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(α_z) (j = 1,…,m) together with its measure having the exponent μ = O (m), which sharpens the known exponent μ = O (m²) obtained by a certain refined version of Siegel's lemma (cf. [1]). Let p be a prime number. Then Theorem 1 further implies the linear independence of the p-adic numbers ∏_n=1^∞ (1+kpⁿ), (k = 0,1,…,p-1), over Q with its measure having the exponent μ < 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.