coefficientsDij and Eij are given in Tables (4 and 5).

3F2 



a, b, c

; 1 1 +a−b+i, 1 +a−c+i+j





= 2−2c+i+j Γ (1 +a−b+i) Γ (1 +a−c+i+j)

×

Γ(b−₂i −|j₂|)Γ(

c−1₂(i+j+|i+j|))

Γ (a−2c+i+j+ 1) Γ (a−b−c+i+j+ 1) Γ (b) Γ (c)

×

 



Fij

Γ(1₂a−c+1₂ +[i+j+1₂ ])Γ(1₂a−b−c+i+ 1 +[j+1₂ ])

Γ(₁ 2a+12

)

Γ(₁

2a−b+ 1 + [_i

2 ])

+Gij

Γ(1₂a−c+ 1 +[i+j₂ ])Γ(₂1a−b−c+ 3₂+i+[j₂])

Γ(₁ 2a

)

Γ(₁

2a−b+12 + [_i+1 2 ])    (1.5)

provided that Re(a−2b−2c)>−2−2i−j, fori=−3,−2,−1,0,1,2 and j= 0,1,2,3. The coefficients Fij and Gij appear in Tables (6 and 7) at the end of the paper. Also,

ifFij is the left-hand side of (1.5), the natural symmetry

Fij(a, b, c) =Fi+j,−j(a, c, b) (1.6)

makes it possible to extend the result to j =−1,−2,−3. The case (j = 0 =i) of (1.5) reduces immediately to the well knows Dixon’s theorem.

In 1996, Lavoie et. al. in [4], obtained the following contiguous extension of the well known classical Whipple’s theorem on the sum of a3F2

3F2 



a, b, c ; 1 e, f





= Γ (e) Γ (f) Γ

(

c−1₂(j+|j|))

Γ(

e−c+1₂(i+|i|))

Γ(

a− 1₂(i+j+|i+j|))

22a−i−j_{Γ (e−a) Γ (f −a) Γ (e−c) Γ (a) Γ (c)}

×

 



Hij

Γ(₁ 2e−

1 2a+

1

4(1−(−1)i) )

Γ(₁ 2f−

1 2a

)

Γ(1₂e+1₂a− 1₂i+[−j₂])Γ(₂1f+1₂a−1₂i+1₄(−1)j₍₍₋₁₎j_{−1) +}[₋j 2

])

+Iij

Γ(₁

2e−12a+14(1 + (−1)i) )

Γ(1₂e+1₂a−₂1i+[−₂j +1₂])

× Γ

(₁

2f−12a+12 )

Γ(1₂f+1₂a−1₂ −₂1i+1₄(−1)j_(1−(−1)j_{) +}[₋j 2 +12

])  



(1.7)

wherea+b= 1 +i+j, e+f = 2c+ 1 +ifori, j= 0,±1,±2,±3. The coefficientsHij and

Iij are given in Tables (8,9,10 and 11) at the end of the paper.

It should be remarked here that whenever generalized hypergeometric functions reduce to closed forms, in particular, Gamma function products, the results are very important from an applications point of view. Applications of (1.4), (1.5) and (1.7) are given in the next section.

2

Main Results

In this section, the following four master formulas will be established

• First Double Integral

1 ∫ 0 1 ∫ 0

xb−1yc−1(1−x)12α− 1 2b+

1 2i−

1

2(1−y)c+j−1(1−xy)−αdxdy

=Aij 2a+b+i−2Γ (c) Γ (c+j) Γ (

1 2a−

1 2b+

1 2i+

1 2

)

×

Γ(c+[j₂]+1₂)Γ(

c−1₂a−1₂b−₂1|i+j| −1₂j−1₂)

Γ(₁ 2 )

Γ (a) Γ (2c+j)

×

 



Bij

Γ(₁

2a+14(1−(−1)ij) )

Γ(₁ 2b

)

Γ(c−1₂a+1₂ +[₂j]−(−₄1)j(1−(−1)ij₎)_Γ(_c−1

2b+12 + [

j 2

])

+Cij

Γ(₁

2a+ 14(1 + (−1)ij) )

Γ(₁ 2b+ 12

)

Γ(c−1₂a+[j+1₂ ]+(−₄1)j(1−(−1)ij₎)_Γ(_c−1 2b+

[ j+1 2 ])    (2.8)

fori, j= 0,±1,±2.

The coefficientAij, Bij andCij are given in Tables (1,2 and 3) at the end of the paper.

The conditions for the convergence of this integral are : (i)Re(b)>0, Re(c)>0 for j= 0,1,2; (ii)Re(c)>0 forj=−1,−2 and (iii)Re(a−b+i)>−1 fori= 0,±1,±2.

• Second Double Integral

1 ∫ 0 1 ∫ 0

xb−1yc−1(1−x)a−2b+i(1−y)a−2c+i+j(1−xy)−αdxdy

= 2

−2c+i+j_{Γ (1 +a−2b+i) Γ}(

b− 1₂i−1₂|i|)

Γ(

c−1₂|i+j| −1₂(i+j))

Γ (a−b−c+i+j+ 1)

×

 



Fij

Γ(1₂a−c+1₂ +[i+j+1₂ ])Γ(1₂a−b−c+ 1 +i+[j+1₂ ])

Γ(₁ 2a+12

)

Γ(₁

2a−b+ 1 + [_i

2 ])

+Gij

Γ(1₂a−c+ 1 +[i+j₂ ])Γ(₂1a−b−c+3₂ +i+[₂j])

Γ(₁ 2a

)

Γ(₁

for i = −3,−2,−1,0,1,2 and j = 0,1,2,3. The conditions for the convergence of this integral are : (i)Re(b)>0, Re(a−2b+i)>−1 fori= 0,±1,±2; (ii)Re(c)>0 and Re(a−2c+i+j)>−1, for i, j= 0,±1,±2.

Also because of the property (1.6), the result (2.9) can be extended to the results forj=−1,−2,−3. The coefficients Fij andGij are given in Tables (6 and 7) at the

end of the paper.

• Third Double Integral

1 ∫

0 1 ∫

0

xb−1yc−1(1−x)e−b−1(1−y)f−c−1(1−xy)−αdxdy

= Γ (b) Γ (e−b) Γ (f−c) Γ

(

c− 1

2(j+|i|)

)

× Γ

(

e−c+1₂(i+|i|))

Γ(

a− 1₂(i+j+|i+j|))

22a−i−j_{Γ (e−a) Γ (f −a) Γ (e−c) Γ (a)}

×

 



Hij

Γ(₁

2e−12a+14(1−(−1)i) )

Γ(₁

2f −12a )

Γ(1₂e+1₂a−1₂i+[−j₂])Γ(₂1f +1₂a−1₂i+(−₄1)j((−1)i_{−1) +}[₋j 2

])

+Iij

Γ(₁

2e−12a+14(1 + (−1)i) )

Γ(1₂e+1₂a−₂1 −1₂i+[−j+1₂ ])

× Γ

(₁

2f −12a+12 )

Γ(1₂f +1₂a−1₂ −1₂i+(−₄1)j(1−(−1)i_{) +}[−j+1 2

])  



(2.10)

fori, j = 0,±1,±2,±3. Also, wherea+b=i+j+ 1, e+f = 2c+ 1 +iand the coef-ficientHij andIij are given in Tables (8, 9, 10 and 11) at the end of the paper. The

• Fourth Double Integral 1 ∫ 0 1 ∫ 0

yc+j(1−x)c+j−1(1−y)c−1(1−xy)1−2c−j

× 2F1 



a, b,

; ₁1₋−_xyy

1

2(a+b+i+ 1)



dxdy

=Aij 2a+b+i−2Γ (c) Γ (c+j) Γ ( c+ [ j 2 ] + 1 2 ) × Γ(₁

2a+12b+12i+12 )

Γ(c−1₂a−1₂b−|i+j₂ |−(i+j₂ ))

Γ(₁ 2 )

Γ (a) Γ (b) Γ (2c+j)

×

 



Bij

Γ(₁

2a+14(1−(−1)i) )

Γ(₁ 2b

)

Γ(c−1₂a+1₂ +[j₂]−(−₄1)j(1−(−1)i₎)_Γ(_c−1 2b+

1 2 + [ j 2 ])

+Cij

Γ(₁

2a+14(1 + (−1)i) )

Γ(₁ 2b+12

)

Γ(c−1₂a+[j+1₂ ]+(−₄1)j(1−(−1)i₎)_Γ(_c− 1 2b+

[ −j+1 2 ])    (2.11)

where i, j = 0,±1,±2 and provided that Re(2c−a−b) > −1−i−2j for i, j = 0,±1,±2;j=−1,−2. The coefficientAij, Bij andCij are given in Tables (1,2 and 3)

at the end of the paper. The conditions for the convergence of this integral are : (i) Re(c+j)>0 forj = 0,±1,±2 andRe(2c+ 2j−a−b+i+ 1)>0 fori, j= 0,±1,±2.

3

Derivations

In order to derive the double integral (2.8), let us first evaluate a more general double integral (which is belived to be new)

1 ∫ 0 1 ∫ 0

xb−1yc−1(1−x)d−b−1(1−y)e−c−1(1−xyz)−adxdy

= Γ(b)Γ(c)Γ(d−b)Γ(e−c) Γ(d)Γ(e) 3F2





a, b, c ; z d, e



. (3.12)

To prove (3.12), denoting the LHS of (3.12) byI, expressing (1−xyz)−a _{with the help}

of the Binomial theorem as

(1−xyz)−a=

∞ ∑

n=0

(a)n

n! x

n_yn_zn

and changing the order of summation and integration (which are justified due to the uniform convergence of the series involved in the process), we get after little algebra

I = ∞ ∑ n=0 (a)n n! z n 1 ∫ 0 1 ∫ 0

Now, evaluating the double integrals with the help of the beta integral result viz.

1 ∫

0

xα−1_(1−x)β−1_dx= Γ(α)Γ(β)

Γ(α+β)

provided that Re(α)>0 and Re(β)>0. We have after some simplification

I = Γ(b)Γ(c)Γ(d−b)Γ(e−c) Γ(d)Γ(e)

∞ ∑

n=0

(a)n(b)n(c)n

(d)n(e)n

zn.

Finally, summing up the series, we get

I = Γ(b)Γ(c)Γ(d−b)Γ(e−c) Γ(d)Γ(e) 3F2





a, b, c ; z d, e



.

This completes the proof of (3.12).

Remark: In (3.12), if we setz= 1, b= 1, c−1 =α, d−b=α, e−c=β anda=α+β−1, we at once at Edwards’s result (1.1). Thus the result (3.12) may be regarded as the generalization of (2.8).

3.1 _{Derivation of (2.8)}

In (3.12), if we setz= 1,e= 2c+ 1 and e= 1₂(a+b+i+ 1), then fori, j= 0,±1,±2, it takes the following form

1 ∫

0 1 ∫

0

xb−1yc−1(1−x)12a− 1 2b+

1 2i−

1

2(1−y)c+j−1(1−xy)−adxdy

= Γ(b)Γ(c)Γ(

1

2a−12b+12i+12)Γ(c+j)

Γ(1₂a+1₂b+₂1i+1₂)Γ(2c+j) 3F2





a, b, c

; z

1

2(a+b+i+ 1), 2c+j



.

The 3F2 appearing on the R.H.S. can now be evaluated with the help of the known

result (1.2) and after little simplification, we easily arrive at the right hand side of (2.8). This completes the proof of (2.8).

In exactly the same manner the results (2.9) and (2.10) can be easily established by employing the results (1.3) and (1.4) respectively.

3.2 _{Derivation of (2.11)}

1 ∫

0 1 ∫

0

yα(1−x)α−1(1−y)β−1(1−xy)1−α−β2F1 



a, b

; ₁1₋−_xyy c



dxdy

= Γ(α)Γ(β) Γ(α+β) 3F2





a, b, α ; 1 c, α+β



 (3.13)

provided that Re(α)>0, Re(β)>0 and Re(c+α−a−b)>0.

In order to derive (3.13), express the hypergeometric function as a series, change the order of integration and summation, which is easily seen to be justified due to the uniform convergence of the series in the interval (0,1), evaluate the integral (1.1) due to Edwards, sum the resulting series, we finally arrive at the right-hand side of (3.13).

Now we are ready to evaluate (3.12). In (3.12), if we set α = c +j, β = c and c= 1₂(a+b+i+ 1), then fori, j = 0,±1,±2, it takes the following form

1 ∫

0 1 ∫

0

yc+j(1−x)c+j−1(1−y)c−1(1−xy)1−2c−j

× 2F1 



a, b

; ₁1₋−_xyy

1

2(a+b+i+ 1)



dxdy

= Γ(c)Γ(c+j) Γ(2c+j) 3F2





a, b, c

; 1

1

2(a+b+i+ 1), 2c+j



 (3.14)

The 3F2 on the right-hand side of (3.14) can now be summed with the help of the

known result (1.2) and after little simplification, we easily arrived at the right-hand side of (3.12).

For a similar investigation on a class of single integrals involving hypergeometric func-tion, see [9] and [10].

4

Special Cases

1 ∫ 0 1 ∫ 0

yc+j(1−x)c+j−1(1−y)c−1(1−xy)1−2c−j

× 2F1 



−2n, a+ 2n

; ₁1₋−_xyy

1

2(a+i+ 1)



dxdy

=Dij

Γ (c) Γ (c+j)(₁ 2 )

n (

1

2a−c+34 − (−1)i

4 −

[ j

2 +14(1−(−1)i) ]

n )

Γ (2c+j)(c+1₂ +[j₂])

n (₁

2a+14(1 + (−1)i) ) n (4.15) and 1 ∫ 0 1 ∫ 0

yc+j(1−x)c+j−1(1−y)c−1(1−xy)1−2c−j

× 2F1 



−2n−1, a+ 2n+ 1

; ₁1₋−_xyy

1

2(a+i+ 1)



dxdy

=Eij

Γ (c) Γ (c+j)(₃ 2 )

n (

1

2a−c+ 54+ (−1)i

4 −

[ j

2+ 14(1 + (−1)i) ]

n )

Γ (2c+j)(c+1₂+[j+1₂ ])

n (₁

2a+ 14(3−(−1)i) )

n

(4.16)

where i, j = 0,±1,±2; the coefficients Dij and Eij are given in Tables (4 and 5) at the

end of the paper.

On the other hand, if the Gegenbauer polynomials [5] are considered in the following form

C_na(1−2x) = (2a)n n! 2F1





−n, n+ 2a ; x

1 2 +a





= ( n!

a+1₂) n

P(a−

1 2,a−

1 2)

n (1−2x) (4.17)

wherePn(a,b)(x) denotes the Jacobi polynomials [5], then the cases (i= 0, j=−1) of (4.15)

and (4.16) given by the following interesting double integrals

1 ∫ 0 1 ∫ 0

yc−1(1−x)c−2(1−y)c−1(1−xy)2−2cC

a

2

2n (

1−2 1−y 1−xy

)

dxdy

= Γ (c) Γ (c−1) (a)2n

(

a−c+ 3₂) n

Γ (2c−1)(

c−1₂) n(2n)!

and

1 ∫

0 1 ∫

0

yc−1(1−x)c−2(1−y)c−1(1−xy)2−2cC

a

2

2n+1 (

1−2 1−y 1−xy

)

dxdy

=−Γ

(₁ 2 )

Γ (c−1) (a)_2n+1(

a−c+ 3₂) n

22c−2_Γ(

c+1₂) (

c+1₂)

n(2n+ 1)!

(4.19)

These are two typical illustrations. Similarly many other results can also be obtained from our main results but we shall not record them due to lack of space.

Acknowledgments

M. A. Rakha was supported by the research grant (IG/SCI/DOMS/12/06) funded by Sultan Qaboos University - Oman. The author is highly grateful to the referee for carefully reading the manuscript and providing certain very useful suggestions which led to a better presentation of this research article.

References

[1] J. Edwards, A treatise on the integral calculus with applications, examples and prob-lems II, Chelsea Publishing Company, New York (1954).

[2] J. L. Lavoie, F. Grondin and A. K. Rathie, Generalizations of Watson theorem on the sum of3F2, Indian J. Math.,32 1 (1992) 23 - 32.

[3] J. L. Lavoie, F. Grondin, A. K. Rathie and K. Arora, Generalizations of Dixon theo-rem on the sum a 3F2, Math. Comp., 63 (1994) 367 – 376.

[4] J. L. Lavoie, F. Grondin and A. K. Rathie, Generalizations of Whipple theorem on the sum of a3F2, J. Comput. Appl. Math., 72 (1996) 293 - 300.

http://dx.doi.org/10.1016/0377-0427(95)00279-0

[5] E. D. Rainville, Special Functions, The Macmillan Company, New York, (1960).

[6] M. A. Rakha, A new proof of the classical Watson’s summation theorem, App. Math. E-Notes, 11 (2011), 278 - 282.

[7] M. A. Rakha and A. K. Rathie, Generalization of classical summation theorems for the series2F1 and 3F2 with applications, Int. Trans. & Spec. Func., 2

¯2 11 (2011) 823 - 840.

http://dx.doi.org/10.1080/10652469.2010.549487

[8] Y. S. Kim, M. A. Rakha, and A. K. Rathie, Extensions of Certain Classical Sum-mation Theorems for the Series2F1,3F2, and4F3 with Applications in Ramanujan’s

Summations, Int. J. Math. Math. Sci., (2010) Article ID 309503, 26 pages.

i\j −2 −1 0 1 2

2 1

2(c−1)(a−b−1)(a−b+ 1)

1

2(a−b−1)(a−b+ 1)

1

4(a−b−1)(a−b+ 1)

1

4(a−b−1)(a−b+ 1)

1

8(c+ 1)(a−b−1)(a−b+ 1)

1 1

(c−1)(a−b)

1 (a−b)

1 (a−b)

1 2(a−b)

1 2(c+ 1)(a−b)

0 1

2(c−1) 1 1 1

1 2(c+ 1)

−1 1

(c−1) 1 2 2

2 (c+ 1)

−2 1

2(c−1) 1 1 2

2 (c+ 1) Table 1: Table for the coefficientsAi,j

i\ j −2 −1 0 1 2

2 c(a+b−1)

−(a+ 1)(b+ 1) + 2

a+b−1 +b(2c−b)−2c+ 1

a(2c−a)

−(a−b)2+ 1 2c(a+b−1)

2c(c+ 1)

× {(2c+ 1)(a+b−1)

−a(a−1)−b(b−1)} −(a−b−1)(a−b+ 1)

× {(c+ 1)(2c−a−b+ 1) +ab}

1 c−a−1 1 1 2c−a+b

−(a−b)(c−b+ 1) 2c(c+ 1) 0 (c−a−1)(c−b−1)

+(c−1)(c−2) 1 1 1

(c−a+ 1)(c−b+ 1) +c(c+ 1)

−1 2(c−1)(c−2)

−(a−b)(c−b−1) 2c−a+b−2 1 1 c−b+ 1

−2

2(c−1)(c−2)

× {(2c−1)(a+b−1)

−a(a+ 1)−b(b+ 1) + 2} −(a−b−1)(a−b+ 1)

× {(c−1)(2c−a−b−3) +ab}

2(c−1)(a+b−1)

−(a−b)2_{+ 1}

a(2c−a)

+b(2c−b)−2c+ 1 a+b−1

c(a+b−1)

−(a−1)(b−1)

Table 2: Table for the coefficientsBi,j

i\ j −2 −1 0 1 2

2 −4 −(4c−a−b−3) −8 −

(

8c2_{−2c(a+b−1)}

−(a−b)2+ 1)

−4(2c+a−b+ 1) (2c−a+b+ 1)

1 −(c−a−1) −1 −1 −(2c+a−b) −(2c(c+ 1)

+(a−b)(c−a+ 1))

0 4 1 0 −1 −4

−1 2(c−1)(c−2)

+(a−b)(c−b−1) 2c+a−b−2 1 1 c−b+ 1

−2 4(2c−a+b−3) (2c+a−b−3)

8c2−2(c−1)(a+b+ 7)

−(a−b)2₋₇ 8 4c−a−b+ 1 4

Table 3: Table for the coefficientsCi,j

i\j −2 −1 0 1 2 2

{(c−1)(a−1)+2n(a+2n)}

(c−1)(a+4n−1)

×_(a+4(a+1)_n+1)

(a+1)(a−1) (a+4n+1)(a+4n−1)

{(a−1)(2c−a−1)−4n(a+2n)}

(2c−a−1)(a+4n+1)

×_(a+4(a+1)_n−1)

{(a−1)(2c−a−1)−8n(a+2n)}

(2c−a−1)(a+4n+1)

×_(a(₊₄a+1)_n−1) D2,2

1 ₍a_c−(c₁₎₍+2_an₊₄−1)_{n) a+4}a_{n a+4}a_{n (2}a_c(2₋c_a−₎₍a_a−₊₄4n_n)₎

a(c+1)(2c−a) (c+1)(2c−a)(a+4n)

−_(c2₊₁₎₍₂an(2c_c+₋a_a+4_)(an₊₄+2)_n)

0 1− _(c−2₁₎₍₂n(a+2_c−n_a)₋₃₎ 1 1 1 1−_(c+1)(22n(a+2_c−n_a+1))

−1 1−2₍n_c−(2₁₎₍₂c+a_c+4_−an₋−₄₎2) 1− 4n

2c−a−2 1 1 1 +c2+1n

−2 D−2,−2 1−_(a−8₁₎₍₂n(a+2_c−n_a)₋₃₎ 1−_(a−4₁₎₍₂n(a+2_c−n_a)₋₁₎ 1 1 +_(c2₊₁₎₍n(a+2_a−n₁₎)

Table 4: Table for the coefficientsDi,j

D2,2=

(a+ 1){

(a−1)(c+ 1)(2c−a+ 1)−2an(6c+a+ 5)(2c−a+ 1) + 4n2

(5a2

+ 4a−5−4c(3c−a+ 4)) + 64n3

(a+n)}

(c+ 1)(2c−a+ 1)(2c−a−1)(a+ 4n+ 1)(a+ 4n−1)

D−2,−2= 1−

2an(6c+a−7)(2c−a−3)−4n2 {5a2

−4a−21−4c(3c−a−8)} −64n3

(a+n) (c−1)(a−1)(2c−a−3)(2c−a−5)

i\ j −2 −1 0 1 2 2 _(c−1)((a_a+1)(2_+4n+1)(c−a_a−₊₄3)_{n+3) (a+4n}(₊₁₎₍a+1)(4_a+4c−_na₊₃₎₍₂−3) _{c−1) (a+4n+1)(}2(a+1)_a+4n+3) E2,1

(a+1)(2c+a+4n+3) (c+1)(2c−a−1)(a+4n+1)

×(2c_(a−₊₄a−_n4₊₃₎n−1)

1 _(c−c−₁₎₍a−_a+42n−_n+2)2 _(a+42_nc₊₂₎₍₂−a−2_{c−1) a+4}1_{n+2 (2c}2₊₁₎₍c+a+4_a+4n+2_n+2)

(c+a+2)(2c−a) (c+1)(2c−a)(a+4n+2)

−_(c+1)(22n(3a_c−₋2_ac₎₍+4_a+4n+2)_n+2)

0 _c−₋1_{1 2}−_c−1₁ 0 _2c1_{+1 c+1}1

−1 E−1,−2 −(2_a(2c+_ca₋+4₁₎n) −_a1 −_a(2(2_cc₊₁₎−a) −(_ac−_(ca₊₁₎−2n)

−2 −(2c₍+_a−a+4₁₎₍n_c−₋₁₎₍₂1)(2c_c−−_aa₋−₅₎4n−5) E−2,−1 _a−₋2_{1 (}−_a−(4₁₎₍₂c−a_c+1)_{+1) (}−_a(2₋₁₎₍c−a_c+1)₊₁₎

Table 5: Table for the coefficientsEi,j

E2,1 =

(a+ 1){(4c+a+ 3)(2c−a−1)−8n(a+ 2n+ 2)}

(a+ 4n+ 1)(a+ 4n+ 3)(2c+ 1)(2c−a−1)

E−2,−1 =

− {(4c+a−1)(2c−a−3)−8n(a+ 2n+ 2)}

(a−1)(2c−1)(2c−a−3)

E−1,−2 =

− {(c+a)(2c−a−4)−2n(3a−2c+ 4n+ 6)} a(c−1)(2c−a−4)

i/j 0 1 2 3

3 5a−b

2_{+ (a+ 1)}2

−(2a−b+ 1)(b+c) − − −

2

1

2(a−1)(a−4)

−(b2_{−5a+ 1)}

−(a−b+ 1)(b+c)

(b−1)(b−2)

−(a−b+ 1)(a−b−c+ 3)

1

2(a−c+ 2)(a−2b−c+ 5)

[(a−c+ 2)(a−2b+ 2)−a(c−3)]

−(b−1)(b−2)(c−2)(c−3)

−

1 1 −(a−c+ 1) a(a−1)

+(b+c−3)(c−2a−1) −

0 1 −1

1 2

{

(a−b−c+ 1)2 +(c−1)(c−3)−b2+a}

3ab+c(a−b−c+ 4)

−(a+ 1)(a+ 2)−(a−1)(b−1)

−1 1 1 b+c−1 (c−1)(c−2)

−b(a−c+ 1)

−2

1

2(a−1)(a−2b−2)

−c(a−b−1) (a−b−1)

1

2(a−1)(a−2b−2c)

+b(b+c)

(a−b−1)(c−1)

−b(b+ 1)

−3

(a−1)

×(a−2b−2c−4) +bc

(a−b−2)(a−c−1)

−ac

(a−b−1)(a−b−2c−2)

−bc

b(b+ 1) + (a−1)(a−b)

−c(2a−b−2)

Table 6: Table for the coefficientsFi,j

i/j 0 1 2 3

3 −a+ 3b

2_{−(a+ 3)}2

+(2a−3b+ 5)(b+c) − − −

2 −2 −(b−1)(b−2)

+(a−b−2c+ 5)(a−b−c+ 3) −2(a−c+ 2)(a−2b−c+ 5) −

1 −1 a−2b−c+ 3 −(a−b−c+ 2)(a−b−c+ 3)

+(b−1)(b−c+ 1) −

0 0 1 −2 (a+ 2)(a+ 4)−b(2a+ 5)

−3c(a−b−c+ 4) + 3

−1 1 1 −(b−c+ 1) (c−1)(c−2)

+b(a−2b−c+ 1)

−2 2 a−b−2c−1 2 b(a−2c+ 2)

−(b−c+ 1)(a−b−2c+ 1)

−3 (a−2)(a−2b−2c−3)

+3bc

(a−b−2)(a−2b−2c−3) +bc

(a−b−2)(a−b−2c−1) +bc

(a−1)(a−2)

−3b(a−b−2)

−c(2a−3b−4) Table 7: Table for the coefficientsGi,j

i/j −3 −2 −1

3

−(a+ 2)(a−3) +3c(c+ 3)

−e(3c−e+ 5)

−(a−1)(a+e−3) +c(a+c)

−(a−1)(a−2) +c(2c−e+ 2)

2

−(a+ 1)(a−2) +c(a+c+ 3)

−e(2c−e+ 3)

−1

2(a+ 1)(a−2)

+c(c+ 2)

−e

3(2c−e+ 3)

c−a+ 1

1

(a−1)(a+ 2)

−2c(c+ 2) +e(3c−e+ 3)

a−c+e−1 1

0 e(2c−e+ 1)

+a(a−c+ 1)

a(a+ 1)

+e(2c−e+ 1) 1

−1 − e(2c−e+ 1)

+(a+ 2)(c−e) 2c−e

−2 − 1

2p2(a, c, e)

e(2c−e−1)

−ac

−3 − − −

Table 8: Table for the coefficientsHi,j;j=−3,−2,−1

i/j 0 1 2 3

3

e(2c−e)

−(a−6)(a−c+e)

−c−11

− − −

2 −

1

2(a−1)(a−2)

+1

2(e−2)(2c−e+ 1)

e(2c−e+ 3)

−(a+ 3)(c−1)−6

1

2p1(a, c, e) −

1 1 2c−e

e(2c−e+ 4) +a(c−e+ 1)

−7c−1

−

0 1 1 (a−1)(a−2)

+(e−2)(2c−e−1)

(a+ 3)(1 +a−c) +e(2c−e+ 1)

−6a

−1 1 1 a−c+e−1 (a−1)(a−2)

+(e−c)(2c−e−2)

−2 −

a

2(a+ 1)

+e

3(2c−e−1)

c−a−1

−1

2(a+ 1)(a−2)

+c(c−2)

−e

2(2c−e−1)

−(a−1)(a+ 1) +c(a+c−2)

−e(2c−e−1)

−3 e(2c−e−a−3)

−(a+ 2)(a−c+ 1)

−(a−1)(a+ 2) +(c−1)(2c−e)

−2c

−(a−1)(a+ 2) +a(c−e) +c(c−3)

−(a+ 2)(a−3) +3c(c−3)

−e(3c−e−4) Table 9: Table for the coefficientsHi,j;j= 0,1,2,3

p1(a, c, e) =−a(a−5)[2c(c−e) + (e−1)(e−2) + 2] + (e−1)(e−4)(2c−e) 2

i/j −3 −2 −1

3

(a+ 1)(a−2)

−c(c+ 3) +e(c−e+ 3)

(a−1)(a−e+ 1) +c(a−c−2)

(a−1)(a−2)

−c(e−2)

2

−(a−1)(a+ 2) +c(a−c+ 3)

−e(2c−e+ 3)

−2 −(a+c−1)

1 −a(a+ 1)

−e(c−e+ 1) −(a+c−e+ 1) −1

0 e(2c−e+ 1)

+(a+ 2)(a+c+ 1) 2 1

−1 − e(2c−e+ 1)

−a(c−e) e

−2 − 2(e+ 1)(2c−e) c(a+ 2) +e(2c−e−1)

−3 − − −

Table 10: Table for the coefficientsIi,j;j=−3,−2,−1

i/j 0 1 2 3

3

−e(2c−e+a+ 2) +(a+ 3)(a+c+ 1)

−6a

− − −

2 −2 −(a−1)(c−1)

−(e−3)(2c−e) −2(e−3)(2c−e) −

1 −1 −(e−2)

(a+ 3)(c+ 1)

−e(2c−e+a)

−6

−

0 0 −1 −2

−(a−7)(a+c−2) +e(2c−e+ 1)

−3(a−1)

−1 1 1 a+c−e+ 1 (a−1)(a−2)

+(e−2)(c−e)

−2 2 a+c−1 2

(a−1)(a−3)

−c(c−a) +e(2c−e−1)

−3 e(2c−e+a−1)

−a(a+c+ 1)

−a(a+ 1) +e(c−1)

−(a+ 1)(a−2)

−a(c−e) +c(c−3)