q-Karamata functions and second order q-difference equations

q-Karamata functions and second order q-difference equations

In this paper we introduce and study q-rapidly varying functions on the lattice q0 := {qk : k ∈ N0}, q > 1, which naturally extend the recently established concept of q-regularly varying functions. These types of functions together form the class of the so-called q-Karamata functions. The theory of q-Karamata functions is then applied to half-linear q-difference equations to get information abo...

Second Order Linear Difference Equations and Karamata Sequences

We establish necessary and sufficient conditions for all positive solutions of a linear second order difference equation to be Karamata sequences, i.e., slowly varying or regularly varying or rapidly varying. Moreover, we discuss relations with the standard classification of nonoscillatory solutions and with the notion of recessive solutions. Our results lead to a complete characterization of p...

Variational Iteration Method for q-Difference Equations of Second Order

q-Hypergeometric solutions of q-difference equations

We present algorithm qHyper for finding all solutions y(x) of a linear homogeneous q-difference equation, such that y(qx) = r(x)y(x) where r(x) is a rational function of x. Applications include construction of basic hypergeometric series solutions, and definite q-hypergeometric summation in closed form. ∗The research described in this publication was made possible in part by Grant J12100 from t...

Analytic q-difference equations

A complex number q with 0 < |q| < 1 is fixed. By an analytic q-difference equation we mean an equation which can be represented by a matrix equation Y (z) = A(z)Y (qz) where A(z) is an invertible n× n-matrix with coefficients in the field K = C({z}) of the convergent Laurent series and where Y (z) is a vector of size n. The aim of this paper is to give an overview of our present knowledge of th...