Transcendence and algebraic independence of the values of some hypergeometric $E$-functions

I. I. Belogrivov

Sbornik. Mathematics, Tome 11 (1970) no. 3, pp. 355-376

Résumé

We investigate the arithmetical character of the values of the functions \begin{gather*} A_{m,s}(z)=\sum_{n=0}^\infty[\lambda_1+1,n]^{-m_1}[\lambda_2+1,n]^{-m_2}\cdots[\lambda_s+1,n]^{-m_s}\biggl(\frac zm\biggr)^{mn},\\ \lambda_1,\lambda_2,\dots,\lambda_s\ne-1,-2,\dots,\\ A_{m,s,\mu}(z)=1+\sum_{n=1}^\infty[\lambda_1+1,n]^{m_1}\cdots[\lambda_{i-1}+1,n]^{-m_{i-1}}\cdots[\lambda_i+1,n-1]^{-m_i}\cdots\\ \cdots[\lambda_s+1,n-1]^{-m_s}[\lambda_i+n]^{q^{i-1}-\mu}\biggl(\frac zm\biggr)^{mn}, \end{gather*} $\lambda_1,\lambda_2,\dots,\lambda_s\ne-1,-2,\dots$; $\mu=q_{i-1}+1$, $q_{i-1}+2,\dots,q_i$, $i=1,2,\dots,s,$ where $s\geqslant1$; $\lambda_1,\lambda_2,\dots,\lambda_s$ are rational numbers; $[\lambda,0]=1$, $[\lambda,n]=\lambda(\lambda+1)\cdots(\lambda+n-1)$, $n\geqslant1$, $m_1,m_2,\dots,m_s$ are arbitrary nonnegative rational integers, $m_0=0$, $m=m_1+m_2+\dots+m_s$, $m\geqslant1$; $q_i=m_1+m_2+\dots+m_i$, $i=1,2,\dots,s-1$, $q_0=0$, $q=q_s=m_1+m_2+\dots+m_{s-1}+t^s$, $t_s\geqslant m_s$, $t_s$ a natural number. The function $A_{m,s}(z)$ is the solution of a linear differential equation of order $m$ with polynomial coefficients. The system of functions $A_{m,s,\mu}(z)$, $\mu=1,2,\dots,q$, constitutes the solution of a system of $q$ linear differential equations whose coefficients are rational functions of $z$. By means of the general theorem of Shidlovskii on the transcendence and algebraic independence of the values of the $E$-functions we prove six theorems on the mutual transcendence of the values of the functions in each aggregate $A_{m,s}(z), A'_{m,s}(z),\dots,A^{(m-1)}_{m,s}(z)$ and $A_{m,s,\mu}(z)$, $\mu=1,2,\dots,q$, at arbitrary algebraic points $a\ne0$ for various rational values of the parameters $\lambda_1,\lambda_2,\dots,\lambda_s$, $s\geqslant1$, and arbitrary values $m_1,m_2,\dots,m_s$. Bibliography: 8 titles.