On algebraic relations for Ramanujan’s functions

Let P,Q, and R denote the Ramanujan Eisenstein series. We compute algebraic relations in terms of P(qi) (i=1,2,3,4), Q(qi) (i=1,2,3), and R(qi) (i=1,2,3). For complex algebraic numbers q with 0<|q|<1 we prove the algebraic independence over ℚ of any three-element subset of {P(q),P(q2),P(q3),P(q4)} and of any two-element subset of {Q(q),Q(q2),Q(q3)} and {R(q),R(q2),R(q3)}, respectively. For all the results we use some expressions of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$P(q^{i_{1}}), Q(q^{i_{2}}) $\end{document}, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R(q^{i_{3}}) $\end{document} in terms of theta constants. Computer-assisted computations of functional determinants and resultants are essential parts of our proofs.