The discrete Painlevé I equation (dPI) is an integrable difference equation which has the classical first Painlevé equation (PI) as a continuum limit. dPI is believed to be integrable because it is the discrete isomonodromy condition for an associated (single-valued) linear problem. In this paper, we derive higher-order difference equations as isomonodromy conditions that are associated to the ...

Remark : An alternative approach, advocated in [4], is to use the infinitesimal invariance criteria, which requires solving a linear system of first order partial differential equations based on the prolonged infinitesimal generators of the transformation group. In contrast, the moving frame method is completely algebraic, typically much simpler, and, moreover provides significantly more inform...

We consider the half-linear second-order difference equation Δ rkΦ Δxk ckΦ xk 1 0, Φ x : |x|p−2x, p > 1, where r, c are real-valued sequences. We associate with the above-mentioned equation a linear second-order difference equation and we show that oscillatory properties of the abovementioned one can be investigated using properties of this associated linear equation. Themain tool we use is a l...

Abstract Discrete systems are often described by difference equations. With some attributes of initial state type, a difference equation defines a system, i.e. a set of (input,output) pairs. For systems defined by linear difference equations (with constant or variable coefficients) on arbitrary sets A ⊂ Z sufficient conditions of their stability are formulated. It is also shown that these condi...

We study a scalar linear difference equation with several delays by transforming it to a system of Volterra equations without delays. The results obtained for this system are then used to establish oscillation criteria and asymptotic properties of solutions of the considered equation.

In this article, we present a general solution for linear divide-and-conquer recurrences of the form

We present an automatic method for mapping a system of linear recurrence equations onto systolic architectures. First, we show that systolic architectures can be derived from linear recurrence equations using the notion of directed recurrence equations. Next, we provide a procedure called CUBIZATION to achieve better performance while mapping such equations. The CUBIZATION procedure is complete...

We introduce a new family of sequences {tk(n)}n=−∞ for given positive integer k > 3. We call these new sequences as generalized Alcuin’s sequences because we get Alcuin’s sequence which has several interesting properties when k = 3. Also, {tk(n)}n=0 counts the number of partitions of n − k with parts being k, (k − 1), 2 (k − 1), 3 (k − 1), . . . , (k − 1) (k − 1). We find an explicit linear rec...