triple Infinite Integral Representation For The Polynomial SetRn(x1, x2, x3)

(1)

Rajnish Kumar Singh et al., IJSRR 2019, 8(2), 2711-2719

IJSRR, 8(2) April. – June., 2019 Page 2711

Research article Available online www.ijsrr.org

ISSN: 2279–0543

International Journal of Scientific Research and Reviews

Triple Infinite Integral Representation For The Polynomial

Set R

n

(x

1

, x

2

, x

3

)

Rajnish Kumar Singh

1

and Brijendra Kumar Singh

2*

1

Department of Mathematics, Jai Prakash University, Chapra, Bihar (India)

1

Email:srajnish093@gmail.com

2

Department of Mathematics, Jai Prakash University, Chapra, Bihar (India)

2

Email:brijendrakumarsingh111956@gmail.com

ABSTRACT

In the present paper, an attempt has been made to express a Triple Infinite Integral Representation for the polynomial set R_{n(x1, x2, x3)}. Many interesting new results may be

obtained as particular cases on separating the parameter.

KEYWORDS:

Appell Function, Generalized Hyper geometric Polynomial, Integral Representation, Lauricella function.

*Corresponding author

Brijendra Kumar Singh

Department of Mathematics Jai Prakash University Chapra, Bihar, India

(2)

1. INTRODUCTION

We defined the generalized hyper geometric polynomial set _Rn(x_1,x_2,x_{3) by means of}

generating relation,

… (1.1)

Where, , ₁, ,₁,₂, ₃are real and r, r₁are non-negative integer and r₂, r₃are natural numbers.

The left hand side of (1.1) contains the product of generalized hyper geometric function and Lauricella function in the notation of Burchanall and Chaundy1.

The polynomial set contains number of parameters, for simplicity we shall denote

by _Rn(x_1,x_2,x_3).

where n denotes the order of the polynomial set. After little simplification (1.1) gives





 

_

_



   1 1 - 1

1 2

1 _t 1- _{x t}r r





     





     

 

 

 

  

 

 

2 2 3 3 1

1 2 2 3 3 v

; ; , ,

; ; ;

p u h m

r r r r

r

q k w

A C E G

F x t x t x t

B D F H

 







 



 



 







 _{      }



 

1 1 2 3

1 2 3 v

; ; ; ; ; ; ; ; ; ; ;

1 2 3

, ; ; ; ; ; ; ; 0

,

p u h m

q k w

A C E G _n

n r r r r B D F H n

R

x x x

t

 

    





     





      ₁ ₁ ₂ ₃

1 2 3 v

; ; ; ; ; ; ; ; ; ; ;

1 2 3

, ; ; ; ; ; ; ;

,

p u h m

q k w

A C E G

n r r r r B D F H

R

x x x





    

   

  

  _ _{ } _

   

       

   



  



1 1 2 2 1 1

2 3

1

1 2 3

0 0 0 0

, ,

n r r s r s

n r r s n r

n

e e

r r

n

s s r s

R x x x





_ _ _ _





_ _ _ _

           

 

 



 

 

1 1 2 1 3 3 1 1 2 2 3 3

1 1 1 1

p

n r r s r s r s q

n r r s r s r s

A B













   













   

           

     



   

 

     

2 3 1 1 1 1 2 2 3 3 2 3

1 1 2 2 3 3 2 3

1 1 2 3

v ! 1! 2! 3!

s s s s

u n r r s r s r s h s m s s s

k w

n r r s r s r s s s

C E G

(3)

… (1.2)

2. NOTATIONS

I.(i) (n) = 1, 2, 3, ……… n – 1, n.

(ii) _{(ap) = a1, a2, a3, ……… ap.}

(iii)(ap ;i) = a1, a2, …… ai–1, ai+1 ……… ap.

II.(i) _{[(ap)] = a1}_a2_{a3,……… ap.}

(ii)

III.(i)

(ii)

(iii)

IV.(i)

(ii)

(iii)

(iv)

 





   







 



1 1 2 2 3 3

1 1 2 2 4

3 3

1 2

1 1 2 2 3 3

!

3 n r r s r s r s _{r s r s} r

r s

x

n r r s r s

r s

x





     







    

 



1 2

1

.

p

p n i n n n p n

i

a a a a a

    

 a b, b b, 1, b a 1.

a a a

 

  







 



k ;  1  1

k k k

b b b a

a b

a a a







 





1

a

k r

b r

a

 

 

     

_ _ _ _  



1 1

1

; .

p m

i p

k

i r

a r m a

m

   



 

_ _





1 .

p

p i

i

a a

   

 

 

_ _





1

; .

p

p i

i s

a s a

 







 

_  _  

 



1

.

m

r

m _r

a a

m m

 







 

 _ _





1

; .

a

r

b r a b

(4)

V.(i)

(ii)

VI.(i)

3. THEOREM For r_{2> 1 and}r_{3> 1}

… (3.1) Proof : we have







 





a



b

 

a



b



a



b

.

     

_* ab   ab  ab .





 









 

  _ _   



  _ _  

4

1 1

v !

n r

p _n u _n

q _n _n

A C x

M

B D n











 



 

1 1 2 3

0

8 1 2 , ,

1 1 ₁

2 2 2 2

n

M R x x x

i f t J t dt





   



 

       

_



2 2

 

21 2 2 2



2



2 2 2



2 2

2 2 0 0 0

exp x y

x y x y z i x y z

x y

 

   _ _ _ _

    _   _



 

  





2 2

1 2 2 2

cos 2 tan

2

x y

f x y z 

     

       

 

    

 





 





  

       

 

 

 _     _



1 2 3 1 2 3

1 v : : : :

1 2 3

: , , , , 1 : , , 1, 1 ——————, 1 : , , 1, 1

q q h m

p u k w

h

n r r r r B n r r r r F

A n r r r r

 





   



 







 





   



 



v 1 2 3 1

1 2 3

1 : , , , , :1 :1 :1 , :1 , 1 :1

2 1

1 : , , , , :1 :1 , :1 1 :1

2 2 2

h m

u k w

D n r r r r E G u

C n r r r r F H



 

    _  _{ } _    

  _ _

 

   

    _  _{ } _  

  _  _{ } _





 









 





       

   

 

1 1

1

4 4

v 1 v 1

1 2

1 1

, ,

r p q u _r r p q u

r r

x

x x





 









 





            

    _



  _

2 3

2

2 3

4 4

v 1 v 1

2 2 3

1 1 3

1 1

,

r p q u p q r p q u p q

r

r r

x

dxdydz

x x x





2 2

1 2 2 2

cos 2 tan

2

x y

f x y z 

     

       

 

    

(5)

1 1 2 2 1 1

2 3

1

1 2 3

0 0 0 0

n r r s r s n r r s

n r n

r r

s s s s

              _ _{ } _                 

 







_ _ _ _













_ _ _ _









1 1 2 2 3 3 2 1 1 2 2 3 3

1 1

v

1 1

p u _{n r r s} _{r s} _{r s} h _s

n r r s r s r s

q _{n r r s} _{r s} _{r s} k _s

n r r s r s r s

A C E

B D F

                      _ _ _ _      _ _ _ _        





   

1 1 2 2 3 3 1 1 2 2 3

1 4

1 3

3 3 3

1 1 2 3 1

1 2 3 1 1 2 2 3 3

! ! ! !

n r r s r s r s r s r s s

s r

s

m _s _s _s

r s w _s

G x x

H s s s x n r r s r s r s

                       







 



1 1 1 1 2 1 ₁

2 2 2

s s s         



2 2



2211



2 2 2



221



2 2 2



2 2

2 2 0 0 0

exp

s s

x y

x y x y z i x y z

x y       _ _ _ _     _   _   

  





      _ _ __       2 2

1 2 2 2

cos 2 tan

2 x y

f x y z

                                  

  



1 1 2 2 1 1

2 3

1

1 2 3

0 0 0 0

n r r s r s

n r r s

n r n

r r

s s s s





_ _ _ _









_ _ _ _





1 1 2 2 3 3 1 1 2 2 3 3

1 1

v

1 1

p u _{n r} _{r s} _{r s} _{r s}

n r r s r s r s

q _{n r} _{r s} _r _s _r _s _{n r r s} _{r s} _{r s}

A C B D                       _ _      _ _  









 









1 1 2 2 2 3 1

1

2 3

3 3

2 3

1 2 2 3

1 2 3 3

! ! ! !

r s r s s s s

s

h s m s s s

r s

k _s w _s

E G x

F H s s s x s

                       









 



 



1 1 2 2 3 3 4

1

1 1

1

1 1 2 2 3 3

1 2

1

! 1

2 2 2

n r r s r s r s r

s

s s

x dxdydz

n r r s r s r s

    _             





_ _ _ _





_ _ _ _

1 1 2 2 1 1

2 3

1

1 1 2 2 3 3

1 2 3

1 1 2 2 3 3

1 1

0 0 0 0

1 1

n r r s r s n r r s

n r n

r r

r

r _p

n r r s r s r s

s s s s q _{n r r s} _r _s _r _s

A B               _ __ _{ } _                                   

  















 













1 1 2 2 1

1 1 1 2 2 3 3 2 3

1 1 2 2 3 3 2 3

1 1 2

v ! 1! 2!

r s r s s

s

u _{n r r s} _{r s} _{r s} h _s m _s _s _s

k w

n r r s r s r s s s

C E G x

D F H s s s

                                     















 



1 1 2 2 3 3 2 3 4

1 3 3

1 1

2 3 1

3 3 1 1 2 2 3 3

1 2 1

! ! 1

2 2 2

n r r s r s r s

s s r

s r s

s s

x

x s n r r s r s r s

    _

     



 

(6)

IJSRR, 8(2) April. – June., 2019 Page 2716 … (3.2)

The single terminating factor makes all summation in (3.2) run upto and we finally achieve

on using2

PARTICULAR CASES(3.1)

On specializing the values of the parameters involved in Lauricella form, a number of known and unknown polynomials can be obtained as the particular cases of polynomial set R_n(x₁,x₂,x₃). Some of them, which are well known, are listed below.

(I) Hermite Polynomials



 











   



 

         

 

 

     



1

1 1

0 1

1 1 ₁

2 2 2 2

1 8 1

2 2

n

n s

i s s

f t J t dt s



 







1 1 ₁

2 2 2 2

8 1 2

i           

    

    

   

  

  _ _{ } _

   

       

   



  



1 1 2 2 1 1

2 3

1

1 2 3

1

0 0 0 0

n r r s r s n r r s

n r n

r r

s s s s

M





_ _ _ _









_ _ _ _





1 1 2 2 3 3 1 1 2 2 3 3

v

1 1

q _{r r s} _{r s} _{r s}

r r s r s r s

p _{r r s} _r _s _r _s u _{r r s} _{r s} _{r s}

B n D n

A n C n

       

    _   _

 



    _   _

 









 





 





 









_

_

_

_

2 1

1 1

2 3

1 1 1

4 4

2 3

v 1 v 1

1 1

1 1 1 2

1 1

! ! !

r p q u r p q u s s

s

h s m s s s

r r s

r r

k _s w _s

E G

F H s s x x s

       

         

   



     

   





   









2 2

2 2 2 1 1

1 1 2 2 3 3 2 2

3 3 4

v 1 2 2 2

1 3 3

1

!

r p q u p q s s r s

r s

r r s r s r s

r s _{r s}

r

x x n

x s x

     

  

  









   





 

3 3

3

3 3 4

v 1 3

0 1 3

1 r p q u p q s

s

n r s

r f t J t dt

x x

      _

 









 







 1 2 3    

0

1 1 ₁

2 2 2 2 _, _,

8 1 2

n

n n

i

R x x x f t J t dt



 

       



   





2 2

 

21 2 2 2



2



2 2 2



2 2 2 2 0 0 0

exp x y

x y x y z i x y z

x y

 

   _ _ _ _

   _   _



 

  





2 2

1 2 2 2

cos 2 tan

2

x y

f x y z dxdydz



    

  

   

 

   

 



 







1 1 ₁

2 2 2

8 1 2

n

i          

(7)

On setting p = 0 = q = u = v = s; = 1 = = _{1 = r = r4; r1 = 2 =}; _{1= – 4}

and writingxfor_x1in (3.1) we achieve

where _Hn(x) are the Hermite Polynomials.

(II) Bedient Polynomials3

If we set q = 0 = s; p = 1 = u = v =  = _{1 = r4; r1 = 2 =}; _{1 = –4, A1 =},

C1 = _{, D1 =} + and writing_x1forxin (3.1), we get

WhereGm(, , x) are Bedient Polynomials.

(III). Sylvester Polynomials4

If we putp = 0 = q = u = v = s; _{= 1 = r1 =}; = _{1 =}= r; _{1 = x}and writing

xfor_x1in (3.1) we arrive at

 





 



 



   

0

8 1 2

2

1 1

1

2 2 2 2

n n

n

x H x

i f t J t dt



     

 

       

_



2 2

 

21 2 2 2



2



2 2 2



2 2

2 2 0 0 0

exp x y

x y x y z i x y z

x y

 

   _ _ _ _

    _   _



 

  





2 2

1 2 2 2

cos 2 tan

2

x y

f x y z 

    

      

 

   

 







 



2

1

, , 1 :1 ;

2 2 2 2

1 ×

1 _{:1 , 1} _{:1 ;}

2 2 2

n n

F dxdydz

x

_ _ _  __{ }  

 _ _ 

 

  

 

 _ _ 

 

 

 

 





     

 



 



   

0

8 1 2

2 , ,

1 1

! 1

2 2 2 2

n n n m

n

n n

x

G x

n i f t J t dt





     

  

 

          

_



2 2

 

21 2 2 2



2



2 2 2



2 2

2 2 0 0 0

exp x y

x y x y z i x y z

x y

 

   _ _ _



    _   _



 

  





2 2

1 2 2 2

cos 2 tan

2

x y

f x y z



    

      

 

   

 







 



2

1

, ,1 , 1 :1 ;

2 2 2 2

1 ×

1

1 ,1 , :1 , 1 :1 ;

2 2 2

n n _n

F dxdydz

x

n n

 _ _ _ _{    }  __{ }  

 _ _ 

 

  

 

 _ _ 

       

 

(8)

IJSRR, 8(2) April. – June., 2019 Page 2718 where_n(x)are the Sylvester Polynomials.

(IV) Laguerre Polynomials

On puttingp = 0 = q = u = v = h = s; k = 1 =  =  = _{2 =}_{= r4 = r2 = x1;}

x2 = y,F1 = 1 +  in (3.1), we obtained

where are the Laguerre Polynomials. (V) Jacobi Polynomials

On making the substitution_{p = 0 = q = u = h = s = r; r2 = v = 1 = k =} =  = _{2 = r4;} =

1 = _{1; D1 = 1 +}_{; F1 = 1 +}; and instead of in (3.1), we get

 







 



   

0

8 1

2

1 1

! 1

2 2 2 2

n

x x

i n f t J t dt



      

         

_



2 2

 

21 2 2 2



2



2 2 2



2 2

2 2 0 0 0

exp x y

x y x y z i x y z

x y

 

   _ _ _ _

    _   _



 

  





2 2

1 2 2 2

cos 2 tan

2

x y

f x y z 

    

      

 

   

 













 _ __{ } 

 

_ _ _ _ 

 

_ _ 

 

, , 1 :1 ; 2

1 ×

1 _{:1 , 1} _{:1 ;}

2 2 2

n x

F dxdydz

x

 





 



 



   

0

8 1 1

2

1 1

! 1

2 2 2 2

n n

n

L x

n i f t J t dt







      

 

       

_



2 2

 

21 2 2 2



2



2 2 2



2 2

2 2 0 0 0

exp x y

x y x y z i x y z

x y

 

   _ _ _ _

    _   _



 

  





2 2

1 2 2 2

cos 2 tan

2

x y

f x y z 

    

      

 

   

 







 





 _ _ _{ } 

 

 _{ } _ _ 

 

 

, , 1 :1; 2 ×

1

1 , :1 , 1 :1 ;

2 2 2

n x

F y dxdydz

 



n

L y

 



2 x 1₁

x x

 _{ }  





 



 



   

 

 

      





   

  _  _   

 



,

0

1 8 1 1

2

1 1

! 1

2 2 2 2 2

n n

n

n n

n

x

P x

(9)

IJSRR, 8(2) April. – June., 2019 Page 2719 where are the Jacobi Polynomials.

4. CONCLUSION

These polynomials are of at most important for scientists, engineers and physical sciences, because these occurs in the solution of integral equation and analytic function, which describe physical problems. The newly defined polynomials may be immense use in new phase of Mathematics relevant to Physics, Chemistry, Engineering and Social Sciences.

REFERENCES

1. Burchnall J L. &Chaundy T. W.,“Expansions of Appell's double hypergeometric functions” (II). Quart. J. Math. Oxford ser.1941;12, 112 - 128.

2. Szego G. “Orthogonal polynomials. Amer. Math. Soc. ColloquiumPubl”. Amer. Math. Soc. Providence, Rhode Island.1967.Vol. 23, 3rd Ed.

3. ErdelyiA., et al “Higher transcendental functions” Bateman Manuscript project McGraw Hill.1953.Vol. I, II and III.

4. Rainville E. D. “Special functions. The Mc. Millan Company” New York.1960.



2 2

 

21 2 2 2



2



2 2 2



2 2

2 2 0 0 0

exp x y

x y x y z i x y z

x y

 

   _ _ _ _

    _   _



 

  





2 2

1 2 2 2

cos 2 tan

2

x y

f x y z



    

      

 

   

 







 





 _ _{ } _ _{ } 

 



 



 

 _{ } _ _ 

 

 

, , 1 :1;

2

1 ×

1 1

1 , :1 , 1 :1 ;

2 2 2

n n

x

F dxdydz

x

 

 ,

n