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A \(q\)-analogue of the Wronskian and a second solution of the Hahn-Exton \(q\)-Bessel difference equation. (English) Zbl 0822.33009

First the authors define a natural \(q\)-analogue of the Wronskian for two solutions of a general second order linear \(q\)-difference equation. They show that this \(q\)-Wronskian satisfies a first order \(q\)-difference equation, which is a \(q\)-analogue of Abel’s theorem.
In the second part of the paper the \(q\)-Wronskian is used to obtain a second solution to the second order \(q\)-difference equation satisfied by the Hahn-Exton \(q\)-Bessel function, which is a \(q\)-analogue of the Bessel function of the first kind \(J_ \nu (x)\). Moreover, it is shown that this second solution is a \(q\)-analogue of the Bessel function of the second kind \(Y_ \nu (x)\).
Reviewer: R.Koekoek (Delft)

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
33D70 Other basic hypergeometric functions and integrals in several variables
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