q-Difference Equations
This chapter includes proofs of the existence and uniqueness of the solutions of first order systems of q-difference equations in a neighborhood of a point a, \(a \geq 0\) . Then, as applications of the main results, we study linear q-difference equations
- PDF / 363,411 Bytes
- 31 Pages / 439.36 x 666.15 pts Page_size
- 17 Downloads / 243 Views
q-Difference Equations
Abstract This chapter includes proofs of the existence and uniqueness of the solutions of first order systems of q-difference equations in a neighborhood of a point a, a 0. Then, as applications of the main results, we study linear q-difference equations as well as the q-type Wronskian. These results are mainly based on (Mansour, q-Difference Equations, Master’s thesis, Faculty of Science, Cairo University, Giza, Egypt, 2001). This chapter also includes a section on the asymptotics of zeros of some q-functions.
2.1 Introduction The study of q-difference equations have been initiated by Jackson in [158]. The paper of Carmichael [73], to the best of our knowledge, is the first study of the problem of existence of solutions of linear q-difference equations using the technique established by Birkhoff in [56]. Mason [209] studied the existence of entire function solutions of homogeneous (f D 0) and nonhomogeneous linear q-difference equations of order n of the form n X
aj .x/y.q nj x/ D f .x/;
(2.1)
j D0
where the coefficients aj are taken to be entire functions. Then Adams in [11– 13] studied extensively the existence of solutions of (2.1) when the coefficients are analytic or have pole of finite order at the origin. More recently, Trjitzinsky [285] has developed an analytic theory of existence of solutions of homogenous linear q-difference equations and for their properties. The existence and uniqueness of solutions of first order linear q-difference equations in the space C Œ0; 1/ and Lp .RC / are discussed in [189]. 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. It is hard to encompass all such axes in a short notice, but to give the reader a reasonable idea we will mention some M.H. Annaby and Z.S. Mansour, q-Fractional Calculus and Equations, Lecture Notes in Mathematics 2056, DOI 10.1007/978-3-642-30898-7 2, © Springer-Verlag Berlin Heidelberg 2012
41
2 q-Difference Equations
42
of these aspects. In the papers [136, 188, 190, 192, 198, 257, 258, 298] one finds general study of the theory of q-difference equations. The spectral analysis has also attracted the attention of many authors, like e.g. [14, 30, 35, 50, 66, 67, 98, 138, 140– 142, 187, 194, 280, 281]. Since investigating q-difference equations using function theory tools explores more properties, this direction is also considered in many works, like [48, 184, 287, 299]. The solutions of difference equations and qanalogues of existing classical ones, especially orthogonal polynomials could be found in [38,39,43,58,74,86,107,108,144,146]. Applicable problems involving qdifference equations and q-analogues of mathematical physical problems are studied extensively, see e.g. [1, 23, 63, 86, 87, 97, 100, 165, 194, 274, 286] for dynamical system, q-oscillator, q-classical and quantum models; [51, 101, 195, 225] for qanalogues of mathematical physical problems including heat and wave equations; [6, 8, 20
Data Loading...
