## On linear independence of the values of generalized Heine series.(English)Zbl 1025.11023

Lower bounds in both the Archimedean and the $$p$$-adic case are given for linear forms in the values 1, $$f_1(\alpha),\dots,f_m(\alpha)$$ of functions satisfying functional equations of the form $\alpha_i z^sf_i(z)= p(z) f_i(qz)+q_i(z), \quad i=1,\dots,m.$ Here, $$s$$ is a positive integer, $$K$$ an algebraic number field, $$q,\alpha,\alpha_i\in K^*$$ and $$p(z)$$, $$q_i(z)\in K[z]$$. Interesting applications concern $$q$$-analogues of the Lindemann-Weierstraß theorem and Bessel functions. The proof follows the lines of the method of Siegel-Shidlovskii, which has to be adapted to this kind of functional equations.

### MSC:

 11J82 Measures of irrationality and of transcendence 11J91 Transcendence theory of other special functions 11J72 Irrationality; linear independence over a field 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$
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