TITLE: Simplify $\Gamma\left(\frac27\right) \Gamma\left(\frac{11}{42}\right)/\;\Gamma\left(\frac1{21}\right)$ to elementary terms
QUESTION [14 upvotes]: How can we prove the following identity?
$$\large \frac{\Gamma\left(\frac27\right) \Gamma\left(\frac{11}{42}\right)}{\Gamma\left(\frac1{21}\right)} = \frac{8 \sin\left(\frac\pi7\right) \sqrt{\pi \, \sin\left(\frac\pi{21}\right) \sin\left(\frac{4\pi}{21}\right) \sin\left(\frac{5\pi}{21}\right)}}{\sqrt[42]2 \; \sqrt[3]7 \; \sqrt[28]{19683}}$$
I guess we could use the Gauss multiplication formula, but how?

REPLY [5 votes]: I am typing some (informal) book on sagemath, with focus on structural application, and this post was an ideal occasion / setting to use sage in a natural, non-trivial way, here i am copy+pasting the full (computer aided, quick) proof, so that in similar situations one has a straight line to the solution.

It is handy, and it makes typing easier to use a better notation for $\Gamma(k/42)$ for all potentially relevant values of $k$, so let us set formally
$$
\color{red}{
[k] =
\Gamma\left(\frac k{42}\right) }\ ,\qquad k=1,2,\dots, 20,21, 22, \dots, 41,\dots\ \ .
$$
Then a duplication formula is
$$
[k]\;[k+21]=\text{(Known constant)}\cdot[2k]\ ,
$$
Note: In the programming section we consider the purely multiplicative functional equations (FEs),
which are the  Gauss multiplication FEs, and the reflexion FEs, 
written additively, 
then  use notationally $x_k$ instead of $[k]$ (in the additive world), and the above relation reads
$x_k+x_{k+21}-x_{2k}$ is "known", this leads to a  system of linear equations, and for our purposes it is enough to pass to the homogeneous version and search for a writing of $x_{11}+x_{12}-x_2$ in terms of the collected equations.

Sage brings us straightforward to the finish, and suggests then (code in the sequel) to look at the relations:
$$
\begin{aligned}{}
[4]
\cdot[4+6]
\cdot[4+12]
\cdot[4+18]
\cdot[4+24]
\cdot[4+30]
\cdot[4+36]
\cdot[7\cdot 4]^{-1}
&=(2\pi)^{6/2}\cdot 7^{1/2-7\cdot 4/42}
&&\text{ to power $+1$}\ ,
\\
%
[6]
\cdot\color{magenta}{[6+6]}
\cdot[6+12]
\cdot[6+18]
\cdot[6+24]
\cdot[6+30]
\cdot[6+36]
\cdot[7\cdot 6]^{-1}
&=(2\pi)^{6/2}\cdot 7^{1/2-7\cdot 6/42}
&&\text{ to power $+1$}\ ,
\\[3mm]
%
\color{blue}{[2]}
\cdot[2+14]
\cdot[2+28]
\cdot[3\cdot 2]^{-1}
&=(2\pi)^{2/2}\cdot 3^{1/2-3\cdot 2/42}
&&\text{ to power $-1$}\ ,
\\
%
[4]
\cdot[4+14]
\cdot[4+28]
\cdot\color{magenta}{[3\cdot 4]^{-1}}
&=(2\pi)^{2/2}\cdot 3^{1/2-3\cdot 4/42}
&&\text{ to power $-1$}\ ,
\\
%
[8]
\cdot[8+14]
\cdot[8+28]
\cdot[3\cdot 8]^{-1}
&=(2\pi)^{2/2}\cdot 3^{1/2-3\cdot 8/42}
&&\text{ to power $+1$}\ ,
\\[3mm]
%
\color{red}{[11]}
\cdot[11+21]
\cdot[2\cdot 11]^{-1}
&=(2\pi)^{1/2}\cdot 2^{1/2-2\cdot 11/42}
&&\text{ to power $+2$}\ ,
\\[3mm]
%
\color{blue}{[2]}
\cdot[42-2]
&=\pi\cdot \left(\sin\frac{2\pi}{42}\right)^{-1}
&&\text{ to power $-1$}\ ,
\\
%
[6]
\cdot[42-6]
&=\pi\cdot \left(\sin\frac{6\pi}{42}\right)^{-1}
&&\text{ to power $-2$}\ ,
\\
%
[8]
\cdot[42-8]
&=\pi\cdot \left(\sin\frac{8\pi}{42}\right)^{-1}
&&\text{ to power $-1$}\ ,
\\
%
[10]
\cdot[42-10]
&=\pi\cdot \left(\sin\frac{10\pi}{42}\right)^{-1}
&&\text{ to power $-1$}\ .
%
\end{aligned}
$$
We multiply the equations above, taken to the mentioned powers,
only the colored factors survive:
$$
\begin{aligned}
\color{red}{[11]^2}
\cdot
\color{magenta}{[12]^2}
\cdot
\color{blue}{[2]^{-2}}
&=
(2\pi)^6
\cdot
\pi^{-5}
\cdot
2^{1-22/21}
\cdot
3^{-1/2-1/7}
\cdot
7^{-2/3}
\cdot
\\
&\qquad\qquad\cdot
\sin^2\frac{6\pi}{42}
\cdot
\sin\frac{2\pi}{42}
\cdot
\sin\frac{8\pi}{42}
\cdot
\sin\frac{10\pi}{42}
\end{aligned}
$$
which is our relation. One can separate the algebraic number, and rewrite
putting in evidence 
$
\color{red}{\frac{11}{42}}
+
\color{magenta}{\frac{12}{42}}
=
\color{blue}{\frac{2}{42}}
+
\color{green}{\frac{21}{42}}
$:
$$
\begin{aligned}
\frac
{
\color{red}{\Gamma\left(\frac{11}{42}\right)}
\color{magenta}{\Gamma\left(\frac{12}{42}\right)}
}
{
\color{blue}{\Gamma\left(\frac{2}{42}\right)}
\color{green}{\Gamma\left(\frac{21}{42}\right)}
}
&=
\frac
{\displaystyle
2^3\cdot\sin\frac{\pi}7
\sqrt{
\sin\frac{\pi}{21}
\cdot
\sin\frac{4\pi}{21}
\cdot
\sin\frac{5\pi}{21}}
}
{\displaystyle2^{1/42}\cdot 3^{9/28}\cdot 7^{1/3}}
\\
&=
\frac
{\displaystyle
2^{5/4}
(5-\sqrt{21})^{1/4}
\sin\frac{\pi}7
}
{\displaystyle
2^{1/42}\cdot 3^{9/28}\cdot 7^{1/3}}
\ .
\end{aligned}
$$
That's all.
$\square$

As promised, some computer art work support the above.


(1)

Numerical check of the equalities above:
LHS = gamma(11/42) * gamma(12/42) / gamma(2/42) / gamma(21/42)
RHS1 = 8 * sin(pi/7) * sqrt(prod([sin(k*pi/21) for k in [1,4,5]])) / 2^(1/42) / 3^(9/28) / 7^(1/3)
RHS2 = 2^(5/4) * sin(pi/7) * (5-sqrt(21))^(1/4) / 2^(1/42) / 3^(9/28) / 7^(1/3)

print "LHS =", LHS.n()
print "RHS1 =", RHS1.n()
print "RHS2 =", RHS2.n()

which gives
LHS = 0.299625085598959
RHS1 = 0.299625085598959
RHS2 = 0.299625085598959


(2)

Minimal polynomial of the expression (sine product) under the square root:
K.<u> = CyclotomicField(84)        # 84 = 7*4*3
u42, u21, u4 = u^2, u^4, u^21
s1 = (u42^1 - 1/u42^1) / 2 / u4    # sine of 2.1pi/42 =  pi/21
s4 = (u42^4 - 1/u42^4) / 2 / u4    # sine of 2.4pi/42 = 4pi/21
s5 = (u42^5 - 1/u42^5) / 2 / u4    # sine of 2.5pi/42 = 5pi/21
print (8*s1*s4*s5).minpoly()

which gives, leading then immediately to the factor $(5-\sqrt{21})$:
x^4 - 5*x^2 + 1


(3) 

As promised, the code that found the needed "linear" relation to be used:
R = PolynomialRing(QQ, 'x', 42 )     # quick init, we do not use x0
v = R.gens()
for k in range(42):
    exec('x%s = v[%s]' % (k, k) )    # so x1 is indeed x1, x2 is indeed x2...

def normalized_variable(k):
    return v[ k % 42 ]

def eq(times, k):
    """times should divide 42 in the caller, Gauss FE linearized, no constant."""
    inc = ZZ(42/times)
    return ( sum([ normalized_variable(k + j*inc)
                   for j in range(times) ])
             - normalized_variable(times*k) )

equations = (
    [ eq(times, k)
      for times in [7, 3, 2] # ZZ(42).divisors()[ ::-1 ]
      for k in [ 1..ZZ(42/times) ]
      if  times > 1
      # and ( times != 7 or k == 1 )
    ]
    +
    [ v[k] + v[42-k] for k in [1..21]
      if k in [2, 6, 8, 10] ]
    # last is a psychological condition, knowing the result,
    # exactly this condition makes us go straightforward to the result,  
    # else we have a hard work in the cyclotomic field of all sine values,
    # which is harder than parking the gamma values in each other...
)

J = R.ideal(equations)
f = (x11 + x12 - x2)

print "Is f = %s in J? %s" % ( f, f in J )
lif = f.lift(J)

for r in range(len(equations)):
    if lif[r]:
        print '%+4s TIMES %s' % (lif[r], equations[r])

The above gives:
Is f = -x2 + x11 + x12 in J? True

 1/2 TIMES x4 + x10 + x16 + x22 + x34 + x40
 1/2 TIMES x6 + x12 + x18 + x24 + x30 + x36
-1/2 TIMES x2 - x6 + x16 + x30
-1/2 TIMES x4 - x12 + x18 + x32
 1/2 TIMES x8 + x22 - x24 + x36
   1 TIMES x11 - x22 + x32
-1/2 TIMES x2 + x40
  -1 TIMES x6 + x36
-1/2 TIMES x8 + x34
-1/2 TIMES x10 + x32
sage: 

which "is" the solution.