confluent hypergeometric functions of two variables denoted by. 2. 1. 2. 1. 2. 1. ,,,,,Î ..... analogues of certain fractional integral operators ., Pro Mathematica Vol.
SOME EXPANSIONS OF CONFLUENT HYPERGEOMETRIC FUNCTIONS OF TWO VARIABLES AND APPELL’S TYPE FUNCTIONS BY K.S. NISAR AND M.A. KHAN
Presented @
ABSTRACT The present paper gives some relation between Appell’s functions and confluent hypergeometric functions of two variables denoted by 1 , 2 , 1 , 2 , 1 , 2 . The paper contains symbolic forms and some
expansion formulae for 1 , 2 , 1 , 2 , 1 , 2 and Appell’s type functions using the method of J.L. Burchnall and T.W.Chaundy [2,3].
INTRODUCTION In [4,7] gave numerous formulae involving the elementary hypergeometric functions of Appell’s type. It would be tedious to put all these on record and we collect here typical examples that should be sufficiently representative. For the most part of these formulae appear as a special cases of corresponding formulae of [4] arising from special values of the arguments or the parameters.
We recall, in slightly modified notation, the four Appell’s functions defined by their expansions.
F [a; b, b' ; c; x, y ] (1)
a m n b m b'n m n x y m!n!c m n
m 0 n 0
F
( 2)
[a; b, b' ; c, c' ; x, y ]
m 0 n 0
F (3) [a, a'; b, b'; c; x, y ]
m0 n 0
F
( 4)
[a; b; c, c'; x, y ]
m 0 n 0
a m n b m b'n m n x y m!n!c m c'n
a m a'n b m b'n x m y n m!n!c m n
a mn b mn m n x y m!n!c m c'n
Confluent hypergeometric functions of two variables: Seven confluent form of four Appell’s functions were defined in 1920 by P. Humbert and is defined as
1[ ; ; ; x, y ]
m0 n0
2 [ , ' ; ; x, y ]
m0 n 0
3[ ; ; x, y ]
m0 n 0
m n m m n x y m! n! m n
, x 1, y
m 'm x m y n , m!n! m n
x , y
m x m y n , m!n! m n
x , y
1[ , ; , ' ; x, y ]
m0 n0
2 [ ; , ' ; x, y ]
m0 n 0
1[ , ' , ; ; x, y ]
m n x m y n , m!n! m 'n
m0 n 0
2 [ , ; ; x, y ]
m0 n 0
m n m x m y n , m! n! m 'n
x , y
m 'n m m n x y , m! n! m n
m m m n x y , m! n! m n
x 1, y
x 1, y
x 1, y
Symbolic operators defined as
(h)( ' h) ( h) ( ' h)
(1)
( h) ( 'h) (h) (h)( 'h)
(2)
( h)
and
x
and ' y x y
then (h)(h)m(h)nxmyn = (h)m+nxmyn and so, if (h)m(h)n occurs in the numerator of the coefficient of xmyn, it is changed in to (h)m+n by the operator (h). The operator (h) effects a similar change in the denominator.
Now we give symbolic representations-
1[ , ; , ' ; x, y ] ( )1[ , ; ; x, y ]
(3)
1[ , , ; ; x, y ] ( )1[ , ; ; x, y ]
(4)
F ( 2) [ ; , ; , ; x, y] ( )( )( )1[ , ; ; x, y]
(5)
F (1) [ ; , ' ; ; x, y] ( )1[ , ' , ; ; x, y]
(6)
1[ , , ; ; x, y ] ( )( ) 1[ , ; , ; x, y ]
(7)
2 [ , ' ; ; x, y ] ( )( ) 2 [ , ; ; x, y ]
(8)
Expansions: Following procedure adopted by Burchnall and Chaundy, we obtain the following expansions
1r r r x r y r [ r , r; 2r; x, y] 1 r! 2 r
1[ , , ; x, y ] r 0
F (1) [ ; , '; ; x, y] r 0
1[ , ; , '; x, y ] r 0
'r r r r x y 1[ r , 'r; r '; 2r; x, y] r! '2r
2r r r r x y 1[ 2r , 2r; 2r; x, y ] r! r 2 r
(9)
(10)
(11)
We illustrate the method by considering one example. In (11) replace m, n by , ' so that symbolically
( h) r 0
( ) r ( ' ) r r!(h) r
thus in (3) we have
r 0
r 0
r 0
( ) r ( ' ) r 1[ , ; ; x, y ] r!( ) r
( ) 2 r ( ) r r r x y r!( ) r ( ) 2 r m 0
n 0
2r mn ( r ) m m!n!( 2r ) mn
( ) 2 r ( ) r 1[ 2r , r; 2r; x, y ] r!( ) r ( ) 2 r
xm yn
1.
Extension of functions of higher orders: Appell’s type functions have been defined in terms of product of two 3F2- functions the same in a way that M.A. Khan and G.S. Abukhammash [4] have written symbolically asM 1 a, a ' , b, b' , c, c' ; d , e, e' ; x, y (d ) 3 F2 (a, b, c; d , e; x)
3 F2 ( a ' , b' , c ' ; d , e' ;
y)
(1)
M 2 a, a ' , b, b' , c, c' ; d , e; x, y (d ) (e) 3 F2 (a, b, c; d , e; x) 3 F2 (a ' , b' , c '; d , e'; y )
(2)
M 3 a, a ' , b, b' , c, c ' ; d , e, e' ; x, y (a ) 3 F2 (a, b, c; d , e; x ) 3 F2 ( a ' , b' , c' ; d ' , e' ; y )
(3)
M 4 a, a ' , b, b' , c, c ' ; d , e, e' ; x, y (a )( d ) 3 F2 (a, b, c; d , e; x ) 3 F2 ( a, b' , c' ; d , e' ; y )
(4)
M 6 a, a ' , b, b' , c, c ' ; d , e, e' ; x, y (a )(b) 3 F2 (a, b, c; d , e; x) 3 F2 (a ' , b, c' ; d ' , e' ; y )
(5)
If we extend these formulae so that on the right the parameters , and 3F2 all are different, we get on the left, double hypergeometric function of higher order. Thus, (h) 3 F2 (a, b, c; d , e; x) 3 F2 (a ' , b' , c ' ; d ' , e' ; y )
m0 n0
(h) m (a ) m (b) m (c ) m (h) n (a ' ) n (b' ) n (c ' ) n x m y n ( h ) m n ( d ) m ( e) m ( d ' ) n ( e ' ) n m! n!
h, a, b, c; h, a' , b' , c' ; F x, y d ' e' ; h : d , e;
(8)
(h)(k ) 3 F2 (a, b, c; d , e; x) 3 F2 (a ' , b' , c' ; d ' , e' ; y )
m0
n0
(h) m (k ) m (a ) m (b) m (c) m (h) n (k ) n (a ' ) n (b' ) n (c' ) n x m y n ( h ) m n ( k ) m n ( d ) m (e) m ( d ' ) n (e' ) n m! n!
h, a, b, c; h, a ' , b' , c ' ; F x, y d ' e' ; h, k : d , e;
(9)
(h) 3 F2 (a, b, c; d , e; x) 3 F2 (a' , b' , c' ; d ' , e' ; y )
m0 n0
(h) m n (a) m (b) m (c) m (h) n (k ) n (a' ) n (b' ) n (c' ) n x m y n ( h ) m ( d ) m ( e) m ( h ) n ( d ' ) n ( e ' ) n m! n!
h, a, b, c; a ' , b' , c' ; F x, y h, d , e; h, d ' , e' ; (h)(k ) 3 F2 (a, b, c; d , e; x)
m 0 n 0
3 F2 ( a ' , b' , c ' ; d ' , e' ;
(10)
y)
(h) m n (k ) m (a) m (b) m (c) m (h) n (k ) n (a' ) n (b' ) n (c' ) n x m y n ( k ) m n ( h ) n ( d ) m ( e) m ( h ) n ( d ' ) n ( e' ) n m! n!
h : k , a, b, c; k , a' , b' , c'; F x, y h, d ' , e'; k : h, d , e;
(11)
(h)(k ) 3 F2 (a, b, c; d , e; x) 3 F2 (a' , b' , c' ; d ' , e' ; y )
m 0 n 0
(h) m n (k ) m n (a) m (b) m (c) m (a' ) n (b' ) n (c' ) n x m y n m! n! ( h ) m ( k ) m ( d ) m ( e) m ( h ) n ( d ' ) n ( e' ) n
h, k : a, b, c; a' , b' , c' ; x, y F h, d , e; h, k , d ' , e' ;
(12)
in a notation that seems more economical than that suggested by Appell and Kampéde Fériet.
The expansion for ,, given in [4] are still valid, and we obtain expansion analogues to those of [4] §5. h, a, b, c; h, a' , b' , c'; F x, y d ' e' ; h : d , e;
r 0
(1) r (h) r (a ) r (b) r (c) r (a ' ) r (b' ) r (c' ) r r r x y r!(h) 2 r (h r 1) r (d ) r (e) r (d ' ) r (e' ) r 2
h r , a r , b r , c r ; 4 F3 x h 2r , d r , e r
h r , a' r , b' r , c' r; y 4 F3 h 2r , d ' r , e' r
(13)
h : a, b, c; a ' , b' , c' ; F x, y h, d , e; h, d ' , e' ;
r 0
(a) r (b) r (c) r (a' ) r (b' ) r (c' ) r r r x y r!(h) r (d ) r (e) r (d ' ) r (e' ) r
a r , b r , c r ; 3 F2 x d r, e r
a' r , b' r , c' r ; F y 3 2 d ' r , e' r
(14)
h : k , a, b, c; k , a' , b' , c' ; F x, y h, d ' , e' ; k : h, d , e;
r 0
(k )r 3 (k h)r (a)r (b)r (c)r (a' )r (b' )r (c' )r r!(k r 1) r [( k ) 2 r ]2 (h) r (d ) r (e) r (d ' ) r (e' ) r
k r , a r , b r , c r ; 4 F3 x k 2r , d r , e r
xr y r
k r , a' r , b' r , c' r ; y 4 F3 k 2r , d ' r , e' r
(15)
REFERENCES 1. APPELL. P.: Sur une class de polynomes., Ann. Sci. Ecole Norm. Sup. (2), Vol.9, pp.119-144, 1980 2. BURCHNALL, J.L. AND CHAUNDY, T.W.: Expansions of Appell’s double hypergeometric functions, Quart. J. Math. Oxford Ser., Vol.12, pp.112-128, 1940 3. BURCHNALL, J.L. AND CHAUNDY, T.W.: Expansions of Appell’s double hypergeometric functions (II), Quart. J. Math. Oxford Ser., Vol.12, pp.112-128, 1941. 4. KHAN, M.A. AND ABUKHAMMASH, G.S: On a generalizations of Appell’s functions of two variables., Pro Mathematica Vol XVI, Nos. 3132, pp. 61-83, 2002. 5. KHAN,M.A. AND ABUKHAMMASH,G.S : A study of two variable analogues of certain fractional integral operators ., Pro Mathematica Vol XVII, Nos.33,pp. , 2003. 6. KHAN,M.A. AND ABUKHAMMASH,G.S : On a certain fractional integral operators of two variable and integral transform., Pro Mathematica Vol XVII, Nos.34,pp. , 2003.
7. KHAN,M.A. AND NISAR K.S : On a generalizations of Appell's type functions of two variables and their expansions., communicated for publications. 8. LAURICELLA,G. : Sulle funzioni ipergeometriche a piu variabili., Rend.Circ. Mat.palermo, Vol.7,pp. 111-158, 1893. 9. RAINVILLE, E.D. : Special Functions, Macmillan, New York; Reprinted by Chelsea Publ. Co., Bronx., New York, (1971). 10.SAIGO,M : A Remark on integral operators involving Hypergeometric functions ., math .Rep.kuyshu Univ.Vol.ll,pp 135-143, 1978.
11.SRIVASTAVA, H.M.: Certain double integrals involving hypergeoraetric functions, Jnanbha Sect.A.,vol.1,pp.1-10,1984. 12.SRIVASTAVA, H.M. AND MANOCHA, H.L. : A Treatise on Generating Functions, John Wiley & Sons (Halsted Press), New York; Ellis Horwood, Chichester, 1984 13. SRIVASTAVA, H.M. AND KARLSON, P.W. : Multiple Gaussian Hypergeometric series, Ellis Horwood Limited Chichester, 1985.
