Gen. Math. Notes, Vol. 14, No. 1, January 2013, pp. 21-34
ISSN 2219-7184; Copyright © ICSRS Publication, 2013
www.i-csrs.org
Available free online at http://www.geman.in
Generalized Elliptic-Type Integrals and
Generating Functions with Aleph-Function
V.B.L. Chaurasia1 and Vinod Gill2
1
Department of Mathematics, University of Rajasthan,
Jaipur-302004, Rajasthan, India
E-mail: drvblc@yahoo.com
2
Department of Mathematics, Arya Institute of Engineering and Technology,
Kukas, Jaipur- 302028, Rajasthan, India
E-mail: vinod.gill08@gmail.com
(Received: 7-10-12 / Accepted: 7-12-12)
Abstract
In view of the great importance and applications of elliptic-type integrals in
certain problems of radiation physics and nuclear technology, in this paper we
obtain certain new theorems on generating functions. The results obtained in the
present paper are of manifold generality and basic in nature. Besides deriving
various known and new elliptic-type integrals and their generalizations these
theorems can be used to evaluate various Euler-type integrals involving a number
of generating functions.
Keywords: Elliptic-type integrals, Euler-type integrals, Generating functions,
Aleph (ℵ)-function.
1
Introduction
Ω j ( k) =
π
∫0
(1 − k cos θ )
2
− j−
1
2
d θ,
(1)
22
V.B.L. Chaurasia at al.
where j = 0,1,2,… and 0 ≤ k < 1 was studied by Epstein-Hubbell [10], for the first
time. Due to its occurrence in a number of physical problems [3,4,11,13,20,21,28],
in the form of single and multiple integrals, several authors notably Kalla [14,15]
and Kalla et al. [17], Kalla and Al-Saqabi [16], Kalla et al. [18], Salman [23],
Saxena et al. [26] and Srivastava and Bromberg [31], have investigated various
interesting unifications (and generalizations) of elliptic-type integrals (1).
Some of the generalizations of elliptic-type integral (1) are as follows:
Kalla [14,15] introduced the generalization of the form:
R µ ( k, α, γ ) = ∫
θ
θ
cos 2α −1   sin 2 γ −2α −1  
2
 2  dθ,
π
0
(1 − k 2 cosθ)
µ+
(2)
1
2
1
where 0 ≤ k < 1, Re( γ ) > Re(α ) > 0, Re(µ ) > − .
2
Results for this generalization are also derived by Glasser and Kalla [12].
Srivastava and Siddiqi [30] have given an interesting unification and extension of
the families of elliptic-type integrals in the following form:
Λ((αλ,,βµ)) (ρ; k) = ∫
π
0
θ
θ
cos 2α −1  sin 2β−1  
−λ
2
 2  1 − ρ sin 2  θ  dθ,
 

1
 2 
µ+

(1 − k 2 cos θ ) 2
(3)
where 0 ≤ k < 1, Re(α ) > 0, Re(β ) > 0, λ, µ ∈C, | ρ | < 1.
To study generalized form of equation (3) and its asymptotic expansion, see Kalla
and Tuan [19]. Saxena and Kalla [27] have studied a family of elliptic-type
integrals of the form:
)
Ω (( ασ ,β,...,
σ
1
=∫
n −2 :δ,µ )
π
0
cos
(ρ1 ,..., ρ n −2 , δ; k)
n −2

 2 β −1  θ 
2  θ 
 sin
  ∏ 1 − ρ j sin  
2
 2  j=1 
 2 
2α −1  θ
⋅ (1 − k cosθ )
2
−µ −
−σ j

2  θ 
1 + δ cos  2 
 

1
2 dθ ,
where 0 ≤ k < 1, Re(α ) > 0, Re (β ) > 0 ; σ j ( j = 1,..., n − 2 ),

δ
2k 2
γ, µ ∈ C ; max  | ρ j | ,
, 2
1+ δ k −1


 < 1.

−γ
(4)
Generalized Elliptic-Type Integrals and…
23
To study extension form of equation (4), see Saxena and Pathan [24].
In a recent paper [8], Chaurasia and Pandey investigated a new family of unified
and generalized elliptic-type integrals:
,β )
Ω ((λα,,βτ) ) ((ρ i ), ( δ i ); k j ) = Ω λ( α,...,
λ
i j
=∫
1
π
cos
0
2α −1  θ
N ,τ1,...,τM
 2 β −1  θ 
 sin
 ∏
2
 2  i =1
(ρ1 ,..., ρ N , δ1 ,..., δ N ; k 1 ,..., k Μ )

2θ 
2  θ 
1 + ρ i sin  2  + δ i cos  2 
 
 

N
M
⋅ ∏ [1 − k 2j cosθ ]
−τ j
− λi
dθ ,
(5)
j=1
where min (Re(α ), Re(β )) > 0, | k j | <1; λ i , τ j ∈C;

2 k 2j
δ − ρi

max  | ρ i |, | δ i | , 2
, i
1 + δi
k j −1




 < 1 (i = 1,..., N and j = 1,..., M),


which includes most of the known generalized and unified families of elliptic-type
integrals (including those discussed in (1) through (4). But due to lack of space we
can not able to discuss here various interesting unifications (and generalizations)
of the Elliptic-type Integrals investigated by several authors. For more details also
see [15,25,24,2,1,22] ).Upon a closer examination of the above equation (5), it can
be seen that the family of elliptic-type integral Ω ((λα,,βτ) ) ((ρ i ), ( δ i ); k j ) can be put
i j
into the following form involving Euler-type integral:
)
Ω((λα,β,...,
λ
1
M
=∏
j=1
N ;τ1,..., τM )
−τ
(1 − k 2j ) j
(ρ1 ,..., ρ N , δ1 ,..., δ N ; k 1 ,..., k Μ )
N
∏
i =1
M
.∏
j=1
(1 + δ i )
−λ i
1
∫0
 2ω k 2 
j 
1 −
2
 k − 1
j


−τ j
ωβ−1 (1 − ω) α −1
N
∏
i =1
 ( δ i − ρ i )ω 
1 −

1 + δ i 

−λi
dω.
(6)
A two-variable generating function F(x,t) possess a formal (not necessarily
convergent for t ≠ 0) power series representation in t, can be written in the
following form
24
V.B.L. Chaurasia at al.
∞
F(x, t) = ∑ C n f n ( x) t n ,
(7)
n =0
where each member of the generalized set {f n ( x)}∞
is independent of x and t.
n =0
Special functions have been around for centuries. No one can imagine
mathematics without Gaussian and confluent hypergeometric function, associated
Legendre and Laguerre polynomials, Bessel functions and many more. The most
well known application areas are in physics, engineering, chemistry, computer
science and statistics. On several occasions, the solution of enumeration problems
involving combinatorial objects requires knowledge from special function theory.
Earlier the emphasis was on special functions satisfying linear differential
equations, but this has now been extended to difference equations, partial
differential equations, non linear differential equations and fractional differential
equations[5,6].
The Aleph (ℵ)-function, introduced by Südland et al. [32], however the notation
and complete definition is presented here in the following manner in terms of the
Mellin-Barnes type integrals [see also 33]:
n
ℵ[z] = ℵm,
pi ,qi ,ci ;r

z


=
(a j ,A j )1,n , [ ci ( a ji ,A ji )]n +1,p ;r 
i

(b j ,B j )1,m ,[ci ( b ji ,B ji )]m +1,qi ;r 

1
Ω m,n
(s) z −s ds.
2πi ∫L pi ,qi ,ci ;r
(8)
for all z ≠ 0, where i = − 1 and
m
∏
n
Ω m,
(s) =
p , q , c ;r
i i i
j=1
pi
r
∑
i =1
ci
n
Γ( b j + β js) ∏ Γ(1 − a j − A js)
∏
j=n +1
j=1
Γ( a ji + A ji s)
qi
∏
j= m +1
,
Γ(1 − b ji − B ji s)
for convergence conditions and other details of Aleph (ℵ)-function, see Südland
et al. [32].
2
Theorems
In this section we derive two new theorems and their corollaries on generating
functions associated with ℵ-function and the families of elliptic-type integrals.
Generalized Elliptic-Type Integrals and…
25
These theorem and corollaries can be used to establish various known and new
elliptic-type integrals. Some of the significant applications of the results derived
in this section are discussed in the section 3.
Theorem 1. Consider the generating function F(x,t) and ℵ-function defined in (7)
and (8) respectively, then
1
∫0

,n'
( zω) ξ
ωα −1 (1 − ω) γ −α −1 ℵm'
pi ,qi ,ci ;r 

η
(a j ,A j )1,n' , [ ci ( a ji , A ji )]n'+1,p ;r
i
( b j, B j )1,m' ,[ ci ( b ji , B ji )]m'+1,q ;r
i




∞
µ
. F[x, tω (1 − ω ) ] dω = Γ(γ − α ) ∑ C n f n ( x) t n (γ − α ) µn
n =0
.
ℵpm',+n'1,+q1 +1,c ;r
i
i
i

z ξ


(1−α −ηn,ξ ),( a j,A j )1,n' ,[ci ( a ji ,A ji )]n'+1,p ;r

i
,
( b j, B j )1,m' ,[ ci ( b ji , B ji )]m'+1,q ;r , (1− γ −ηn −µη,ξ ) 
i

(9)
Provided that

 b j 
Re α + ξ min   > 0, Re [ γ − α] > 0, Re(η) > 0, Re(µ ) > 0, j = 1,..., m' and
 B 

 j 

π
ψ i > 0 , | arg z ξ | < ψ i ; i = 1,..., r
2
π
ψ i ≥ 0 , | arg z ξ | < ψ i and R {Λ i } + 1 < 0,
2
where
qi
n'
m'
 pi


ψ i = ∑ A j + ∑ B j − c i ∑ A ji + ∑ B ji  ,


j=1
j=1
j= m'+1
 j=n'+1

pi
m'
n'
 qi
 1
Λ i = ∑ b j − ∑ a j + c i  ∑ b ji − ∑ a ji  + ( p i − q i ).

 2
j=1
j=1
j= n'+1
 j=1

(10)
(11)
Corollary 1. Let the generating function F(x,t) and ℵ-function defined in (7) and
(8) respectively, then
1
∫0
ω
α −1
(1 − ω)
γ −α −1
(a j ,A j )1,n' , [ ci ( a ji ,A ji )]n'+1,p ;r

i
m',n'
ξ
ℵp ,q ,c ;r {z(1 − ω)} ( b ,B ) ,[c ( b ,B )]
i i i 
j j 1,m' i ji ji m'+1,qi ;r





26
V.B.L. Chaurasia at al.
∞
. F[x, tωη (1 − ω) µ ] dω = Γ( α ) ∑ C n f n ( x) t n ( α ) ηn
n =0
.
(1− γ + α −µn,ξ ),( a j, A j )1,n' ,[ci ( a ji ,A ji )]n'+1,p ;r 

i
m', n'+1
ξ
,
ℵp +1,q +1,c ;r z ( b ,B ) ,[ c ( b ,B )]
,
(
1
−
γ
−
η
n
−
µη
,ξ ) 
i
i
i 
j j 1,m' i ji ji m'+1,qi ;r


(12)
provided that

 b j 
Re[α] > 0, Re  γ − α + ξ min   > 0, Re(η) > 0, Re(µ ) > 0, j = 1,..., m' and
 B 

 j 

remaining condition is same as we already obtained in Equation (10) and (11).
Corollary 2. Let the generating function F(x,t) and ℵ -function defined in (7) and
(8) respectively, then
1
∫0

,n'
{z(1 − ω)ω}ξ
ωα −1 (1 − ω) γ −α −1 ℵm'
pi ,qi ,ci ;r 

η
µ
. F[x, tω (1 − ω) ] dω =
∞
∑
n =0

, n'+2
z ξ
. ℵm'
pi + 2,qi +1,ci ;r 

(a j, A j )1,n' , [ ci ( a ji , A ji )]n'+1,p ;r
i
( b j,B j )1,m' ,[ ci ( b ji , B ji )]m' +1,q ;r
i




C n f n ( x) t n
(1−α −ηn,ξ ),(1+ α − γ −µn,ξ ),( a j, A j )1,n' ,[ci ( a ji ,A ji )]n'+1,p ;r
i
( b j, B j )1,m' ,[ci ( b ji ,B ji )]m'+1,q ;r , (1− γ −ηn −µη, 2 ξ )
i

,


(13)
provided that


 b j 
 b j 





Re α + ξ min
> 0, Re γ − α + ξ min   > 0, Re(η) > 0, Re(µ ) > 0, j = 1,..., m'
 B 
 B 


 j 
 j 


and remaining condition is same as we already obtained in (10) and (11).
Now, we state another modification of the theorem 1 which can be used to obtain
various new interesting generalizations of Elliptic-type integrals.
Theorem 2. Consider the generating function F(x,t) and ℵ-function defined in (7)
and (8) respectively, then
1
∫0
ω
β−1
(1 − ω)
α −1
(a j,A j )1,n' , [ci ( a ji ,A ji )]n'+1,p ;r

i
m',n'
ξ

ℵp ,q ,c ;r ( zω) ( b ,B ) ,[ c ( b ,B )]
i i i 
j j 1,m' i ji ji m'+1,qi ;r





Generalized Elliptic-Type Integrals and…
 2ω k 2 
ℓ 
1 −
 k 2 − 1
ℓ


M
∏
.
ℓ =1
− τℓ
N
∏
ℓ '=1
∞
M
= Γ(α ) ∑ C n f n ( x) t (α ) µn ∏
n
n =0
.
(λℓ ' ) n ℓ'
n ℓ' !
ℓ =1
, n'+1
ℵm'
p +1, q +1,c
i
i
i ;r
 ξ
z

27
 ( δ ℓ ' − ρ ℓ ' )ω 
1 −

1 + δ ℓ ' 

∞
∑
n ℓ =0
−λ ℓ '
(τ ℓ ) n ℓ  2k ℓ2 


n ℓ !  k ℓ2 − 1 
nℓ
F [x, tωη (1 − ω) µ ] dω
N
∞
ℓ '=1
n ℓ ' =0
∏ ∑
 (δ ℓ ' − ρ ℓ ' 


 1 + δ ℓ' 
nℓ'
(1− β −ηℓ −η ℓ ' −ηn,ξ ),( a j , A j )1, n' ,[ ci (a ji , A ji )]n'+1, p ;r

i
( b j , B j )1, m' ,[ ci ( b ji , B ji )]m'+1,q ;r , (1−α − β − n ℓ − n ℓ ' −ηn − µη ,ξ ) 
i

,

(14)
Provided that

 b j 
Re β + ξ min   > 0, Re [α] > 0, Re(η) > 0, Re(µ ) > 0; δ ℓ ' , ρ ℓ ' , λ ℓ ' , τ ℓ ∈C; | k ℓ | < 1
 B 

 j 


2k ℓ2
δ − ρ ℓ'

and
max  | ρ ℓ ' |, | δ ℓ ' | , 2
, ℓ'
1 + δ ℓ'
kℓ −1

where j = 1,..., m' , ℓ' = 1,2,..., N, ℓ = 1,2,..., M,
we already obtained in (10) and (11).


 < 1,

and remaining condition is same as
Proof. Expressing F(x,t) by its power series form (7) in the integral (9) changing
the order of integration and summation, which is permissible due to uniform
convergence of the series involved. Using the definition (8) of the ℵ-function in
the evaluation of the resulting integral, we get the result (9), which proves
theorem 1.
The proof of Theorem 2 and Corollaries 1 and 2 are similar to that of theorem 1.
3
Applications
In view of the importance and usefulness of the theorems and corollaries
discussed in the last section, we mention some interesting applications, which
indicates manifold generality of the results obtained in this article.
(i)
Consider the generating function [29]
F(x, t) = (1 − xt)
−σ
∞
= ∑ (σ) n
n =0
xn tn
,
n!
(15)
and by the use of the theorem 1, under the stated condition, we get the following
interesting results:
28
1
∫0
V.B.L. Chaurasia at al.

ω α −1 (1 − ω ) γ −α −1 ℵpm',,qn' ,c ;r (zω ) ξ

i i i
(a j ,A j )1,n' , [ci ( a ji , A ji )]n'+1,p ;r
i
( b j ,B j )1,m' ,[ci ( b ji ,B ji )]m'+1,q ;r
i



⋅ [1 − xt ω η (1 − ω ) µ ]−σ dω
∞
(σ ) n x n t n (γ − α ) µn
n =0
n!
= Γ(γ − α ) ∑
.
ℵpm',+n'1,+q1 +1,c ;r
i
i
i

z ξ


(1−α −ηn,ξ ),( a j,A j )1,n' ,[ci ( a ji ,A ji )]n'+1,p ;r

i
,
( b j, B j )1,m' ,[ ci ( b ji , B ji )]m'+1,q ;r , (1− γ −ηn −µη,ξ ) 
i

(16)
θ
θ
when we put ω = cos 2   and cos θ = 2 cos 2 − 1 the above equation (16) gives
2
2
the following generalization of the elliptic-type integral
π
∫0
cos
2α −1  θ 
  sin
2
2 γ − 2α −1  θ 


m',n'
2
  ℵp ,q ,c ;r z cos
i i i 
2
 

 θ 
 
 2 
ξ
(a j ,A j )1,n' , [ ci ( a ji ,A ji )]n'+1,p ;r
i
( b j ,B j )1,m' ,[ ci ( b ji ,B ji )]m'+1,q ;r
i




.
∞
( σ ) n x n t n ( γ − α ) µn
n =0
n!
= Γ( γ − α ) ∑
.
ℵpm'+,n'1,+q1 +1,c ;r
i
i
i

z ξ


(1−α −ηn,ξ ),( a j,A j )1,n' ,[ ci ( a ji ,A ji )]n'+1,p ;r

i

( b j, B j )1,m' ,[ ci ( b ji , B ji )m'+1,q ;r , (1− γ −ηn −µη,ξ ) 
i

.
(17)
θ
θ
If we setting ω = sin 2   and using cosθ =1 − 2 sin 2   and σ → 0 in (16), we get
2
2
the following result:
π
∫0
sin
= Γ( γ
2α −1  θ 
  cos
2
2 γ − 2α −1  θ 
− α ) ℵpm'+,n'1,+q1 +1,c ;r
i
i
i


m', n'
2
  ℵp ,q ,c ;r z sin
2
i i i 
 

z ξ



 θ 
 
 2 
ξ
(a j ,A j )1,n' , [ ci ( a ji , A ji )]n'+1,p ;r
i
( b j ,B j )1,m' ,[ ci ( b ji ,B ji )]m'+1,q ;r
i
(1−α ,ξ ),( a j ,A j )1,n' ,[ ci ( a ji ,A ji )]n'+1,p ;r 
i
.
( b j, B j )1,m' ,[ci ( b ji , B ji )]m'+1,q ;r , (1− γ ,ξ ) 
i

(18)
It can be seen that the above elliptic-type integral (16) also provides
generalization to a number of new families of elliptic-type integrals, which also

 dθ


Generalized Elliptic-Type Integrals and…
29
generalizes known families of elliptic integrals. Also by using the generating
function (15) and by the application of the theorem 2, under the stated conditions,
we have obtained the following new family of elliptic-type integrals, which also
generalizes known families of elliptic-type integrals.
1
∫0
ω
β−1
(1 − ω)
α −1
(a j,A j )1,n' , [ci ( a ji ,A ji )]n'+1,p ;r

i
m',n'
ξ
ℵp ,q ,c ;r ( zω) ( b ,B ) ,[ c ( b ,B )]
i i i 
j j 1,m' i ji ji m'+1,qi ;r

M
∏
.
ℓ =1
− τℓ
N
∏
ℓ '=1
∞
(σ ) n x n t n (α ) µn
n =0
n!
= Γ(α ) ∑
⋅
 2ω k 2 
ℓ 
1 −
2
 k − 1
ℓ


M
 ( δ ℓ ' − ρ ℓ ' )ω 
1 −

1 + δ ℓ' 

∞
∏ ∑
ℓ =1 n ℓ =0
(λ ℓ ' ) n ℓ '
−λ ℓ '




[1 − xt ωη (1 − ω) µ ]−σ dω
(τ ℓ ) n ℓ  2k ℓ2 


n ℓ !  k ℓ2 − 1 
nℓ
N
∏
ℓ '=1
∞
 (δ − ρ 
∑  1ℓ+' δ ℓ' 
ℓ' 
n ℓ ' =0 
nℓ'
(1− β −ηℓ −ηℓ ' −ηn,ξ ),( a j ,A j )1,n' ,[ci ( a ji ,A ji )]n'+1,p ;r

i
( b j ,B j )1,m' ,[ ci ( b ji , B ji )]m'+1,q ;r , (1−α − β − n ℓ −n ℓ ' −ηn − µη ,ξ )  ,
i

 ξ
,n'+1
z
ℵm'
pi +1,qi +1,ci ;r 
n ℓ' !


(19)
θ
For σ → 0 and ω = sin 2   above (19) gives the following explicit representation
2
of generalized family of the Elliptic-type Integral.
π
∫0
sin
2β−1  θ 
  cos
2

m',n'
2
  ℵp ,q ,c ;r z sin
2
i i i 
 
M
∏
.
ℓ =1
M
=Γ(α)∏
ℓ=1
.
( λ ℓ' ) n
n ℓ' !
ℓ'
[1 − k ℓ2
−τ
[1− kℓ2 ] ℓ

2α −1  θ 
cos θ]

− τℓ
ℵpm',+n'1,+q1 +1,c ;r
i
i
i
∏
−λℓ'
M
(τℓ )n  2k nℓ N
ℓ
ℓ 
∏
2

nℓ ! k −1 ℓ'=1
 ℓ 
∞
∞
ℓ=1 nℓ =0

z ξ


(a j , A j )1,n' , [ ci ( a ji ,A ji )]n'+1,p ;r
i
( b j ,B j )1,m' ,[ ci ( b ji ,B ji )]m'+1,q ;r
i
−λ ℓ '
∏ [1+ δℓ' ] ∏ ∑
ℓ'=1
ξ

2 θ
2 θ
1 + ρ ℓ ' sin 2 + δ ℓ ' cos 2 


N
ℓ '=1
N
 θ 
 
 2 




dθ
nℓ'
(δ − ρ ) 
∑  1ℓ'+ δ ℓ' 
nℓ' =0 
ℓ' 
(1−β−ηℓ −ηℓ ' ,ξ ),( a j, A j )1,n' ,[ci ( a ji ,A ji )]n'+1,p ;r

i
.
( b j,B j )1,m' ,[ ci ( b ji ,B ji )m'+1,q ;r , (1−α −β− n ℓ − n ℓ ' ,ξ ) 
i

(20)
30
V.B.L. Chaurasia at al.
(ii)
Consider the generating function [29]
F(x, t) = (1 − x 1 t)
α ,α 2
Where g n 1
−α1
(1 − x 2 t)
−α 2
∞
α ,α 2
(X
= ∑ g n1
n =0
1
, X2 ) t2,
(21)
is the Lagrange polynomial defined by
α ,α 2
( x, y) =
g n1
∞
( α1 ) r ( α 2 ) n − r
∑
r ! (n − r) !
n =0
x r y n −r ,
(22)
and by the application of the theorem 1, under the stated condition, we get

,n'
( zω) ξ
ωα −1 (1 − ω) γ −α −1 ℵm'
pi ,qi ,ci ;r 

1
∫0
2
.
∏
j*=1
[1 − X j* t ωη (1 − ω) µ ]
∞
σ ,σ 2
(X
= Γ ( γ − α ) ∑ g n1
n =0
.
ℵpm'+,n'1,+q1 +1,c ;r
i
i
i
1
−σ j*
(a j ,A j )1,n' , [ ci ( a ji ,A ji )]n'+1,p ;r
i
( b j ,B j )1,m' ,[ ci ( b ji ,B ji )]m'+1,q ;r
i




dω
, X 2 ) t n ( γ − α ) µn

z ξ


(1−α −ηn,ξ ),( a j,A j )1,n' ,[ ci ( a ji ,A ji )]n'+1,p ;r

i

( b j, B j )1,m' ,[ ci ( b ji , B ji )m'+1,q ;r , (1− γ −ηn −µη,ξ ) 
i

.
(23)
Also by the application of the theorem 2, under the stated conditions and by the
use of the generating function (21), we get
1
∫0
M
∏
ℓ =1

,n'
( zω) ξ
ωβ−1 (1 − ω) α−1 ℵm'
pi ,qi ,ci ;r 

.
 2ω k 2 
ℓ 
1 −
2
 k − 1
ℓ


− τℓ
N
∏
ℓ '=1
(a j,A j )1,n' , [ci ( a ji ,A ji )]n'+1,p ;r
i
( b j ,B j )1,m' ,[ ci ( b ji ,B ji )]m'+1,q ;r
i
 ( δ ℓ ' − ρ ℓ ' )ω 
1 −

1 + δ ℓ ' 

−λ ℓ '
2
∏
j*=1




[1 − X j* t ωη (1 − ω) µ ]
−σ j*
dω
Generalized Elliptic-Type Integrals and…
∞
σ1,σ 2
= Γ(α ) ∑ g n
n =0
⋅
M
(x1 , x 2 ) t (α ) µn ∏
n
∞
∑
ℓ=1 n ℓ =0
(λ ℓ ' ) n ℓ '
31
(τ ℓ ) nℓ  2k ℓ2 


n ℓ !  k ℓ2 − 1 
nℓ
N
∏
ℓ'=1
∞
 (δ − ρ 
∑  1ℓ+' δ ℓ' 
ℓ' 
nℓ ' =0 
nℓ'
(1− β −ηℓ −ηℓ ' −ηn,ξ ),( a j ,A j )1,n' ,[ci ( a ji ,A ji )]n'+1,p ;r

i
( b j ,B j )1,m' ,[ ci ( b ji , B ji )]m'+1,q ;r , (1−α − β − n ℓ −n ℓ ' −ηn − µη ,ξ )  ,
i

 ξ
,n'+1
ℵm'
z
pi +1,qi +1,ci ;r 
n ℓ' !


(24)
(iii)
Consider the well-known generating function
∞
(−1) n x n t n
,
n!
F(x, t) = e − xt = ∑
n =0
(25)
and by the use of theorem 1, under the stated conditions, we get
1
∫0
.e
ω
α −1
(1 − ω)
γ −α −1
(a j ,A j )1,n' , [ ci ( a ji ,A ji )]n'+1,p ;r

i
m',n'
ξ
ℵp ,q ,c ;r ( zω) ( b ,B ) ,[c ( b ,B )]
i i i 
j j 1,m' i ji ji m'+1,qi ;r

− xt[ ωη (1−ω)µ ]
.
∞
( γ − α ) µn ( − x) n t n
n =0
n!
dω = Γ ( γ − α ) ∑
ℵpm'+,n'1,+q1 +1,c ;r
i
i
i

z ξ


(1−α −ηn,ξ ),( a j,A j )1,n' ,[ ci ( a ji ,A ji )]n'+1,p ;r

i

( b j, B j )1,m' ,[ ci ( b ji , B ji )m'+1,q ;r , (1− γ −ηn −µη,ξ ) 
i





.
(26)
Also by application of theorem 2, under the stated conditions and by the use of the
generating function (iii), we get the following useful integral:
1
∫0
ω
(1 − ω)
α −1
(a j,A j )1,n' , [ci ( a ji ,A ji )]n'+1,p ;r

i
m',n'
ξ

ℵp ,q ,c ;r ( zω) ( b ,B ) ,[ c ( b ,B )]
i i i 
j j 1,m' i ji ji m'+1,qi ;r

M
.
β−1
∏
ℓ =1
 2ω k 2 
ℓ 
1 −
 k 2 − 1
ℓ


− τℓ
N
∏
ℓ '=1
 ( δ ℓ ' − ρ ℓ ' )ω 
1 −

1 + δ ℓ ' 

∞
(− x) n t n (α ) µn
n =0
n!
= Γ(α ) ∑
M
∞
∏ ∑
ℓ =1 n ℓ =0
−λ ℓ '
η (1−ω)µ ]
e − xt[ ω
(τ ℓ ) n ℓ  2k ℓ2 


n ℓ !  k ℓ2 − 1 
nℓ
N
∏
ℓ '=1




dω
∞
 (δ − ρ 
∑  1ℓ+' δ ℓ' 
ℓ' 
n ℓ ' =0 
n ℓ'
32
⋅
V.B.L. Chaurasia at al.
(λ ℓ ' ) n ℓ '
4
 ξ
,n'+1
z
ℵm'
pi +1,qi +1,ci ;r 
n ℓ' !

(27)
(1− β −ηℓ −ηℓ ' −ηn,ξ ),( a j ,A j )1,n' ,[ci ( a ji ,A ji )]n'+1,p ;r

i
( b j ,B j )1,m' ,[ ci ( b ji , B ji )]m'+1,q ;r , (1−α − β − n ℓ −n ℓ ' −ηn − µη ,ξ )  .
i


Special Case
(i)
If we set c1 = c 2 = ... = c r = 1(i.e. c i = 1, i = 1,..., r) in result (9), (12) through
(ii)
(14), results (17) through (20), (23), (24), (26) and (27) reduced to the well
known I-function of Saxena recently obtained by Chaurasia and Singh [9]
when R = 1.
If we take c1 = c 2 = ... = c r = 1and r = 1 in result (9), (12) through (14),
results (17) through (20), (23), (24), (26) and (27) reduced to the well known
Fox H-function recently obtained by Chaurasia and Meghwal [7] when R =
1.
5
Conclusion
In this paper, we have presented a solution of generalized elliptic type integral
with Aleph (ℵ)-function. The solution has been developed in a compact and
elegant form with the help of generating functions. Aleph (ℵ)-function is general
in nature and includes a number of known and new results as particular cases.
This extended elliptic type integral used to compute the certain problems of
radiation physics, nuclear technology and may be utilized in other branch of
mathematics. Most of the results obtained are in a form suitable for numerical
computation. The results obtained in the present paper provides an extension of
the results given by Chaurasia and Singh [9] and Chaurasia and Meghwal[7].
Acknowledgements
The author is thankful to the worthy referee for his valuable suggestions for the
improvement of the paper.
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