simplification of products of gamma functions & bfloat bug

I have not yet puzzled through this.

I have a bit of code that "knows" the Abramowitz & Stegun identities
15.3.3--15.3.9
(linear transformations of the 2F1 functions). Identity 15.3.7 is an
involution, I think:

(%i42) load("irene_and_milton.mac")$

Apply 15.3.7 once:

(%i43) abramowitz_id(hypergeometric([a,b],[c],x), "15.3.7");
(%o43) (gamma(a-b)*hypergeometric([b,-c+b+1],[b-a+1],1/x)*gamma(c))/(gamma
(a)*gamma(c-b)*(-x)^b)+(hypergeometric([a,-c+a+1],[-b+a+1],1/x)*gamma
(b-a)*gamma(c))/(gamma(b)*gamma(c-a)*(-x)^a)

Apply 15.3.7 twice:

(%i44) expand(abramowitz_id(abramowitz_id(hypergeometric([a,b],[c],x),
"15.3.7"),"15.3.7"));
(%o44) (gamma(a-b)*gamma(b-a+1)*hypergeometric([-c+a+1,-c+b
+1],[2-c],x)*gamma(c-1)*gamma(c)*(-x)^(1-c))/(gamma(a)*gamma(b)*gamma
(c-a)*gamma(c-b))+(gamma(-b+a+1)*gamma(b-a)*hypergeometric([-c+a+1,-c+b
+1],[2-c],x)*gamma(c-1)*gamma(c)*(-x)^(1-c))/(gamma(a)*gamma(b)*gamma
(c-a)*gamma(c-b))+(hypergeometric([a,b],[c],x)*gamma(a-b)*gamma(b-a
+1)*gamma(1-c)*gamma(c))/(gamma(1-a)*gamma(a)*gamma(-c+b+1)*gamma(c-b))+
(hypergeometric([a,b],[c],x)*gamma(-b+a+1)*gamma(b-a)*gamma(1-c)*gamma
(c))/(gamma(1-b)*gamma(b)*gamma(-c+a+1)*gamma(c-a))

Try specific values:

(%i45) sublis([a=1/2, b = 2/3, c = 1/7],%);
(%o45) (gamma(1/7)*gamma(1/6)*gamma(5/6)*gamma(6/7)*hypergeometric
([1/2,2/3],[1/7],x))/(%pi*gamma(10/21)*gamma(11/21))-(gamma(1/7)*gamma
(1/6)*gamma(5/6)*gamma(6/7)*hypergeometric([1/2,2/3],[1/7],x))/(gamma
(1/3)*gamma(5/14)*gamma(9/14)*gamma(2/3))
(%i46) float(%);
(%o46) 1.000000000000002*hypergeometric
([0.5,0.66666666666667],[0.14285714285714],x)

Barton

-----maxima-bounces at math.utexas.edu wrote: -----

>To:?Dieter?Kaiser?<drdieterkaiser at web.de>
>From:?Raymond?Toy?<raymond.toy at stericsson.com>
>Sent?by:?maxima-bounces at math.utexas.edu
>Date:?02/23/2009?03:41PM
>cc:?"maxima at math.utexas.edu"?<maxima at math.utexas.edu>,?Barton?Willis
><willisb at unk.edu>
>Subject:?Re:?[Maxima]?simplification?of?products?of?gamma?functions?&
>bfloat?bug
>
>Dieter?Kaiser?wrote:
>>?Am?Sonntag,?den?22.02.2009,?18:50?-0600?schrieb?Barton?Willis:
>>
>>>?Is?there?a?Maxima?function?that?simplifies?(%o47)?to?1??The?composition
>>>?minfactorial(makefact(...))?doesn't?simplify?(%o47)?to?1.?Also,?maybe
>it
>>>?has?already?been?fixed,?but?(%o48)?shows?a?bug:
>>>
>>>??(%o47)
>(gamma(1/7)*gamma(4/21)*gamma(17/21)*gamma(6/7)-gamma(4/21)*gamma
>>>??(10/21)*gamma(11/21)*gamma(17/21))/(gamma(1/7)*gamma(10/21)*gamma
>>>??(11/21)*gamma(6/7))
>>>
>>
>>?The?Maxima?functions?I?know?can?not?simplify?products?of?gamma
>>?functions.?Perhaps?we?can?implement?some?rules?for?the?product?of?gamma
>>?functions.?These?are?some?examples
>>
>>?(1)?gamma(z)*gamma(w)?=?factorial(z+w-2)/binomial(w+z-2,z-1)
>>?(2)?gamma(z)*gamma(w)?=?gamma(z+w)*beta(z,w)
>>
>>?(3)?gamma(z)/gamma(w)?=?factorial(z-w)/binomial(z-1,z-w)
>>?(4)?gamma(z)/gamma(w)?=?pochhammer(w,z-w)
>>
>>?But?the?first?rule?will?not?work?in?the?example?above,?because?we?get?an
>>?undefined?factorial(-1).?The?second?rule?will?simplify?to?an?expression
>>?with?the?sin?function.
>>
>>
>I?did?the?transformation?to?beta?functions?by?hand?yesterday.??I
>couldn't?get?the?trig?functions?to?simplify?to?1?either.???The?trig
>functions?involved?terms?like?sin(n*%pi/21)?and?cos(n*%pi/21).
>
>Ray
>
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