TRANSCENDENCE OF p-ADIC VALUES OF GENERALIZED HYPERGEOMETRIC SERIES WITH TRANSCENDENTAL POLYADIC PARAMETERS

It is established that if \({{\alpha }_{1}}, \ldots ,{{\alpha }_{m}}\) are polyadic Liouville numbers, and the number \(\xi \) is a positive integer or Ξ is a polyadic Liouville number and if \({{\Psi }_{0}}\left( z \right) = \sum\limits_{n = 0}^\infty {{{{\left( {{{\alpha }_{1}}} \right)}}_{n}} \ldots {{{\left( {{{\alpha }_{m}}} \right)}}_{n}}{{z}^{n}}} ,\) \({{\Psi }_{1}}(z)\, = \,\sum\limits_{n = 0}^\infty {{{{\left( {{{\alpha }_{1}} + 1} \right)}}_{n}} \ldots {{{\left( {{{\alpha }_{m}} + 1} \right)}}_{n}}{{z}^{n}}} \), then there are infinitely many primes p such that the at least one of the p-adic integers \({{\Psi }_{0}}\left( \xi \right),\) \({{\Psi }_{1}}\left( \xi \right)\) (respectively, \({{\Psi }_{0}}\left. {\left( {\text{\Xi }} \right)} \right),\) \({{\Psi }_{1}}\left( {\text{\Xi }} \right)\) is transcendental.