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\begin{center}
\vspace*{1cm}
\epsfxsize=4in
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\mbox{\null}
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\begin{center}
\vskip 1cm{\LARGE\bf Full Description of Ramanujan\\
\vskip .1in
Cubic Polynomials}
\vskip 1cm
\large
Roman Witu{\l}a \\
Institute of Mathematics \\
Silesian University of Technology \\
Kaszubska 23 \\
Gliwice 44-100 \\
Poland \\
\href{mailto:roman.witula@polsl.pl}{\tt roman.witula@polsl.pl} \\
\end{center}


\vskip .2in

\centerline{\textit{Dedicated to Vladimir Shevelev~-- for his inspiration}}

\vskip .2in

\begin{abstract}
We give a full description of the Ramanujan cubic polynomials,
introduced and first investigated by V.~Shevelev.
We also present some applications of this result.
\end{abstract}


\vskip .2in

\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{defin}[theorem]{Definition}
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\newenvironment{remark}{\begin{rema}\normalfont\quad}{\end{rema}}






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\section{Introduction}

Shevelev \cite{Shevelev2007} called the cubic polynomial
\begin{equation}\label{rcp-w1}
x^3 +p\, x^2 +q\, x +r
\end{equation}
a {\it Ramanujan cubic polynomial} (RCP),
if it has real roots $x_1, x_2, x_3$ and the condition
\begin{equation}\label{rcp-w2}
p\, r^{1/3} + 3\, r^{2/3} + q = 0
\end{equation}
is satisfied.
It should be noticed, that if
$x_1, x_2, x_3$ are roots of RCP
of the form~(\ref{rcp-w1}),
then the following formulas hold (see~\cite{Shevelev2007,Witula-R}):
\begin{align}
x_{1}^{1/3} + x_{2}^{1/3} + x_{3}^{1/3} &=
\Big( {-}p - 6\, r^{1/3} + 3\, (9\, r - p\, q)^{1/3} \Big)^{1/3},\label{wzorA}\\
(x_{1}\, x_{2})^{1/3} + (x_{1}\, x_{3})^{1/3} + (x_{2}\, x_{3})^{1/3} &=
\Big( q + 6\, r^{2/3} - 3\, (9\, r^2 - p\, q\, r)^{1/3} \Big)^{1/3},\label{wzorB}
\end{align}
and Shevelev's  formula~\cite{Shevelev2007}:
\begin{equation}\label{wzorC}
\Big( \frac{x_{1}}{x_{2}} \Big)^{1/3} +
\Big( \frac{x_{2}}{x_{1}} \Big)^{1/3} +
\Big( \frac{x_{1}}{x_{3}} \Big)^{1/3} +
\Big( \frac{x_{3}}{x_{1}} \Big)^{1/3} +
\Big( \frac{x_{2}}{x_{3}} \Big)^{1/3} +
\Big( \frac{x_{3}}{x_{2}} \Big)^{1/3} =
\Big( \frac{p\, q}{r} - 9 \Big)^{1/3}.
\end{equation}
We note that~(\ref{wzorA}) easily implies all three Ramanujan equalities
\begin{align}
&\Big(\frac{1}{9}\Big)^{\!1/3} -
\Big(\frac{2}{9}\Big)^{\!1/3} +
\Big(\frac{4}{9}\Big)^{\!1/3}
= \big(\sqrt[3]{2}-1\big)^{\!1/3},\label{rcp-wA}\\
&
\Big( \cos \frac{2\, \pi}{7} \Big)^{\!1/3} +
\Big( \cos \frac{4\, \pi}{7} \Big)^{\!1/3} +
\Big( \cos \frac{8\, \pi}{7} \Big)^{\!1/3}
= \Big( \frac{5-3\, \sqrt[3]{7}}{2} \Big)^{\!1/3},\label{rcp-wB}\\
&
\Big( \cos \frac{2\, \pi}{9} \Big)^{\!1/3} +
\Big( \cos \frac{4\, \pi}{9} \Big)^{\!1/3} +
\Big( \cos \frac{8\, \pi}{9} \Big)^{\!1/3}
= \Big( \frac{3\, \sqrt[3]{9}-6}{2} \Big)^{\!1/3},\label{rcp-wC}
\end{align}
since the following decompositions of polynomials hold:
(\ref{rcp-w100}), which implies~(\ref{rcp-wA})
after some algebraic transformations
for every $r\in \mathbb{R}\setminus \{0\}$
(the equality~(\ref{rcp-wA}) we obtain by setting $r=8/729$),
(\ref{nr100}), which implies~(\ref{rcp-wB})
and at last~(\ref{rcp-w4}), which implies~(\ref{rcp-wC}).

In~\cite{Shevelev2007} many interesting and fundamental properties of RCP's are presented.
\medskip

The object of this paper is to prove the following fact

\begin{theorem}\label{rcp-th1}
All RCP's have the following form
\begin{multline}\label{rcp-w3}
x^3 - \frac{P(\gamma-1)}{(\gamma-1)\, (\gamma-2)}\, r^{1/3}\, x^2 -
\frac{P(2-\gamma)}{(1-\gamma)\, (2-\gamma)}\, r^{2/3}\, x  + r = {}\\
{}=
\Big( x - \frac{r^{1/3}}{2-\gamma} \Big)\,
\Big( x - (\gamma -1)\, r^{1/3}  \Big)\,
\Big( x - \frac{2-\gamma}{1-\gamma}\,r^{1/3} \Big),
\end{multline}
where $r\in \mathbb{R}\setminus \{0\}$,
$\gamma\in \mathbb{R}\setminus \{1,2\}$, and
\begin{equation}\label{rcp-w4}
P(\gamma):=
\gamma^3 - 3\, \gamma + 1 =
\Big( \gamma - 2\, \cos \frac{2\, \pi}{9} \Big)\,
\Big( \gamma - 2\, \cos \frac{4\, \pi}{9} \Big)\,
\Big( \gamma - 2\, \cos \frac{8\, \pi}{9} \Big).
\end{equation}
\end{theorem}

\begin{corollary}
From formulas~(\ref{wzorA}), (\ref{wzorB}) and~(\ref{wzorC})  for the sums of the real cube root of the roots
of polynomial (\ref{rcp-w3}), the following equalities can be generated
\begin{multline}\label{rcp-w5a}
\gamma^3 -9\, \big( \gamma - 1 \big)^2  +
3\, \big( \gamma^2 - 3\, \gamma + 3 \big)\,
\sqrt[3]{(\gamma-1)\,(\gamma-2)} ={}\\
{}=
\Big(
\sqrt[3]{1-\gamma} -
\sqrt[3]{(2-\gamma)\, (1-\gamma)^2} +
\sqrt[3]{(2-\gamma)^2}
\Big)^3,
\end{multline}
\begin{multline}\label{rcp-w6}
\gamma^3 - 9\, \gamma + 9 -
3\, \big( \gamma^2 - 3\, \gamma + 3 \big)\,
\sqrt[3]{(\gamma-1)\,(\gamma-2)} ={}\\
{}=
\Big(
\sqrt[3]{2-\gamma} -
\sqrt[3]{(1-\gamma)\, (2-\gamma)^2} -
\sqrt[3]{(1-\gamma)^2}
\Big)^3,
\end{multline}
which, after replacing $\gamma:=3-\gamma$, is equivalent to (\ref{rcp-w5a});
\begin{multline}\label{rcp-w7}
\Bigg(
\frac{1}{\sqrt[3]{(2-\gamma)\,(1-\gamma)}}
+ \sqrt[3]{(2-\gamma)\,(1-\gamma)}
- \sqrt[3]{\frac{1-\gamma}{(2-\gamma)^2}}
+ \sqrt[3]{\frac{2-\gamma}{(1-\gamma)^2}} +{}\\
{}
+ \sqrt[3]{\frac{(1-\gamma)^2}{2-\gamma}}
- \sqrt[3]{\frac{(2-\gamma)^2}{1-\gamma}}\,\,
\Bigg)^{\!\!3}
=
9 -
\frac{P(\gamma-1)\, P(2-\gamma)}{(\gamma-1)^2\, (2-\gamma)^2},
\end{multline}
i.e.,
\begin{equation}\label{rcp-w8}
\big( \gamma^2 - 3\, \gamma + 3 \big)^3 =
9\, (\gamma-1)^2\, (2-\gamma)^2 - P(\gamma-1)\, P(2-\gamma).
\end{equation}
\end{corollary}

The above relations essentially supplement the set of identities presented
in~\cite{Berndt2}.
Furthermore, (\ref{rcp-w5a})--(\ref{rcp-w8}) entail Ramanujan's equalities
(\ref{rcp-wA})--(\ref{rcp-wC}), as well as all the other expressions
of this type discussed in~\cite{Shevelev2007,WitulaSlota-A7s,Witula-R}.


In the second part of this paper we will discuss an important Shevelev
parameter $\frac{p\, q}{r}$ of RCP's having the form~(\ref{rcp-w1}).
We note, that from~(\ref{rcp-w10}) the following
Shevelev inequality follows:
\begin{equation}\label{wzor-gw}
\frac{p\, q}{r} \leq \frac{9}{4}.
\end{equation}
We remark that for every $a\in \mathbb{R}$,
$a\leq \frac{9}{4}$,
there exist at most six different sets of RCP's,
depending only on values~$r$
and having the same value of $\frac{p\, q}{r}$, equal to~$a$.
In the sequel, there exist only two sets of RCP's,
depending on $r\in \mathbb{R}$,
having the value $\frac{p\, q}{r}=2$
(see the descriptions~(\ref{rcp-wI}) and~(\ref{rcp-wII})).
However, there is only one family of RCP's,
depending on $r\in \mathbb{R}$
with $\frac{p\, q}{r}=\frac{9}{4}$
(see the descriptions~(\ref{rcp-w100})).
This fact is proven in Section~\ref{sec2},
but it independently results from~(\ref{rcp-w23}), (\ref{rcp-w3}), (\ref{rcp-w8})
and from the following identity
\begin{equation}\label{wzor-2gw}
\frac{p\, q}{r} =
9 -
\frac{\big( (\gamma-1)\, (\gamma-2) + 1 \big)^3}{\big( (\gamma-1)\, (\gamma-2) \big)^2}.
\end{equation}
From~(\ref{wzor-2gw}) we get
$$
\frac{p\, q}{r}=\frac{9}{4}
\ \ \Leftrightarrow\ \
t:= (\gamma-1)\, (\gamma-2) \in \Big\{ {-}\frac{1}{4}, 2 \Big\}
\ \ \Leftrightarrow\ \
\gamma \in \Big\{ 0, \frac{3}{2}, 3 \Big\},
$$
since we have
$$
\frac{d}{dt} \Big( 9 - \frac{(t+1)^3}{t^2} \Big) =
t\, (2-t)\, \frac{(t+1)^2}{t^4}.
$$
All three values $\gamma \in \big\{ 0, \frac{3}{2}, 3 \big\}$
generate the same  set of RCP's of the form~(\ref{rcp-w100}).


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\section{Proof of Theorem~\ref{rcp-th1}}\label{sec2}

Let us indicate
that from condition (\ref{rcp-w2}) the following equality follows
(see~\cite{Shevelev2007}):
\begin{equation}\label{rcp-w10}
\frac{9}{4} - \frac{p\, q}{r} = \Big( \frac{3}{2} + \frac{p}{r^{1/3}} \Big)^2.
\end{equation}
Let
$$
p:= \Big( \alpha - \frac{3}{2} \Big) \, r^{1/3}.
$$
By (\ref{rcp-w10}) we have
\begin{align*}
\frac{p\, q}{r} &= \frac{9}{4} - \alpha^2,\\
q &= {-} \Big( \alpha + \frac{3}{2} \Big)\, r^{2/3}.
\end{align*}
In other words, an RCP has the form
\begin{equation}\label{rcp-w11}
x^3 +  \Big( \alpha - \frac{3}{2} \Big) \, r^{1/3}\, x^2 -
\Big( \alpha + \frac{3}{2} \Big)\, r^{2/3}\, x + r
\end{equation}
for some $\alpha,r\in\mathbb{R}$.
If $\alpha=0$, the following decomposition holds
\begin{equation}\label{rcp-w100}
x^3 - \frac{3}{2} \, r^{1/3}\, x^2 - \frac{3}{2}\, r^{2/3}\, x + r =
\Big( x -\frac{1}{2}\, r^{1/3} \Big)\,
\big(x + r^{1/3} \big) \,
\big( x -2\, r^{1/3} \big).
\end{equation}
Accordingly, the roots $x_1,x_2,x_3$ of the polynomial (\ref{rcp-w11})
have the form ($r\neq 0$):
\begin{equation}\label{rcp-w12}
x_1 = \Big( \frac{1}{2} + \beta \Big)\, r^{1/3},
\quad
x_2 = \big( {-}1 + \gamma \big)\, r^{1/3},
\quad
x_3 = \big( 2 + \delta \big)\, r^{1/3}
\end{equation}
for certain $\beta,\gamma,\delta\in \mathbb{R}$.
Then from Vieta's formulae the following equations can be obtained
\begin{align}
&\alpha = {-} \big( \beta+\gamma+\delta\big),\label{rcp-w13}\\
&\Big( \frac{1}{2} + \beta \Big)\,\big( {-}1 + \gamma \big) +
\Big( \frac{1}{2} + \beta \Big)\,\big( 2 + \delta \big) +
\big( {-}1 + \gamma \big)\,\big( 2 + \delta \big) =
{-}\alpha - \frac{3}{2},\label{rcp-w14}\\
&\Big( \frac{1}{2} + \beta \Big)\,\big( {-}1 + \gamma \big)\,
\big( 2 + \delta\big) = 1.\label{rcp-w15}
\end{align}
From~(\ref{rcp-w13}) and~(\ref{rcp-w14}) we receive
\begin{equation}\label{rcp-w16}
\beta =
\frac{\frac{3}{2}\, (\delta-\gamma) -\delta\, \gamma}{\delta+\gamma},
\end{equation}
which, by~(\ref{rcp-w15}), implies
$$
\delta^2\, \big( \gamma^2 -3\, \gamma + 2 \big) +
\delta\, \big( 3\, \gamma^2 - 7\, \gamma + 3 \big) + 2\, \gamma^2 -3\, \gamma = 0.
$$
Hence, after some manipulations, we get
$$
\Delta_{\delta} = \big( \gamma^2 - 3\, \gamma + 3 \big)^{2},
$$
and next
\begin{equation}\label{rcp-w17}
\delta =\frac{\gamma}{1-\gamma}
\qquad\mbox{ or }\qquad
\delta=\frac{3-2\,\gamma}{\gamma-2}.
\end{equation}
If we choose $\delta=\frac{\gamma}{1-\gamma}$, then by (\ref{rcp-w16})
we have $\beta=\frac{\gamma}{2\, (2-\gamma)}$, and by (\ref{rcp-w12}) we obtain
\begin{equation}\label{rcp-w18}
\begin{array}{l}
\displaystyle
x_1=\frac{r^{1/3}}{2-\gamma},
\quad
x_2 = (\gamma-1)\, r^{1/3},
\quad
x_3 = \frac{2-\gamma}{1-\gamma}\, r^{1/3},\\[2ex]
\displaystyle
\alpha =
{-}\Big(
\frac{\gamma}{2\, (2-\gamma)}
+\gamma +
\frac{\gamma}{1-\gamma}
\Big) =
\frac{{-}2\,\gamma^3+9\,\gamma^2-9\,\gamma}{2\, (\gamma-1)\, (\gamma-2)}.
\end{array}
\end{equation}
Finally
\begin{multline}\label{rcp-w19}
x^3 + \frac{{-}\gamma^3+3\,\gamma^2-3}{(\gamma-1)\, (\gamma-2)}\, r^{1/3}\, x^2 +
\frac{\gamma^3-6\,\gamma^2+9\,\gamma-3}{(\gamma-1)\, (\gamma-2)}\, r^{2/3}\, x  + r = {}\\
{}=
\Big( x - \frac{r^{1/3}}{2-\gamma} \Big)\,
\Big( x - (\gamma -1)\, r^{1/3}  \Big)\,
\Big( x - \frac{2-\gamma}{1-\gamma}\,r^{1/3} \Big),
\end{multline}
which is compatible with~(\ref{rcp-w3}).

On the other hand, if we choose $\delta=\frac{3-2\,\gamma}{\gamma-2}$, then
$\beta=\frac{\gamma-3}{2\, (\gamma-1)}$,
and we obtain the same values of $x_1,x_2,x_3$ and $\alpha$ as in (\ref{rcp-w18}) above.
\hfill{$\Box$}



\begin{example}
Since
\begin{equation}\label{nr100}
\Big( x - 2\, \cos\frac{2\, \pi}{7}\Big)\,
\Big( x - 2\, \cos\frac{4\, \pi}{7}\Big)\,
\Big( x - 2\, \cos\frac{8\, \pi}{7}\Big)
=
x^3+x^2-2\, x-1
\end{equation}
is the RCP~\cite{WitulaSlota-A7s}, then, from (\ref{rcp-w3}) the following relations can be deduced
\begin{equation*}
\begin{array}{l}
\displaystyle
\gamma = 1 - 2\, \cos\frac{2\, \pi}{7},
\quad r={-}1,\\[2ex]
\displaystyle
\frac{P(\gamma-1)}{(1-\gamma)\,(2-\gamma)} =1
\quad
\mbox{and}
\quad
\frac{P(2-\gamma)}{(1-\gamma)\,(2-\gamma)} = 2,
\end{array}
\end{equation*}
which implies the equalities
$$
\frac{1}{\gamma-2} = 2\, \cos\frac{4\, \pi}{7},
\qquad
\frac{\gamma-2}{1-\gamma} = 2\, \cos\frac{8\, \pi}{7},
$$
\begin{equation}\label{rcp-w21}
\frac{\big( \cos\frac{2\, \pi}{7} + \cos\frac{2\, \pi}{9} \big)\,
\big( \cos\frac{2\, \pi}{7} + \cos\frac{4\, \pi}{9} \big)\,
\big( \cos\frac{2\, \pi}{7} + \cos\frac{8\, \pi}{9} \big)
}{\cos\frac{2\, \pi}{7}\, \big( 1+2\, \cos\frac{2\, \pi}{7}\big)}
= {-}\frac{1}{4},
\end{equation}
and the equivalent one
\begin{equation}\label{rcp-w22}
\frac{
\big( \frac{1}{2} + \cos\frac{2\, \pi}{7} - \cos\frac{2\, \pi}{9} \big)\,
\big( \frac{1}{2} + \cos\frac{2\, \pi}{7} - \cos\frac{4\, \pi}{9} \big)\,
\big( \frac{1}{2} + \cos\frac{2\, \pi}{7} - \cos\frac{8\, \pi}{9} \big)
}{\cos\frac{2\, \pi}{7}\, \big( 1+2\, \cos\frac{2\, \pi}{7}\big)}
= \frac{1}{2}.
\end{equation}

\end{example}

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\section{Values of $\displaystyle \frac{p\, q}{r}$ for RCP's}

By~(\ref{rcp-w3}) we obtain
\begin{equation}\label{rcp-w23}
\frac{p\, q}{r} =
\frac{P(\gamma-1)\,P(2-\gamma)}{(\gamma-1)^2\,(2-\gamma)^2}.
\end{equation}
The examples of RCP's, which are given in \cite{WitulaSlota-A7s,Witula-R}
(see also \cite{Shevelev2007}),
are produced by $\frac{p\, q}{r}$ equal only to $2$, ${-}40$, ${-}180$.

The following theorem holds.

\begin{theorem}
For every $a\leq \frac{9}{4}$
there exist at most six different sets of RCP's,
depending on $r\in \mathbb{R}$, having the same value
of $\frac{p\, q}{r}$, equal to $a$.
\end{theorem}

\begin{proof}
The proof of this theorem results easily from inequality~(\ref{wzor-gw})
and from relation~(\ref{rcp-w23}).
\end{proof}

We will present now a series of remarks, connected with the parameter
$a=\frac{p\,q}{r}$.

\begin{remark}\label{rcp-r2}
Let us consider the following equation
\begin{equation}\label{rcp-w24}
\frac{P(\gamma-1)\,P(2-\gamma)}{(\gamma-1)^2\,(2-\gamma)^2} = a
\qquad (a\in \mathbb{R}).
\end{equation}
This equation, by~(\ref{wzor-2gw}), after substitution $t:=(\gamma-1)\,(\gamma-2)$,
is equivalent to the following one
\begin{equation}\label{rcp-w25}
R(t):=t^3 + (a-6)\, t^2 + 3\, t +1 =0.
\end{equation}
If we replace $t$ in (\ref{rcp-w25}) by $\tau-\frac{a-6}{3}$,
then the canonical form of $R(t)$ can be generated
\begin{equation}\label{rcp-w30}
\tau^3 +
\big( 3 - \frac{1}{3}\, \big(a-6\big)^2\,\big)\, \tau +
\frac{2}{27}\, (a-6)^3 - (a-6) + 1.
\end{equation}
But the polynomial (\ref{rcp-w30}) has only one real root, if and only if
\begin{multline*}
\frac{1}{4}\, \big( \frac{2}{27}\, (a-6)^3 - (a-6) + 1 \big)^2 +
\frac{1}{27}\, \big( 3 - \frac{1}{3}\, \big(a-6\big)^2\, \big)^3 > 0
\ \Longleftrightarrow {}\\
{} \Longleftrightarrow \
\frac{4}{27}\, (a-6)^3 - \frac{1}{3}\, (a-6)^2 -2\, (a-6) + 5 > 0
\ \Longleftrightarrow \
(a-9)^2 \Big(a-\frac{9}{4}\Big) >0.
\end{multline*}
Since the case $a=\frac{9}{4}$ was discussed in (\ref{rcp-w100}),
the polynomial $R(t)$ has three real roots for every  $a\leq \frac{9}{4}$.
\end{remark}


\begin{remark}
If $\gamma_0\in\mathbb{C}$
is a~root of equation~(\ref{rcp-w24})
(for fixed $a\in \mathbb{C}$)
then also $\gamma=3-\gamma_0$ and
$\gamma= \frac{\gamma_0}{1-\gamma_0}$
are roots of this one. We note, that the last fact derives from the following identities
$$
(1-\gamma)^3\, P\Big( \frac{1}{\gamma-1}\Big) = P(2-\gamma)
$$
and
$$
(1-\gamma)^3\, P\Big( \frac{\gamma-2}{\gamma-1}\Big) = P(\gamma-1).
$$
Consequently, the roots of~(\ref{rcp-w24}) are also
$$
\gamma =  \frac{3-\gamma_0}{1-(3-\gamma_0)} = \frac{3-\gamma_0}{\gamma_0 - 2},\quad
\gamma =  3-\frac{\gamma_0}{1-\gamma_0} = \frac{3-4\,\gamma_0}{1-\gamma_0},\quad
\gamma =  3-\frac{3-\gamma_0}{\gamma_0-2} = \frac{4\,\gamma_0 - 9}{\gamma_0 - 2}.
$$
\end{remark}

\begin{remark}
Let us separately discuss equation~(\ref{rcp-w24}) for $a=2$.
After substitution $t=1-\tau$ in~(\ref{rcp-w25}), the following equation is derived
\begin{equation}\label{rcp-w26}
\tau^3 + \tau^2 - 2\, \tau - 1 =0,
\end{equation}
i.e.  (see \cite{WitulaSlota-A7s}):
\begin{equation}\label{rcp-w27}
\Big( \tau - 2\, \cos\frac{2\, \pi}{7}\Big)\,
\Big( \tau - 2\, \cos\frac{4\, \pi}{7}\Big)\,
\Big( \tau - 2\, \cos\frac{8\, \pi}{7}\Big)
=0.
\end{equation}
Hence, equation (\ref{rcp-w24}) for $a=2$
is equivalent to each of the following three equations
\begin{equation}\label{rcp-w26a}
(\gamma-1)\,(\gamma-2) =
1 - 2\, \cos\frac{2\, \pi}{7}
\ \Longleftrightarrow\
\gamma - 1 = {-}2\, \cos\frac{4\, \pi}{7}\ \ \vee \ \
\gamma - 2 =   2\, \cos\frac{4\, \pi}{7},
\end{equation}
or
\begin{equation}\label{rcp-w26b}
(\gamma-1)\,(\gamma-2) =
1 - 2\, \cos\frac{4\, \pi}{7}
\ \Longleftrightarrow\
\gamma - 1 = {-}2\, \cos\frac{8\, \pi}{7}\ \ \vee \ \
\gamma - 2 = 2\, \cos\frac{8\, \pi}{7},
\end{equation}
or
\begin{equation}\label{rcp-w26c}
(\gamma-1)\,(\gamma-2) =
1 - 2\, \cos\frac{8\, \pi}{7}
\ \Longleftrightarrow\
\gamma - 1 = {-}2\, \cos\frac{2\, \pi}{7}\ \ \vee \ \
\gamma - 2 =    2\, \cos\frac{2\, \pi}{7}.
\end{equation}
For the values
$$
\gamma \in \Big\{ 1 - 2\, \cos\frac{2^k\, \pi}{7}:\ k=1,2,3 \Big\},
$$
we obtain the same set of RCP's
\begin{equation}\label{rcp-wI}
x^3 + r^{1/3}\, x^2 - 2\, r^{2/3}\, x - r,
\qquad r\in \mathbb{R}.
\end{equation}

On the other hand, for values
$$
\gamma \in \Big\{ 2 + 2\, \cos\frac{2^k\, \pi}{7}:\ k=1,2,3 \Big\},
$$
we obtain the following set of RCP's
\begin{equation}\label{rcp-wII}
x^3 - 2\, r^{1/3}\, x^2 - r^{2/3}\, x + r,
\qquad r\in \mathbb{R}.
\end{equation}
We note, that  RCP of the form (see~\cite{Shevelev2007,WitulaSlota-A7s}):
\begin{multline*}
x^3 + 7\, x^2 - 98\, x - 343 = {}\\
{}=
\bigg(
x - 128\, \cos \frac{2\, \pi}{7}\,
\Big(
\sin\frac{2\, \pi}{7}\,
\sin\frac{8\, \pi}{7}\,
\Big)^{\!3}
\bigg)\,
\bigg(
x - 128\, \cos \frac{4\, \pi}{7}\,
\Big(
\sin\frac{2\, \pi}{7}\,
\sin\frac{4\, \pi}{7}\,
\Big)^{\!3}
\bigg)\,\cdot{}\\
{}\cdot
\bigg(
x - 128\, \cos \frac{8\, \pi}{7}\,
\Big(
\sin\frac{4\, \pi}{7}\,
\sin\frac{8\, \pi}{7}\,
\Big)^{\!3}
\bigg)
\end{multline*}
belongs to the set (\ref{rcp-wI}) of RCP's with $\frac{p\, q}{r}=2$
for $r=7^3$, because of the following remark.
\end{remark}

\begin{remark}
Suppose, that
$\alpha\in \{\frac{2\pi}{7},\frac{4\pi}{7},\frac{8\pi}{7}\}$.
Then, we have $\sin\alpha= \sin 8\alpha$, which implies
\begin{multline*}
14\, \cos\alpha=
7\, \frac{\sin 2\alpha}{\sin \alpha} =
\big( 8\, \sin\alpha\, \sin 2\alpha\, \sin 4\alpha \big)^2\,
\frac{\sin 2\alpha}{\sin \alpha} =
64\, \frac{\sin \alpha}{\sin 4\alpha}
\big(\sin 2\alpha\big)^3\,
\big(\sin 4\alpha\big)^3 ={}\\
{}=
64\, \frac{\sin 8\alpha}{\sin 4\alpha}
\big(\sin 2\alpha\big)^3\,
\big(\sin 4\alpha\big)^3 =
128\, \cos 4\alpha\,
\big( \sin 2\alpha\, \sin 4\alpha \big)^3.
\end{multline*}
\end{remark}








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\begin{thebibliography}{10}


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\bibitem{Witula-R}
R.~Witu{\l}a,
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\end{thebibliography}



\bigskip
\hrule
\bigskip

\noindent 2000 {\it Mathematics Subject Classification}: Primary
11C08; Secondary 11B83, 33B10.

\noindent \emph{Keywords:} Ramanujan cubic polynomial.

\bigskip
\hrule
\bigskip

\vspace*{+.1in} \noindent Received December 1 2009;  revised versions
received March 2 2010; May 4 2010.
Published in {\it Journal of Integer Sequences}, May 5 2010.

\bigskip
\hrule
\bigskip

\noindent Return to \htmladdnormallink{Journal of Integer Sequences
home page}{http://www.math.uwaterloo.ca/JIS/}. \vskip .1in


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