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h43-0300

Connection formula of solutions between 0 and infty.

Typeset by om2tex.xsl

Typeset by Mathematica

Formula in the tfb format:

logic1.implies(((((((((set1.notin(a1 - a2, set1.map(OMLBIND(OMBVAR(x), 
logic1.and(set1.in(x, setname1.Z), relation1.leq(x, 0)))))) ~logic1.and~ 
(set1.notin(a1 - a3, set1.map(OMLBIND(OMBVAR(x), logic1.and(set1.in(x, 
setname1.Z), relation1.leq(x, 0))))))) ~logic1.and~ (set1.notin(a1 - a4, 
set1.map(OMLBIND(OMBVAR(x), logic1.and(set1.in(x, setname1.Z), relation1.leq(x, 
0))))))) ~logic1.and~ (set1.notin(a2 - a3, set1.map(OMLBIND(OMBVAR(x), 
logic1.and(set1.in(x, setname1.Z), relation1.leq(x, 0))))))) ~logic1.and~ 
(set1.notin(a2 - a4, set1.map(OMLBIND(OMBVAR(x), logic1.and(set1.in(x, 
setname1.Z), relation1.leq(x, 0))))))) ~logic1.and~ (set1.notin(a3 - a4, 
set1.map(OMLBIND(OMBVAR(x), logic1.and(set1.in(x, setname1.Z), relation1.leq(x, 
0))))))) ~logic1.and~ (set1.notin(b1, set1.map(OMLBIND(OMBVAR(x), 
logic1.and(set1.in(x, setname1.Z), relation1.leq(x, 0))))))) ~logic1.and~ 
(set1.notin(b2, set1.map(OMLBIND(OMBVAR(x), logic1.and(set1.in(x, setname1.Z), 
relation1.leq(x, 0))))))) ~logic1.and~ (set1.notin(b3, 
set1.map(OMLBIND(OMBVAR(x), logic1.and(set1.in(x, setname1.Z), relation1.leq(x, 
0)))))), (hypergeo1.hypergeometric_pFq(list1.list(a1, a2, a3, a4), 
list1.list(b1, b2, b3), z)) ~relation1.eq~ 
(((((arith1.divide(((((((hypergeo0.gamma(b1) * hypergeo0.gamma(b2)) * 
hypergeo0.gamma(b3)) * hypergeo0.gamma(a2 - a1)) * hypergeo0.gamma(a3 - a1)) * 
hypergeo0.gamma(a4 - a1)))  , ((((((hypergeo0.gamma(b1 - a1) * 
hypergeo0.gamma(b2 - a1)) * hypergeo0.gamma(b3 - a1)) * hypergeo0.gamma(a2)) * 
hypergeo0.gamma(a3)) * hypergeo0.gamma(a4))) ) )  * 
hypergeo1.hypergeometric_pFq(list1.list(a1, (a1 - b1)  + 1, (a1 - b2)  + 1, (a1 
- b3)  + 1), list1.list((a1 - a2)  + 1, (a1 - a3)  + 1, (a1 - a4)  + 1), 
arith1.divide(1 , z) )) * (arith1.power((arith1.unary_minus(z))  , 
(arith1.unary_minus((a1) )) ) ) ) + (((arith1.divide(((((((hypergeo0.gamma(b1) 
* hypergeo0.gamma(b2)) * hypergeo0.gamma(b3)) * hypergeo0.gamma(a1 - a2)) * 
hypergeo0.gamma(a3 - a2)) * hypergeo0.gamma(a4 - a2)))  , 
((((((hypergeo0.gamma(b1 - a2) * hypergeo0.gamma(b2 - a2)) * hypergeo0.gamma(b3 
- a2)) * hypergeo0.gamma(a1)) * hypergeo0.gamma(a3)) * hypergeo0.gamma(a4))) ) 
)  * hypergeo1.hypergeometric_pFq(list1.list(a2, (a2 - b1)  + 1, (a2 - b2)  + 
1, (a2 - b3)  + 1), list1.list((a2 - a1)  + 1, (a2 - a3)  + 1, (a2 - a4)  + 1), 
arith1.divide(1 , z) )) * (arith1.power((arith1.unary_minus(z))  , 
(arith1.unary_minus((a2) )) ) ) ) + (((arith1.divide(((((((hypergeo0.gamma(b1) 
* hypergeo0.gamma(b2)) * hypergeo0.gamma(b3)) * hypergeo0.gamma(a1 - a3)) * 
hypergeo0.gamma(a2 - a3)) * hypergeo0.gamma(a4 - a3)))  , 
((((((hypergeo0.gamma(b1 - a3) * hypergeo0.gamma(b2 - a3)) * hypergeo0.gamma(b3 
- a3)) * hypergeo0.gamma(a1)) * hypergeo0.gamma(a2)) * hypergeo0.gamma(a4))) ) 
)  * hypergeo1.hypergeometric_pFq(list1.list(a3, (a3 - b1)  + 1, (a3 - b2)  + 
1, (a3 - b3)  + 1), list1.list((a3 - a1)  + 1, (a3 - a2)  + 1, (a3 - a4)  + 1), 
arith1.divide(1 , z) )) * (arith1.power((arith1.unary_minus(z))  , 
(arith1.unary_minus((a3) )) ) ) ) + (((arith1.divide(((((((hypergeo0.gamma(b1) 
* hypergeo0.gamma(b2)) * hypergeo0.gamma(b3)) * hypergeo0.gamma(a1 - a4)) * 
hypergeo0.gamma(a2 - a4)) * hypergeo0.gamma(a3 - a4)))  , 
((((((hypergeo0.gamma(b1 - a4) * hypergeo0.gamma(b2 - a4)) * hypergeo0.gamma(b3 
- a4)) * hypergeo0.gamma(a1)) * hypergeo0.gamma(a2)) * hypergeo0.gamma(a3))) ) 
)  * hypergeo1.hypergeometric_pFq(list1.list(a4, (a4 - b1)  + 1, (a4 - b2)  + 
1, (a4 - b3)  + 1), list1.list((a4 - a1)  + 1, (a4 - a2)  + 1, (a4 - a3)  + 1), 
arith1.divide(1 , z) )) * (arith1.power((arith1.unary_minus(z))  , 
(arith1.unary_minus((a4) )) ) ) )) ));

Connection formula of 4F3 between 0 and infty for generaic values of parameters.

Reference: [0]

Proof: [0] Retrieve the formula in Mathematica form h43-0300-math-auto.m Retrieve the formula in Risa/Asir form h43-0300-asir-auto.rr Retrieve the formula in LaTeX form h43-0300-tex-auto.tex Interactive replacement h43-0300-js-auto.html


Nobuki Takayama 2003-02-03