Linear independence and transcendence of values of hypergeometric functions (Q446263)

Let NEWLINE\[NEWLINE\psi_j(z)= 1+ \sum^\infty_{\nu=1} z^\nu \prod^\nu_{x=1} {a_j(x)\over b_j(x)}\quad (j= 1,\dots, t)NEWLINE\]NEWLINE be generalized hypergeometric functions, where \(a_j(x)\), \(b_j(x)\) are polynomials with complex coefficients, \(m_j=\deg b_j(x)> \deg a_j(x)\), \(b_j(x)\neq 0\) \((x= 1,2,\dots)\). The author gives a necessary and sufficient condition for linear independence of functions \(1\), \(\psi^{(s)}_j(z)\) \((j= 1,\dots, t\); \(s= 0,\dots, m_j-1)\) over \(\mathbb C(z)\).NEWLINENEWLINE Furthermore, a necessary and sufficient condition for linear independence of the values \(1\), \(\psi^{(s)}_j(\omega_k)\) \((j= 1,\dots, t\); \(s= 0,\dots, m_j-1;\, k= 1,\dots, r)\) over \(\overline{\mathbb Q}\) is also given, where \(\omega_1,\dots, \omega_r\) be different nonzero algebraic numbers. Therefore, the numbers \(\psi^{(s)}_j(\omega_k)\) all are transcendental under certain conditions connected with \(\psi_j(z)\).