Generalized Elliptic-Type Integrals and Generating Functions with Aleph-Function

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Generalized Elliptic-Type Integrals and Generating Functions with Aleph-Function

V.B.L. Chaurasia¹ and Vinod Gill²

1Department of Mathematics, University of Rajasthan, Jaipur-302004, Rajasthan, India

E-mail: [email protected]

2Department of Mathematics, Arya Institute of Engineering and Technology, Kukas, Jaipur- 302028, Rajasthan, India

E-mail: [email protected] (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

, θ )

θ

− (

= (

Ω ^k)

∫

^{π 1 k cos −−2 d}

j 1 2

j 0 (1)

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:

, θ )

θ

− (



 



θ



 



θ

= ) γ , α

( µ+

− α

− γ

− π α

µ

∫

^d

cos k 1

sin 2 cos 2

k, R

2 1 2

1 2 2 1

2

0

(2)

where ≤ < γ)> α)> , µ)>− . 2 Re( 1

0 Re(

Re(

1, k 0

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:

,

 θ



 



 



 



θ ρ

− )

θ

− (



 



θ



 



θ

=

; ρ ( Λ

λ

− +

µ

− β

− π α

) β , α

((λ,µ)

∫

^{1 sin} ₂ ^d

cos k 1

sin 2 cos 2

k) ²

2 1 2

1 2 1

2

0

(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:

k)

2 n 2 1

n

1 (ρ ,...,ρ ,δ; Ω^(α,β) _{:δ,µ) −}

σ − ,..., σ (

γ β σ

π α θ θ ⁻ ρ θ ⁻ δ θ ⁻

=

−

− 



 



 



 

 + 



 



 



 



− 



 



 



 



=

∫

^cos ₂ ^sin ₂

∏

^{n 2 1 jsin2} ₂ ^{j 1 cos2} ₂

1 j 1 2 1

2 0

(1 k cos ) ²d ,

1

2 θ ^−µ− θ

−

⋅ (4)

where 0≤ k <1,Re(α)> 0,Re(β) >0;σ ( j=1,...,n−2),

j

.

<











, − δ + ,| δ ρ

∈ µ ,

γ 1

1 k

k 2

| 1 max

;

C 2

2 j

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:

) ,...,

; δ ,..., δ , ρ ,..., ρ ( Ω

= ) );

δ ( ), ρ ((

Ω₍⁽_λ^α,β_,τ⁾₎ k _λ^α_,...,^,β)_{λ ,τ ,...,τ} k k_Μ

1 N 1 N 1 (

M 1 N j 1

i j i

i

i 2

i 2

i N

1 i 1 2 1

2

0 cos 2

sin 2 2 1

2 sin cos

β λ

π α θ θ ρ θ δ θ ⁻

=

−

− 



 



 



 

 + 



 

 + 



 



 



 



=

∫



∏

[1 k²_jcos ] ^jd ,

M 1 j

θ θ ^−τ

=

−

⋅

∏

(5)

where min (Re( Re( 0 k 1 C;

j i j|< ;λ ,τ ∈

|,

>

)) β ), α

M), 1,..., j and N 1,..., i 1 1

1 k

k 2

| max

i i i 2

j 2 j i

i < ( = =













δ +

ρ

− , δ ,| −

δ

|

|, ρ

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₍⁽_λ^α,β_,τ⁾₎((ρ_i),(δ_i); _j)

j i

k can be put into the following form involving Euler-type integral:

) ,...,

; δ ,..., δ , ρ ,..., ρ (

Ω₍⁽_λ^α,β_,...,⁾_{λ ;τ,...,τ )} k k_Μ

1 N 1 N M 1

1 N 1

1 1 1

0 i i N

1 i j 2 j M

1 j

1 1

k

1 ^{−λ β− α−}

= τ

−

=

) ω

− ( ω )

δ + ( )

− (

=

∏ ∏ ∫

ω.













δ +

ω ) ρ

− δ

−(













−

− ω .

λ

−

= τ

−

=

∏

∏

¹ ₁ ^d

1 k

k 2 1

i i i i N

1 i j 2

j 2 j M

1 j

(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

, (

=

∑

^∞

=

n n n 0 n

t x) f C t)

F(x, (7)

where each member of the generalized set {f_n(x)}^∞_n=0 is independent of x and t.

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]:











 ℵ

= [

ℵ ^{+ ;}

)]

, ( [, ) ,

; + , )]

, ( [, ) ,

; , ,

ir p 1, n Aji aji ci n 1, Aj (aj

ir 1q m Bji bji ci m 1, Bj (bj n

m, ir ic iq

p z

z]

s)z ds.

i 2

1 _{m,n s}

ir ic iq L p

; − ,

, (

π Ω

=

∫

(8)

for all z ≠ 0, where i= −1 ^and

,

−

− ( Γ +

( Γ

−

− ( Γ β

+ ( Γ

= ( Ω

∏

∏

∑

∏

∏

+

= +

=

=

=

=

; , ,

s) B b 1 s)

A a c

s) A a 1 s)

b s)

ji ji qi

1 m j ji ji pi

1 n j i r

1 i

j j n

1 j j j m

1 n j

m, ir ic iq p

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.

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











( ω) ℵ

) ω

− (

ω ^{, ) [, ( , )] + ;}

; )] + , ( [, ) , ξ (

; ,

− , α

− γ

−

∫

α ^{ji n' 1,pir}

jiA i a n' c 1, Aj (aj

ir q 1, m' Bji bji ci m' 1, Bj bj n'

, m'

ir ic iq p 1 1 1

0

z 1

. _{n n} ⁿ _n

0 n

) ( t x) ( f C ) ( d ] ) 1 ( t

F[x, ω^η −ω ^µ ω=Γ γ −α

∑

^∞ γ −α _µ

=

. 











ℵ ^{+ ;}

)]

, ( [, ) , ( ), ξ η

− α

− (

) ξ , µη

− η

− γ

− (

; , )] + , ( [, ) , ξ ( + +, ; , +

ir p 1, n' Aji aji ci n' 1, Aj aj n, 1

n r 1

qi 1, m' Bji bji ci m' 1, Bj bj 1

n' , m'

ir c i 1 q i 1

p z , (9)

Provided that

and m' 1,..., j 0 Re(

0 Re(

0 [

Re B 0

b min Re

j

j > , γ−α]> , η)> , µ)> , =





















 ξ  + α

r 1,..., 2 i

z arg

0 i

i > |, |< πψ ; =

ψ ^ξ (10)

,

<

+ } Λ πψ

<

|

|,

≥

ψ ^ξ andR{ 1 0

z 2 arg

0 i i

i (11)

where











 +

− +

=

ψ

∑ ∑ ∑ ∑

+

= +

=

=

= ji

qi 1 m' j ji pi

1 n' j i j m'

1 j j n'

1 j

i A B c A B ,

).

− (

+











 −

+

−

=

Λ

∑ ∑ ∑ ∑

+

=

=

=

= ji i i

pi 1 n' j ji qi

1 j i j n'

1 j j m'

1 j

i p q

2 a 1 b

c a b

Corollary 1. Let the generating function F(x,t) and ℵ-function defined in (7) and (8) respectively, then













)}

ω

− { ℵ

) ω

− (

ω ^{+ ;}

)]

, ( [, ) ,

; )] + , ( [, ) , ξ (

; ,

− , α

− γ

−

∫

α ^{ji n' 1,pir}

jiA i a n' c 1, Aj (aj

ir q 1, m' Bji bji ci m' 1, Bj bj n'

, m'

ir ic iq p 1 1 1

0

z(1 1

.

n n n n 0 n

t x) f C d

1 t

F[x, _η

∞

= µ

η( −ω) ] ω=Γ(α) ( (α)

ω

∑

.











ℵ  ^{( −γ+α−µ ξ),( , ) [, ( ,}_{,( −}^)]_γ−η⁺ _−µη^;_,ξ)

; )] + , ( [, ) , ξ ( + + , ; , +

ir p 1, n' Aji aji ci n' 1, Aj aj n, 1

n r 1

qi 1, m' Bji bji ci m' 1, Bj bj 1

n' , m'

ir c i 1 q i 1

p z , (12)

provided that

m' 1,..., j 0 Re(

0 Re(

B 0 b min Re

0 Re[

j

j > , η)> , µ)> , =





















 ξ  + α

− γ ,

>

]

α and

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













} ω ) ω

− { ℵ

) ω

− (

ω ^{+ ;}

)]

, ( [, ) ,

; )] + , ( [, ) , ξ (

; ,

− , α

− γ

−

∫

α ^{ji n' 1,pir}

jiA i a n' c 1, Aj (aj

ir q 1, m' Bji bji ci m' 1, Bj bj n'

, m'

ir ic iq p 1 1 1

0

z(1 1

. ⁿ

n n 0 n

t x) f C d

1 t

F[x, ω ( −ω) ] ω=

∑

^∞ (

= µ

η

.











ℵ  ^{( −α−η ξ),( +α−γ−µ ξ),( , )}_{,( −γ}^[,_−η⁽ _−µη^, _{, ξ}^)]₎ ^{+ ;}

; )] + , ( [, ) , ξ ( +, + , ; +

ir p 1, n' Aji aji ci n' 1, Aj aj n, 1

n, 1

2 n r 1

qi 1, m' Bji bji ci m' 1, Bj bj 2

n' , m'

ir c i 1 q i 2

p z , (13)

provided that

m' 1,..., j 0 Re(

0 Re(

B 0, b min Re

B 0 b min Re

j j j

j > η)> , µ)> , =





















 ξ  + α

− γ ,

>





















 ξ  + α

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











 ) ω ( ℵ

) ω

− (

ω ^{+ ;}

)]

, ( [, ) ,

; )] + , ( [, ) , ξ (

; ,

− , α

−

∫

β ^{ji n' 1,pir}

jiA i a n' c 1, Aj (aj

ir q 1, m' Bji bji ci m' 1, Bj bj n'

, m'

ir ic iq p 1 1 1

0

z 1

. ω ( −ω) ] ω













δ +

ω ) ρ

− δ

−(













−

− ω ^{η µ}

λ

−

= τ

−

=

∏

∏

¹ _k^{2 k}₁ ¹ ₁ ^{'F[x, t 1 d}

' ' ' N

1 ' 2

M 2 1

ℓ

ℓ ℓ ℓ ℓ

ℓ

ℓ ℓ ℓ

'

'

n

' ' ' 0 n N

1 ' n 2

2 n

0 n M

1 n n n n 0

n 1

( 1

k 2k

! n

) ) (

( t x) ( f C ) (

ℓ

ℓ ℓ ℓ

ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ

ℓ

ℓ 



 +

 −







 Γ −

=

∑ ∏ ∑ ∏ ∑

^∞

=

=

∞

=

=

∞

= δ

ρ τ δ

α

α _µ

. 









ℵ ₊ ⁺ ₊  ^{(1− − − '− n, ),(aj,Aj}₊^)1,n',[_;r^c_,ⁱ₍⁽₁^a₋^ji,₋^Aji₋^)]_n^n'+₋^1,_n^pi_'−^;r_{n− , )}

qi 1, )]m' Bji ji, b i( c [ m', )1, Bj j, b ( 1

n' , m'

r i; c , i 1 q , i 1 p '

n '

' z

! n

)

( ^{β η η η ξ}

ξ µη η β

ξ α

λ ^{ℓ ℓ}

ℓ ℓ ℓ

ℓ ℓ

, (14) Provided that

1 k

| C;

0 Re(

0 Re(

0 [ Re B 0

b min

Re ' ' '

j

j > , α]> , η)> , µ)> ;δ ,ρ ,λ ,τ ∈ |<





















 ξ  +

β _{ℓ ℓ ℓ ℓ ℓ}

and < ,













δ +

ρ

− , δ ,| −

δ

|

|,

ρ 1

1 1 k

| 2k max

' ' ' 2

2 '

'

ℓ ℓ ℓ ℓ

ℓ ℓ

ℓ

where j=1,...,m',ℓ '=1,2,...,N,ℓ=1,2,...,M, and remaining condition is same as we already obtained in (10) and (11).

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]

, )

σ (

=

−

=

∑

^∞

= σ

−

! n

t xt) x

(1 t) F(x,

n n n 0 n

(15)

and by the use of the theorem 1, under the stated condition, we get the following interesting results:









 ℵ 

− ^{− − +}₊

∫

− (^(ab^jj^,,B^Aj^j)⁾1,^1,m'^n',,^[[^ccⁱi⁽(^ab^jiji^,,^AB^jiji^)])]^n'm'^1,1,^pqⁱi^;r;r n'

, m'

r i; c i, q i, p 1 1 1

0

) z ( )

1

( ^{γ α ξ}

α ω ω

ω

⋅^[1−^xt ω^η(1−ω^)µ]−σdω

! n

) ( t x ) ) (

( ⁿ

n n n 0 n

α µ

γ α σ

γ − ⁻

Γ

=

∑

^∞

=

. 











ℵ ^{+ ;}

)]

, ( [, ) , ( ), ξ η

− α

− (

) ξ , µη

− η

− γ

− (

; , )] + , ( [, ) , ξ ( + +, ; , +

ir p 1, n' Aji aji ci n' 1, Aj aj n, 1

n r 1

qi 1, m' Bji bji ci m' 1, Bj bj 1

n' , m'

ir c i 1 q i 1

p z , (16)

when we put 1

cos 2 2 cos 2 and

cos²  θ= ² θ−



 



θ

=

ω the above equation (16) gives

the following generalization of the elliptic-type integral



















 



 



θ ℵ



 



θ



 



θ ^{, ) [, ( , )] + ;}

; )] + , ( [, ) , ( ξ

; ,

− , α

− γ

− π α

∫

^{ji n' 1,pir}

ji A i a n' c 1, Aj (aj

ir q 1, m' Bji bji ci m' 1, Bj bj 2

n' , m'

ir ic iq p 1

2 2 1

2

0 zcos 2

sin 2 cos 2

.

! n t

x n

n n n 0 n

∞ µ

=

) α

− γ ( )

σ ) (

α

− γ ( Γ

=

∑

. 









ℵ  ^{( −α−η ξ),( , ) [, ( ,} _,(^)]_−γ⁺_{−η −}^;_µη,ξ)

; ) + , ( [, ) , ξ ( + + , ; , +

ir p 1, n' Aji aji ci n' 1, Aj aj n, 1

n r 1

qi 1, m' Bji bji ci m' 1, Bj bj 1

n' , m'

ir c i 1 q i 1

p z . (17)

If we setting and 0

sin 2 2 1 cos using 2 and

sin^{2 2}  σ→



 



θ

−

= θ



 



θ

=

ω in (16), we get

the following result:

 θ



















 



 



θ ℵ



 



θ



 



θ ^{, ) [, ( , )] + ;}

; )] + , ( [, ) , ( ξ

; ,

− , α

− γ

− π α

∫

^d

sin 2 2 z

2 cos sin

ir p 1, n' Aji aji ci n' 1, Aj (aj

ir q 1, m' Bji bji ci m' 1, Bj bj 2

n' , m'

ir ic iq p 1

2 2 1

2 0

.











 ℵ

) α

− γ ( Γ

= ^{+ ;}

)]

, ( [, ) , ( ), ξ α

− (

) ξ , γ

− (

; , )] + , ( [, ) , ξ ( + + , ; , +

ir p 1, n' Aji aji ci n' 1, Aj aj , 1

r 1 qi 1, m' Bji bji ci m' 1, Bj bj 1

n' , m'

ir c i 1 q i 1

p z (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

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.











 ) ω ( ℵ

) ω

− (

ω ^{+ ;}

)]

, ( [, ) ,

; )] + , ( [, ) , ξ (

; ,

− , α

−

∫

β ^{ji n' 1,pir}

jiA i a n' c 1, Aj (aj

ir q 1, m' Bji bji ci m' 1, Bj bj n'

, m'

ir ic iq p 1 1 1

0

z 1

. − ω ( −ω) ] ω













δ +

ω ) ρ

− δ

−(













−

− ω ^{η µ −σ}

λ

−

= τ

−

=

∏

∏

¹ _k^{2 k}₁ ¹ ₁ ^{' [1 xt 1 d}

' ' ' N

1 2 '

M 2 1

ℓ

ℓ ℓ ℓ ℓ

ℓ

ℓ ℓ ℓ

n ' '

' ' ' 0 n N

1 ' n 2 n 2 0 n M

1 n n n n 0

n 1

( 1

k 2k

! n

) (

! n

) ( t x ) ) (

(

ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ

ℓ ℓ ℓ

ℓ ℓ

ℓ ℓ 



 +

−











 Γ −

=

∑ ∏ ∑ ∏ ∑

^∞

=

=

∞

=

=

∞

= δ

ρ τ δ

α

α σ ^µ









 ℵ 

⋅ ₊ ⁺ ₊ ^{(1− − − '− n, ),(aj,Aj}₊^)1,n',[_;^c_rⁱ_,(⁽₁^a₋^ji,₋^Aji₋^)]_n^n'+₋^1,_n^pi_'−^;r_{n− , )}

qi 1, )]m' Bji ji, b i( c [ m', )1, Bj j, b ( 1

n' , m'

r i; c , i 1 q , i 1 ' p

n '

' z

! n

)

( β η η η ξ

ξ µη η β

ξ α

λ ^{ℓ ℓ}

ℓ ℓ ℓ

ℓ ℓ

,

(19)

For 



 



θ

= ω

→

σ 0and sin² 2 above (19) gives the following explicit representation of generalized family of the Elliptic-type Integral.



















 



 



θ ℵ



 



θ



 



θ ^{, ) [, ( , )] + ;}

; )] + , ( [, ) , ( ξ

; ,

− , α

− π β

∫

^{ji n' 1,pir}

ji A i a n' c 1, Aj (aj

ir q 1, m' Bji bji ci m' 1, Bj bj 2

n' , m'

ir ic iq p 1

2 1

2

0 zsin 2

cos 2 sin 2

.  θ

 



 +ρ θ+δ θ

] θ

−

[ ^−λ

= τ

−

=

∏

∏

^{1 k cos 1 sin} ₂ ' ^cos2 ₂ ^{' d} 2

' N

1 ' 2

M 1

ℓ ℓ

ℓ ℓ

ℓ ℓ

ℓ

n' '

' ' ' 0 n N

1 ' n 2 n 0 n M

1 ' ' N

1 ' 2 M

1 k 1 1

2k 1 n

k 1

ℓ

ℓ ℓ ℓ ℓ ℓ

ℓ

ℓ ℓ ℓ

ℓ ℓ ℓ ℓ

ℓ ℓ ℓ

ℓ ℓ

ℓ 









 δ +

) ρ

− δ (













! − ) τ ] (

δ + [ ]

− [ ) α ( Γ

=

∏ ∏ ∏ ∑ ∏ ∑

^∞

=

=

∞

=

= λ

−

= τ

−

=

. 











! ℵ ) λ

( ^{( −β−η −η ξ),( , ) [, ( , )]} _{+ ;}

) ξ ,

−

− β

− α

− (

; , ) + , ( [, ) , ξ ( + +, ; , +

ir p 1, n' Aji aji ci n' 1, Aj aj ', 1

n ' n r 1

qi 1, m' Bji bji ci m' 1, Bj bj 1

n' , m'

ir c i 1 q i 1 p '

n '

' z

n

ℓ ℓ

ℓ ℓ ℓ

ℓ ℓ

. (20)