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# h43-0300

Connection formula of solutions between 0 and infty.

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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]

Nobuki Takayama 2003-02-03