Q difference equations pdf

Pdf in this paper we investigate the existence of positive solutions of the q difference equation d2qut at f ut with some boundary. In general kth order difference equations going to have an as a linear combination of the kth steps back, and its characteristic equation would be obtained by choosing an as lambda to the n. It is equivalent to katzs conjectural description of the generic galois group. That is, we have looked mainly at sequences for which we could write the nth term as a n fn for some known function f. The general theory of linear qdifference equations jstor. Linear qdifference equations request pdf researchgate. In this paper we prove an analogous statement for arithmetic q. Grothendiecks conjecture on pcurvatures predicts that an arithmetic differential equation has a full set of algebraic solutions if and only if its reduction in positive characteristic has a full set of rational solutions for almost all finite places. Di erence equations relate to di erential equations as discrete mathematics relates to continuous mathematics. Pdf floquet theory for qdifference equations martin. The present paper is devoted to an investigation of the existence and prop erties of solutions of linear qdifference equations.

In this section we will consider the simplest cases. Inaddition,wealso discusssomepropertiesof q polynomials h. Jones function of a knot is a solution of a qdifference equation, as was proven by ttq. Developing a series solution method of qdifference equations. Request pdf linear qdifference equations we prove that a linear q difference equation of order n has a fundamental set of nlinearly independent solutions. And if we simplify we get lambda to the k minus beta 1 lambda k minus 1 and so forth. This equation is called the characteristic equation.

View more articles from american journal of mathematics. Pdf positive solutions of qdifference equation researchgate. Classi cation of di erence equations as with di erential equations, one can refer to the order of a di erence equation and note whether it is linear or nonlinear and whether it is homogeneous or inhomogeneous. In section 4, we solve a family of elementary linear qdifference equations. This chapter includes proofs of the existence and uniqueness of the solutions of first order systems of qdifference equations in a neighborhood of a point a, \a \geq 0\. Anyone who has made a study of di erential equations will know that even supposedly elementary examples can be hard to solve. These solutions allow us, in section 5, to give a formula for the mould w. Elaydi and others published an introduction to difference equation find, read and cite all the research you need on researchgate. Developing a series solution method of q difference equations. Existence results for nonlinear secondorder qdifference equations. We present algorithm qhyper for finding all solutions yx of a linear homogeneous qdifference equation, such that yqxrxyx where rx is a. Pdf in this paper we investigate the existence of positive solutions of the qdifference equation d2qut at f ut with some boundary.