\documentclass[12pt,reqno]{article} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsthm} \usepackage{epsf} \usepackage{fullpage} \usepackage{float} \usepackage[usenames]{color} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.5in} \setlength{\textheight}{8.9in} \newcommand{\seqnum}[1]{\href{http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=#1}{\underline{#1}}} \begin{document} \begin{center} \vspace*{1cm} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \mbox{\null} \hspace*{4cm} \end{center} \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} \newenvironment{definition}{\begin{defin}\normalfont\quad}{\end{defin}} \newtheorem{examp}[theorem]{Example} \newenvironment{example}{\begin{examp}\normalfont\quad}{\end{examp}} \newtheorem{rema}[theorem]{Remark} \newenvironment{remark}{\begin{rema}\normalfont\quad}{\end{rema}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% Section %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% Section %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% Section %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% Section %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{10} \bibitem{Berndt2} B.~C.~Berndt, H.~H.~Chan and L.-Ch.~Zhang, Radicals and units in Ramanujan's work, \textit{Acta Arith.} \textbf{87} (1998), 145--158. \bibitem{Shevelev2007} V.~Shevelev, {On Ramanujan Cubic Polynomials}, preprint, \\ \href{http://arxiv.org/abs/0711.3420}{\tt http://arxiv.org/abs/0711.3420}, 2007. \bibitem{Sloane} N.~J.~A.~Sloane, {\it The On-Line Encyclopedia of Integer Sequences}, available electronically at \href{http://www.research.att.com/~njas/sequences/}{\tt http://www.research.att.com/\char'176njas/sequences/}, 2010. \bibitem{WitulaSlota-A7s} R.~Witu{\l}a and D.~S{\l}ota, {New Ramanujan-type formulas and quasi-Fibonacci numbers of order~7}, \textit{J.~Integer Seq.} \textbf{10} (2007), \htmladdnormallink{Article 07.5.6}{http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Slota/witula13.html}. \bibitem{Witula-R} R.~Witu{\l}a, {Ramanujan type trigonometric formulas: the general form for the argument $\frac{2\, \pi}{7}$}, \textit{J.~Integer Seq.} \textbf{12} (2009), \htmladdnormallink{Article 09.8.5}{http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Witula/witula17.html}. \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 \end{document}