J. Comp. & Math. Sci. Vol.5 (1), 15-21 (2014)
A Summation Formula for Multivariable H-Function Neelam Pandey1 and Reshma Khan2 1
Mathematics, Model Science College Rewa, M.P., INDIA. 2 University Teaching Department, A P S University Rewa, M.P., INDIA. (Received on: January 2, 2014) ABSTRACT In the present work, we establish a summation formula for H-function of `r’ variable defined by Shrivastava and Panda. some particular cases have also been discussed. In one particular case, the formula reduces to a result involving multivariable H-function and Jacobi polynomial or chebyshev polynomial. The formula is of general character and can be applied to obtain a large number of results. Keywords: Chebyshev polynomial, Multivariable H-function, Jacobi polynomial.
I. INTRODUCTION
z1 ....z r was introduced by variables Shrivastava and Panda1. It is represented by
H-function of several complex
z1 . [(a ; 1 .... ( r ) ) : (c ' ' ) ....(c ( r ) 1 ) m r , nr j j j 1, p j j 1, p1 j j 1, pr . H [ z1 ....z r ] H p0,,nq::mp1 ,,nq1 .... 1 (r ) ( r ) (r ) 1 1 .... pr , qr (b j ; j .... j )1,Q ; (d 'j 'j )1,q1 ....(d j j )1, qr . z r
1 (2 ) r
(1)
r
( ) z ....z d ....d
.... (1.... r )
L1
i
i
1
r
r
1
r
where 1
(2)
i 1
Lr mi
and i ( i )
1
ni
(d (ji ) (ji ) i )
(1 d (ji )
j 1 qi
j mi 1
(1 c
(i ) j
(ji ) i )
j 1
where i 1....r
pi
(j i ) i )
(c (ji ) j ni 1
(i ) j i )
Journal of Computer and Mathematical Sciences Vol. 5, Issue 1, 28 February, 2014 Pages (1-122)
(3)
16
Neelam Pandey, et al., J. Comp. & Math. Sci. Vol.5 (1), 15-21 (2014)
The parameters a j , j 1.... p, c (ji ) , j 1.... pi ; b j , j 1....q, d (ji ) , j 1....qi i 1....r are complex numbers and (ji ) j 1.... p, (ji ) , j 1.... pi (j i ) j 1....q, (j i ) , j 1....qi i 1....r pi
p
i
(ji )
j 1
i
j 1
(ji )
j 1
(j i )
j 1
ni
p
qi
q
(ji )
j 1
(i ) j
0
j 1 qi
q
(ji )
(ji )
j 1
(i ) j
1 (6) i i 1....r 2 The contour L i in the complex i plane runs arg( zi )
from to with indentatio ns , if necessary to seperate poles of (d
(i) j i )
from (1
c (ji )
(i) j i )
r
and (1 a (ji )
(5)
0
j 1
The multiple Mellin –Barnes contour integral representing the multivariable Hfunction converges absolutely under the conditions (4) and (5) when
(i ) j
(4)
(i) j i )
.
2 2 u (n k )! (n 2 2u ) (k u ) sin(n 2k 1) (1 u )(u )k!n!(n k 2 u ) k 0
(7) provided u 1,0 and n is a non negative int eger.
(c) Rainville 1 (i ) ( 2 z ) 2 2 z 1 ( z ) ( z ) 2
(8)
n
( 2 ) n m ( z 1) m (9) 1 m m 0 2 m!( n m)!( ) m 2 where (a) n is the pochhamer symbol defined by
(ii ) Pn ( z )
i 1
(a) n a (a 1) ....(a n 1)
II. RESULTS AND FORMULA USED
(a n) ( a )
(d) Shrivastava, Gupta and Goel The following results will be used in our investigation. (a) Richard Askey2
sin 12u pn1u (cos )
(i ) Pn( ) ( x) (1 ) n 1 x ] 2 F1[ n,1 n;1 ; n! 2 provided Re( 1) 0, Re( 1 1) 0,
2 2u ( n k )! ( n 2 2 u) ( k u ) sin(n 2 k 1) (1 u )( u) k! n! (n k 2 u ) k 0 provided u 1,0
Pn( ) ( x ) is the Jacobi Polynomial.
and n is a non negative int eger .
(ii)
U n ( x)
(b) Rainville
sin 1 2u p1nu (cos )
sin[(n 1) cos 1 x] sin(cos 1 x)
,n 0
where
Journal of Computer and Mathematical Sciences Vol. 5, Issue 1, 28 February, 2014 Pages (1-122)
17
Neelam Pandey, et al., J. Comp. & Math. Sci. Vol.5 (1), 15-21 (2014) U n ( x ) is the chebychev
(iii ) U n ( x)
( n 1)! 3 2 n
polynomial
of
sec ond kind .
(e) Erdelyi
1 1 ( , ) pn2 2 ( x)
sin nz 1 n 1 n 3 n sinz 2 F1 , , ; sin 2 z ; sin z 1 2 2 2
III. MAIN RESULT 2
n
(sin ) 2 v 1
(cos ) m 0, N 1:m1 ,n1 ....mr ,nr m H p 1,Q 1: p1 , q1 .... p r , qr 2 m!(n m)! m0
(n k )! m r , nr sin(n 2k 1) H 0p,N1,Q1:m1:1p,n1, .... 1 q1 .... pr , q r n ! k ! k 0
z1 . [1 n 2v;2 ....2 ], ( a ; 1 .... ( r ) ) , [1 v; .... ] : (c ' ' ) ....(c ( r ) 1 ) 1 r j j j 1, p 1 r j j 1, p1 j j 1, pr . [ k 1 v; 1....r ], (b j ; 1j .... (j r ) )1,Q , n k v, 1 ....r : (d 'j 'j )1, q ....(d (jr ) (j r ) )1,q 1 r zr provided ' n' is non negative int eger,
i 0, i 1....r (all i are not zero simul tan eously), 0 arg zi
1 i where i is given by (5) 2
[1 n 2v m;21 ....2r ], ( a j ; 1j .... (jr ) )1, p : (c 'j 'j )1, p ....(c (jr ) 1j )1, p 1 r 21 2 r z1 sin ....z r sin (b ; 1 .... ( r ) ) , 1 m v, .... ; (d ' ' ) ....(d ( r ) ( r ) ) j j j 1,Q 1 r j j 1, q1 j j 1,q r 2
IV. PROOF 22u (n k )!(n 2 2u) (k u) sin(n 2k 1) (1 u)(u)k!n!(n k 2 u) k 0
(sin )12u p1nu (cos ) r
replacingu by (1 v ii ), we get i 1 r
r
2v 1 2
(sin )
r
ii
i1
ii
v
pn
i1
r r ii ) (n 2v 2 ii ) (k 1 v ii )(n k )!sin(n 2k 1) i1 2 i 1 i 1 (cos ) . . r r r k 0 (v ii ) (1 v ii ) k!n!(n k 1 v ii ) 2(1v
i 1
i 1
i 1
Since
Journal of Computer and Mathematical Sciences Vol. 5, Issue 1, 28 February, 2014 Pages (1-122)
18
Neelam Pandey, et al., J. Comp. & Math. Sci. Vol.5 (1), 15-21 (2014)
(sin )1 2u p1n u (cos )
2 2u (n k )! (n 2 2u ) (k u ) sin( n 2k 1) (1 u ) (u)k!n!(n k 2 u ) k 0
r
replacing u by (1 v
), we get i i
i 1 r
r
2 v 1 2
(sin )
2(1 v
r
ii
v
pn
i 1
i i i 1
(cos )
2
i i ) (n 2v 2 r i i ) i 1 .
r
(v i i ) i 1
i 1
r
(1 v i i )
r
(k 1 v i i )(n k )!sin( n 2k 1)
.
k 0
i 1
r
k!n! (n k 1 v i i )
i 1
i 1
Since 2l
(cos 1) m 0, N 1:m ,n ....m , n H p 1,Q 1:1p11, q1 ....rp r ,rqr m m!(2l m)! m 0
2
z sin 21 1 [1 2l 2v m;21....2r ], ( a j ; 1j .... (jr ) )1, p : (c 'j 'j )1, p1 ....(c (jr ) 1j )1, pr . 1 (b j ; 1j .... (j r ) )1,Q , m v, 1....r ; ( d 'j 'j )1, q1 ....(d (jr ) (j r ) )1, qr . 2 z1 sin 21 m r , nr B ( ) H 1p,N1,Q1:m11:,pn1,.... q .... p , q 1
r
1
r
z1 [1 2l 2v m;21....2r ], ( a j ; 1j .... (jr ) )1, p 1 v, : (c 'j 'j )1, p1 ....(c (jr ) 1j )1, pr . 1 . (b j ; 1j .... (j r ) )1,Q , m v, 1 ....r ; ( d 'j 'j )1,q1 ....(d (jr ) (j r ) )1,q r 2 z r n
(2 ) n m ( z 1) m 1 m m 0 2 m! ( n m )!( ) m 2 r r (cos 1) m n 2 (v i i ) v i i i 1 n m Pn i1 (cos ) r 1 m m 0 2 m!( n m)!( v i i ) m 2 i 1 Since Pn ( z )
r
(n m 2v 2 i i )(cos 1) m
n
i 1
r
2v 2 i i 1
r
r 1 m 0 i1 2 (v i i ) ( m v i i ) 2 m m!(n m)! 2 i 1 i 1 (a m) 1 (a) m and (2 z ) 2 2 z 1 ( z ) ( z ) ( a ) 2 r
v
Substituing the value of Pn
i i i1
(cos ) our result becomes
Journal of Computer and Mathematical Sciences Vol. 5, Issue 1, 28 February, 2014 Pages (1-122)
19
Neelam Pandey, et al., J. Comp. & Math. Sci. Vol.5 (1), 15-21 (2014) r
(cos 1) m (n m 2v 2 i i ) i 1
sin 2
2v 1 2
i i i 1
r
r 1 2 m!(n m)!(v i i ) ( m v i i ) 2 i 1 i 1 m
r
n
r
2 v 1 2
m 0
2
r
i i
( n 2v 2 i i )
i 1
i 1
.
r
r
(v 2 i i )
(1 v i i )
i 1
i 1
r ( k 1 v i i ) ( n k )! i 1 . sin( n 2k 1) r k! n! ( k 1 v ) k 0 i i i 1
Multiplying both sides by r
1 ( 2 )
( 1 .... r )
r
r
r
( ). ( v ) z 2 2
i
i
i i
i 1
i 1
i
i
i
, where ( 1 .... r ) and i ( i ) are given by () and ()
i 1
and Integratin g ' r ' times along the contours L1 .... L r between to , we get r
sin ( 2 ) r 2 1
2 v 1
r
.... ( 1 .... r )
L1
Lr
( ) z sin i
i 1
i
i
n
2 i
(cos 1) m ( n 2 v m 2
m0
1 2 m m! ( n m )! ( m v 2
) i i
i 1 r
d 1 .... d r
i i )
i 1
Changing the order of integration and summation and interpreting the result as H-function of r-variables, the result follows. On the L.H.S the change in the order of integration and summation is justified because series is finite and integral exist on R.H.S it is justified r
because the series converges uniformly w.r.t. i ( i ) z i i
(n 2v m 2 i i ) i 1
r
(1 v i i ) i 1
continuous and integral is absolutely convergent when the given conditions are satisfied. V. SPECIAL CASES Case 1: When n is even (n=2l) our result takes the form 2l
(cos 1) m 0, N 1:m , n ....m , n H p 1,Q 11: p11,q1 ....rp r ,rqr m m ! ( 2 l m )! m 0
2
Journal of Computer and Mathematical Sciences Vol. 5, Issue 1, 28 February, 2014 Pages (1-122)
is
20
Neelam Pandey, et al., J. Comp. & Math. Sci. Vol.5 (1), 15-21 (2014)
z sin 21 1 [1 2l 2v m;21....2r ], ( a j ; 1j .... (jr ) )1, p : (c 'j 'j )1, p1 ....(c (jr ) 1j )1, pr . 1 (b j ; 1j .... (j r ) )1,Q , m v, 1....r ; ( d 'j 'j )1, q1 ....(d (jr ) (j r ) )1, qr . 2 z1 sin 21 m r , nr B ( ) H 1p,N1,Q1:m11:,pn11,.... q1 .... pr , qr
z1 [1 2l 2v m;21....2r ], ( a j ; 1j .... (jr ) )1, p 1 v, : (c 'j 'j )1, p1 ....(c (jr ) 1j )1, pr . 1 . (b j ; 1j .... (j r ) )1,Q , m v, 1....r ; ( d 'j 'j )1,q1 ....(d (jr ) (j r ) )1,q r 2 z r
B( )
(2l k )! (sin ) 2 2v U 2l 2 k (cos ) k 0 k! 2l!
2
or 1 1 ( , ) (2l k )!(l k )! 2 2 B( ) 2 (sin ) 2 2v Pl k (cos 2 ) 1 k 0 k! 2l! (l k ) 2
Provided validity conditions (as stated) are satisfied for n=2l. Case 2: When n is odd (say n=2l+1) ,our result takes the form 2l 1 m 0
(cos 1) m
2
m
0 , N 1:m1 , n1 ....mr ,nr H p 1,Q 1: p1 ,q1 .... p r , qr m!(2l 1 m)!
z sin 21 1 [ 2l 2v m;21....2r ], (a j ; 1j .... (jr ) )1, p : (c 'j 'j )1, p1 ....(c (jr ) 1j )1, p r . 1 ' ' 1 (r) (r) (r) (b j ; j .... j )1,Q , m v, 1 ....r ; (d j j )1, q1 ....(d j j )1, qr . 2 z1 sin 21 1, N 1:m ,n ....m , n
A( ) H p 2,Q 12: p11 ,q1 ....r prr ,qr z1 1 (r ) ' ' (r) 1 . [2l 2v m;21 ....2r ], (a j ; j .... j )1, p 1 v, 1....r : (c j j )1, p1 ....(c j j )1, pr . k 1 v, .... (b ; 1 .... (r ) ) , 2l 1 k v, .... ; (d ' ' ) ....(d ( r ) ( r ) ) 1 r j j j 1,Q 1 r j j 1,q1 j j 1, qr z r Journal of Computer and Mathematical Sciences Vol. 5, Issue 1, 28 February, 2014 Pages (1-122)
Neelam Pandey, et al., J. Comp. & Math. Sci. Vol.5 (1), 15-21 (2014)
A( )
21
(2l k 1)! (sin ) 2 2v U 2l 2 k 1 (cos ) k 0 k! ( 2l 1)!
2
or A( )
(2l k 1)! sin 2 (sin )12v U l k (cos 2 ) k 0 k! ( 2l 1)!
2
Provided validity conditions (as stated) are satisfied for n=2l+1. REFERENCES 1. H.M. Shrivastava and R. Panda. Some bilateral generating function for a class of generalized Hypergeometric polynomials. J. Reine Math. 283/284, p.265-274 (1976). 2. Richard Askey:``Orthogonal expansions with positive coefficients’’ Proc. Amer. Math. Soc. 16,1191-1194 (1965).
3. Rainville, E.D. ``Special Functions’’ Macmillan publishers, New York (1963). 4. Shrivastava H.M, Gupta K.C., Goel S.P. The H-function of one and two variables with applications. South Asian publishers, New Delhi (1982). 5. Erdélyi A-Higher Transcendental Functions, Vol1. Macgraw Hill Book company New York.
Journal of Computer and Mathematical Sciences Vol. 5, Issue 1, 28 February, 2014 Pages (1-122)