On $q$ -analoques of divergent and exponential series

Abstract

We shall consider linear independence measures for the values of the functions $D_a(z)$ and $E_a(z)$ given below, which can be regarded as $q$-analogues of Euler's divergent series and the usual exponential series. For the $q$-exponential function $E_q(z)$, our main result (Theorem 1) asserts the linear independence (over any number field) of the values 1 and $E_q(\alpha_j)(j = 1,...,m)$ together with its measure having the exponent $\mu = O(m)$, which sharpens the known exponent $\mu = O(m^2)$ obtained by a certain refined version of Siegel's lemma (cf. [$\mathbf{1}$]). Let p be a prime number. Then Theorem 1 further implies the linear independence of the $p$- adic numbers $\prod_{n=0}^\infty(1 + kp^n), (k = 0,1,...,p-1)$, over $\boldsymbol{Q}$ with its measure having the exponent $\mu < 2p$. Our proof is based on a modification of Maier's method which allows to construct explicit Padé type approximations (of the second kind) for certain $q$-hypergeometric series.