By P. Fitzpatrick
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Extra info for Advanced Calculus [Int'l Student edn]
S/x s ds; where c 2 h˛; ˇi. g/: Fitouhi et al. Rq;C /. 103) is convergent. The authors of  introduced the following sufficient condition that guarantees the existence of the q-Mellin transform. 26. 0; 1/ and let u and v be real numbers such that u > v. x u / as x ! x v / as x ! s/ exists in the strip h u; vi. Proof. See [102, P. 310]. t u The following rules of the q-Mellin transform is proved in . Rule 1. a C s/: Rule 2 . f /. s/: Mq f t Rule 3. For s 2 h1 Ä Mq Rule 4. t / D C n X1 s/: .
1) when the coefficients are analytic or have pole of finite order at the origin. More recently, Trjitzinsky  has developed an analytic theory of existence of solutions of homogenous linear q-difference equations and for their properties. RC / are discussed in . Apart from this old history of q-difference equations, the subject received a considerable interest of many mathematicians and from many aspects, theoretical and practical. H. S. 1007/978-3-642-30898-7 2, © Springer-Verlag Berlin Heidelberg 2012 41 2 q-Difference Equations 42 of these aspects.
1. 32) See Fig. 1. q; 1 and k0 2 N0 . 1 2 1t 2 11 q 1Cq q; q < t Ä 2 t 1Cq q ; 2 Ä t Ä 1: 12 1 Preliminaries Since the value of the limit is dependent on the point x, then g is not q-regular at zero. 0; 1. 33) is not always valid. For example, let us define a function hW Œ0; 1 ! R by, cf. Fig. 4q ˆ ˆ ˆ 1 q ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ : 4 1 q n . 3 The q-Difference Operator 13 Fig. a < b/ are in the form a D xq n and b D xq m ; n, m 2 Z. 12. t; x/ Œ0; a such that for each 1/ are q-integrable on Œ0; a.
Advanced Calculus [Int'l Student edn] by P. Fitzpatrick