Using elliptic modular functions, Kronecker proved a number of recurrence relations for suitable class numbers of positive binary quadratic forms. For instance if F (N) denotes the number of uneven classes of positive binary quadratic forms with determinant −N, then

and Applied Analysis 3

We present an extension of Sharkovsky’s Theorem and its converse to periodic difference equations. In addition, we provide a simple method for constructing a p-periodic difference equation having an r-periodic geometric cycle with or without stability properties.

*Correspondence: [email protected] 2Department of Mathematics, Yanbian University, Yanji, 133002, P.R. China Full list of author information is available at the end of the article Abstract In this paper, the concepts of a right fractional sum and right fractional difference operators are introduced. Some basic properties of a right fractional sum and right fractional difference operators are pr...