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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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