-(q+1)$. But the latter coefficient is of the form \[ \left(\frac{b}{\sigma}w^{-1}\right)^{q+1} - w^{-\sigma} \left(\frac{b}{\sigma}w^{-1}\right)^{p} + \text{ terms of higher degree in $w$}; \] it has $w$-adic valuation $>-(q+1)$ if and only if $(\frac{b}{\sigma})^{q+1}=(\frac{b}{\sigma})^{p}$ if and only if\linebreak $b \in \sigma \mu_{\sigma}$. Therefore, \[ a \in \sigma \mu_{\sigma}. \] In conclusion, the list of determining polynomials\footnote{These ``determining polynomials'' are the ``eigenvalues'' of~\cite[Definition~3.26]{VdPS}.} at $\infty$ of $\hypergeoequa{p}{q}{\und \alpha}{\und \beta}$ is \[ \left(0\; \text{ repeated $p$ times},\; \zeta_{1} \sigma t^{-1/\sigma},\,\ldots,\,\zeta_{\sigma} \sigma t^{-1/\sigma}\right) \] with $\zeta_{j} = e^{\frac{2 \pi i j}{\sigma}}$. \subsubsection{A preliminary remark concerning Proposition~\ref{prop sln} in the \texorpdfstring{$\SL_{2}(C)$}{SL2(C)} case}\label{subsec:rem on SL2 case} The hypothesis of Proposition~\ref{prop sln} is: \begin{quote} Let $f \in K^{\times}$ satisfy a nonzero homogeneous linear differential equation $L(f)=0$ over $k$ of order $n\geq 1$ and assume that the differential Galois group $G_{L}$ over $k$ of the latter equation is simply connected. Let $g \in K$ satisfy a nonzero homogeneous linear differential equation over~$k$. Assume that $f \not \in k$ and that $f$ and $g$ are algebraically dependent over $k$. \end{quote} With the notations of the proof of Proposition~\ref{prop sln}, we have \[ \pi_{M}=u \circ \pi_{L}. \] We shall now focus on the case $G_{L}=\SL_{2}(\CC)$. Then, the classification of the representations of $\SL_{2}(\CC)$ shows that the representation $u: G_{L} \rightarrow G_{M}$ is conjugate to the direct sum of symmetric power representations \[ \Sym^{m_{i}} : G_{L}=\SL_{2}(\CC) \rightarrow \SL_{m_{i}+1}(\CC). \] {Letting $\cM$ and $\cL$ denote the differential modules associated with $Y'= A_MY$ and $Y' = A_LY$, the tannakian correspondence implies that $\cM$ is isomorphic to the direct sum of the $\Sym^{m_{i}}(\cL)$}. We will now use this to study the algebraic relations between generalized hypergeometric series. \subsubsection{Proof of Theorem~\ref{thm intro : application to alg rel hypergeo}} The differential Galois group over $\overline{\CC(x)}$ of $L=\hypergeoequa{0}{1}{-}{\beta}$ is $\SL_{2}(\CC)$; see~\cite[Theorem~3.6]{expsum}. As explained in Section~\ref{subsec:rem on SL2 case}, the {differential module $\cM$ associated to the} minimal nonzero differential equation $M$ with coefficients in $\overline{\CC(x)}$ annihilating $\pFq{p}{q}{\und \gamma}{\und \delta}{x}$ is isomorphic to a direct sum of symmetric powers of $\cL$, say \[ \cM \cong \oplus_{i=1}^{r} \Sym^{m_{i}}(\cL). \] Let us first assume that $q+1 > p$. We have seen in Section~\ref{sec:formal structure hypergeo} that $L$ is irregular at $\infty$: it has exactly one slope at $\infty$, namely $1/2$, and its list of determining polynomials at $\infty$ is $\pm 2 x^{1/2}$. Therefore, the list of the determining polynomials of $\Sym^{m_{i}}(\cL)$ at $\infty$ is \begin{equation}\label{det fact symm} -2m_{i}x^{1/2},2\left(-m_{i}+2\right)x^{1/2},2\left(-m_{i}+4\right)x^{1/2},\,\ldots,\,2\left(m_{i}-2\right)x^{1/2},2m_{i}x^{1/2}. \end{equation} So, the list of determining polynomials of $\cM$ is the concatenation of the lists~\eqref{det fact symm} for $i$ varying in $\{1,\,\ldots,\,r\}$. On the other hand, $M$ is a factor of $\hypergeoequa{p}{q}{\und \gamma}{\und \delta}$ so (see Section~\ref{subsec:rem on SL2 case}) the list of determining polynomials of $\cM$ is a sublist of \[ 0\;\text{ with multiplicity }\; p, \zeta_{1} \sigma x^{1/\sigma},\,\ldots,\,\zeta_{\sigma} \sigma x^{1/\sigma} \] with $\sigma=q-p+1$ and $\zeta_{j}=e^{\frac{2 \pi i j}{\sigma}}$. Comparing the two preceding descriptions of the determining factors of $\cM$, we find that $\sigma=2$ and that we have \begin{itemize} \item either all the $m_{i}$ are equal to $0$; \item or one of the $m_{i}$, say $m_{r}$, is equal to $1$ and the other $m_{i}$ are equal to $0$. \end{itemize} We claim that the first case cannot happen. Indeed, otherwise the entire function $\pFq{p}{q}{\und \gamma}{\und \delta}{x}$ would be algebraic and, hence, polynomial, which is false. Therefore, we have \[ \cM \cong \Sym^{0}(\cL) \oplus \cdots \oplus \Sym^{0}(\cL) \oplus \Sym^{1}(\cL). \] It follows that $\pFq{p}{q}{\und \gamma}{\und \delta}{x}$ is of the form $a+bg+cg'$ for some $a,b,c \in \overline{\CC(x)}$, $b$ or $c \neq 0$, and some nonzero solution $g$ of $L$. Since the differential Galois group of $L$ over $\overline{\CC(x)}$ is $\SL_{2}(\CC)$, the only possibility for $a+bg+cg'$ to be algebraically depend with $\pFq{0}{1}{\alpha}{\beta}{x}$ is that $g= d \cdot \pFq{0}{1}{-}{\beta}{x}$ for some $d \in \CC$ and $c=0$. Therefore, \[ \pFq{p}{q}{\und \gamma}{\und \delta}{x} \in \overline{\CC(x)} \pFq{0}{1}{-}{\beta}{x} + \overline{\CC(x)}. \] The fact that we can descend this linear relation to $\CC(x)$ follows from the fact that $\pFq{p}{q}{\und \gamma}{\und \delta}{x}$ and $\pFq{0}{1}{-}{\beta}{x}$ are entire functions and that $\pFq{0}{1}{-}{\beta}{x}$ and $1$ are linearly independent over $\overline{\CC(x)}$. Let us now consider the case $q+1 \leq p$. In that case, we have seen that $\hypergeoequa{p}{q}{\und \gamma}{\und \delta}$ is regular at $\infty$. Since $M$ is a factor of $\hypergeoequa{p}{q}{\und \gamma}{\und \delta}$, $M$ and hence $\cM$ are regular at $\infty$ as well. Since $\Sym^{m_{i}}(\cL)$ is irregular at $\infty$ if $m_{i}$ is nonzero, we infer that all the $m_{i}$ are equal to $0$. Therefore, $\pFq{p}{q}{\und \gamma}{\und \delta}{x}$ belongs to $\overline{\CC(x)}$. This excludes the case $q+12$ or $c=1$ if $m=2$. In either case, $y = \frac{1}{ma_m}(cx -a_{m-1} +d)\in E$ for some constant $d$, contradicting the fact that $y$ is transcendental over $E$. \end{proof} \begin{lemm}\label{lem:algindep} Let $k\subset K$ be differential fields with the same algebraically closed subfield of constants $C$. Assume that there exists $x \in k$ such that $x' = 1$. \begin{enumerate} \item\label{lem:mult1} If $f$ is an iterated integral over $k$, then $k\langle f\rangle$ is a Picard--Vessiot extension of $k$ with unipotent Galois group. \item\label{lemm7.5.2} If $f \notin k$, then there exist algebraically independent $y_1,\,\ldots \,y_{r-1}, y_r = f$ such that $ k\langle f\rangle = k(y_1,\,\ldots,\,y_r)$. In particular, $k(f)$ is algebraically closed in $ k\langle f\rangle$. \end{enumerate} \end{lemm} \begin{proof} If $f^{(n)} = h \in k$ then $f$ satisfies \begin{equation}\label{eq:int} y^{(n+1)} - \frac{h'}{h} y^{(n)} = 0. \end{equation} \eqref{lem:mult1} If $f \in k$, then clearly $k\langle f\rangle$ is a Picard--Vessiot extension of $k$. If $f \notin k$ then $f, 1, x,\,\ldots\,x^{n-1}$ are linearly independent over $C$ and so form a basis of the solution space of~\eqref{eq:int} and $k\langle f\rangle$ is a Picard--Vessiot extension of $k$. In addition, if we define $k_i = k_{i-1}(f^{(n-i)})$, we have a tower of differential fields $k = k_0 \subset k_1\subset\,\ldots\,\subset k_n = k\langle f\rangle$ where each $k_i$ is a Picard--Vessiot extension whose differential Galois group is either $\{0\}$ or $\Ga$. Therefore the differential Galois group of $k\langle f\rangle$ over $k$ is unipotent. \eqref{lemm7.5.2} Assume that the transcendence degree of $k\langle f\rangle$ over $k$ is $r$. By assumption $r \geq 1$. Using the construction of the $k_i$ we see that from the set $\{ f^{(n-1)},f^{(n-2)}, \ldots, f', f \}$ we may select a subset of elements \[ T= \left\{y_i = f^{(n-n_i)}\, \middle| \, n_1

1$ to conclude that $f_1 = A_{1,\,j} f_j + B_{1,\,j}$ where $A_{1,\,j} \in k$ and $B_{1,\,j} \in F_{1,\,j}$. Therefore $\partial_j(f_1) = A_{1,\,j} \in k$, that is, $f_1 \in k(f_2,\,\ldots,\,f_n)$ is linear in each of the $f_i, 2\leq i \leq n$ and so the conclusion is obtained. \end{proof} \appendix \section{The tannakian correspondence}\label{tannaka} We give an informal description of this correspondence, enough to understand its use in our proofs. For a formal description see~\cite{delignemilne,KatzAlgSolDiffEqua,katzcalculation}, \cite[Chapter~2 and Appendix B]{VdPS}. Let $k$ be a differential field of characteristic zero (not assumed to be algebraically closed) with algebraically closed field of constants $C$. Let \begin{align}\label{eq:app1} Y' =& AY \end{align} with $A \in \gl_n(k)$ and let $K$ be its Picard--Vessiot extension. If $\frkY \in \GL_n(K)$ is a fundamental solution matrix of~\eqref{eq:app1}, we refer to the ring $k[\frkY, \det(\frkY)^{-1}]$ generated by the entries of $\frkY$ and the inverse of its determinant as the Picard--Vessiot ring associated with~\eqref{eq:app1}. It is independent of our choice of fundamental solution matrix. One can associate to~\eqref{eq:app1} a differential module $M$, that is, a finite dimensional $k$-vector space $M$ endowed with a map $\partial : M\rightarrow M$ such that for all $f \in k$ and $m,n \in M$, $\partial (m+n) = \partial m + \partial n$ and $\partial(fm) = f'm + f \partial m$. This is done in the following way. Let $M = k^n$ and denote the standard basis by $\{e_i\}$. We define $\partial$ by setting $\partial(e_i):= -\sum_j a_{j,\,i} e_j$ where $A = (a_{i,\,j})$. Given any differential module together with a basis $\bfe =\{e_i\}$, one can reverse this process and produce a differential equation $Y'= A_{\bfe} Y$. If ${\bff} = B{\bfe}$ is another basis of $M$ with $B \in \GL_n(k)$ then $A_{\bfe}$ and $A_{\bff}$ are related by $A_{\bfe} = B^{-1}A_{\bff}B- B^{-1}B'$. In this case we say that the equations are \emph{gauge equivalent} or just \emph{equivalent}. Given differential modules $M_1,M_2$ one can define a differential module homomorphism as well as sub and quotient differential modules in the obvious way. One can also form direct sums, tensor products, and duals as in the following table: \vspace{4pt} %\newcommand*\arraystretch{1.5} \[ \begin{tabu}{ c | c | c } \hline \text{Construction} & \partial & \text{Equation} \\ \hline M_1 \oplus M_2 &\partial(m_1\oplus m_2) = \partial_1m_1 \oplus \partial_2m_2 & Y' = \begin{pmatrix} A_1 & 0 \\ 0 & A_2 \end{pmatrix}Y\\ \hline M_1 \otimes M_2 & \partial(m_1\otimes m_2) =& Y' = \\ & \partial_1m_1 \otimes m_2 + m_1 \otimes \partial_2m_2 & \left(A_1 \otimes I_{m_2} + I_{m_1}\otimes A_2\right)Y\\[0.15em] \hline M_1^* & \partial(f)(m) = f(\partial_1(m)) & Y' = -A^TY\\ \hline \end{tabu} \] \vspace{4pt} The collection of all differential modules forms a category closed under differential module homomorphism and the constructions of submodules, quotients, direct sums, tensor products, and duals, which we refer to as \emph{the constructions of linear algebra}. We denote by $\{\mkern-2.7mu\{M\}\mkern-2.7mu\}$ the smallest subcategory containing $M$ and closed under these five constructions. Given any $N \in \{\mkern-2.7mu\{M\}\mkern-2.7mu\}$ and any basis ${\bfe}$ of $N$, we can form the associated differential equation $Y' = A_{N,\,{\bfe}} Y$, $A_{N,\,{\bfe}}\in \gl_m(k)$. It is known (\cite[Chapter~2.4]{VdPS}) that this equation has a fundamental matrix $\frkY_{N,\,\bfe} \in \GL_m(K)$ and, a fortiori, in the Picard--Vessiot ring $k[\frkY, \det(\frkY)^{-1}]$ (\cite[Corollary~1.38]{VdPS}). The differential Galois group $G$ of $K$ over $k$ acts on the $C$-space $V$ spanned by the columns of $\frkY$ and so induces a representation of $G$. The space $V$ depends on our selection of bases of $N$ but it can be shown that the representation of $G$ is independent, up to $G$-isomorphism, of this choice of bases (see~\cite[Chapter~2.4]{VdPS} for a basis free way of defining this representation). We will denote this representation of $G$ by $S(N)$. We denote by $\rep_G$ the category of representations of $G$, that is the category of finite dimensional $C$-vector spaces on which $G$ acts. In this category one also has a notion of the five constructions above. The tannakian correspondence~\cite[Theorem~2.22]{VdPS} says that \begin{quotation} The map $N \mapsto S(N)$ is a bijective map between elements of $\{\mkern-2.7mu\{M\}\mkern-2.7mu\}$ {(modulo isomorphism of differential modules)} and representations of $G$ {(modulo conjugacy of representations)}. This map is compatible with homomorphisms and the five constructions above. \end{quotation} One consequence of this is: if $k$ is algebraically closed, then there is an isomorphism $\varphi$ of the Picard--Vessiot ring $R = k[\frkY, \det(\frkY)^{-1}]$ and the coordinate ring $k[G\otimes k]$ such that for $\sigma \in G$ and $z \in R$ we have \[ \varphi(\sigma(z)) = \rho^*_\sigma(\varphi(z)) \] where $\rho^*_\sigma: k[G\otimes k] \rightarrow k[G\otimes k]$ is the isomorphism which, when applied to a regular function $F \in k[G\otimes k]$, yields the regular function $\rho^*_\sigma(F): G\otimes k \rightarrow G\otimes k$ given by $\rho^*_\sigma(F)(g) = F(g\cdot \sigma)$. \begin{exam}\label{ex:appendix} Consider $Y' = AY$ with $A \in \gl_n(k)$ and let $\tilde{A} = A -(\tr(A)/n)I_n$. Since $\tr(\tilde{A}) = 0$, the differential Galois group of the equation $Y' = \tilde{A}Y$ is unimodular (\cite[Exercise~1.35.5]{VdPS}) and this construction is often used to reduce questions about general linear differential equations to ones that have unimodular Galois groups. We now form the $2n\times 2n$ differential system \begin{align} Y' =& \diag\left((\tr(A)/n)I_n, \tilde{A}\right) Y \end{align} and let $\tilde{K}$ be its Picard-Vessiot extension with differential Galois group $H$. Let $K$ be the Picard--Vessiot extension for $Y'=AY$ and let $G$ be its differential Galois group. If $\frkY$ is the fundamental solution matrix of $Y'=AY$, then $\frkY = \sqrt[n]{\det \frkY}$ is a solution of {$y'= (\tr{A}/n)y$}. Note that $\frkY$ need not lie in $K$ but $\frkY^n \in K$. A computation shows that if $\tilde{\frkY}$ is a fundamental solution matrix of $y' = \tilde{A}y$, then $\frkY I_n\cdot \tilde{\frkY}$ is a fundamental solution matrix of $Y' = AY$. Therefore $K \subset \tilde{K}$. The differential Galois theory implies that restricting $H$ to $K$ gives a short exact sequence \begin{align*} 0\rightarrow H_0 \rightarrow H \xrightarrow[]{\rho} G \rightarrow 0 \end{align*} where $H_0$ is a finite subgroup of $\GL_n(C)$ (since $\tilde{K}$ is a simple radical extension of $K$) and the map $\rho$ is given by \begin{align*} \rho: \diag (t,h) \rightarrow th. \end{align*} This defines a homomorphism of $H \subset \GL_{2n}(C)$ to $\GL_n(C)$ and so defines a representation of $H$. This example appears in the proof of Proposition~\ref{prop: case GL}. \end{exam} \subsection*{Acknowledgements} We would like to thank the anonymous referees for their useful suggestions and comments. \vspace{50pt} \bibliography{roques} \end{document}