Analytic theory of linear \(q\)-difference equations. (English) Zbl 0007.21103

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[1] The author began this work while he was a National Research Fellow at Harvard University.
[2] R. D. Carmichael,The general theory of linear q-difference equations. Amer. Journ. Math., Vol. 34 (1912), pp. 147–168. This paper will be referred to as (C). G. D. Birkhoff,The Generalized Riemann Problem for Linear Differential Equations and the Allied Problems for Linear Difference and q-Difference Equations, Proc. Am. Acad. Arts and Sciences, vol. 49 (1914), pp. 521–568. This paper will be referred to as (B). · doi:10.2307/2369887
[3] He supposes also that no root of (2) is equal to the product of another by an integral power ofq. This restriction, however, is not fundamental.
[4] C. R. Adams,On the Linear Ordinary q-Difference Equation, Annals of Math., vol. 30, No. 2, April, 1929, pp. 195–205. This paper will be referred to as (A). · doi:10.2307/1968274
[5] G. D. Birkhoff and W. J. Trjitzinsky,Analytic Theory og Singular Difference Equations, Acta mathematica, 60: 1–2, pp. 1–89, hereafter referred to as (BT)
[6] The case when |q|=1 has been satisfactorily treated in (C).
[7] It will be said thatg 1(x),g 2(x),...g n (x) are linearly independent if the determinant |(g j (q i x)){\(\beta\)} is not identically zero.
[8] Cf. (C).
[9] Cf. (BT; § 1; (6), (6a)).
[10] Cf. (BT; § 3; (9), (9a), (10)). Fork=1 we have the system (8).
[11] The latter condition is assumed only for convenience.
[12] Unless stated to the contrary, asymptotic relations are taken in the ordinary sense, that is, to infinitely many terms.
[13] Compare with certain analogous situations in (BT).
[14] Provided that the coefficients of the equation are of the nature, in the vicinity ofx= specified in § 1.
[15] This follows by the reasoning employed in (BT) for an analogous purpose.
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