In this paper, we study a class of second order difference equations with three paremeters. With positive initial values, the asymptotic behavior of positive solutions are investigated.

The asymptotic behavior of the solutions to a neutral difference equation of the form

Consider the sequence of polesA = {α1, α2, . . .}, and suppose the rational functions φj with poles in A form an orthonormal system with respect to a Hermitian positive-definite inner product. Further, assume the φj satisfy a three-term recurrence relation. Let the rational function φ j\1 with poles in {α2, α3, . . .} represent the associated rational function of φj of order 1; i.e. the φ (1) j...

An (infinite) sequence is a function from the set IN = {0, 1, 2, . . .} of natural numbers to some set S. If a : IN → S is a sequence, we often denote a(n) by an. The values a0, a1, a2, . . . are called the elements or terms of the sequence. A recurrence relation is a way of defining a sequence. A few of the first elements of the sequence are given explicitly. Then, the recurrence relation give...

orthogonal zero interpolants (ozi) are polynomials which interpolate the “zero-function” at a finite number of pre-assigned nodes and satisfy orthogonality condition. ozi’s can be constructed by the 3-term recurrence relation. these interpolants are found useful in the solution of constrained approximation problems and in the structure of gauss-type quadrature rules. we present some theoretical...

Quantitative recurrence indicators are defined by measuring the first entrance time of the orbit of a point x in a decreasing sequence of neighborhoods of another point y. It is proved that these recurrence indicators are a.e. greater or equal to the local dimension at y, then these recurrence indicators can be used to have a numerical upper bound on the local dimension of an invariant measure.

In this paper, we define a new kind of the hypergeometric function, say, Hermite-Tricomi functions. An explicit representation, recurrence relations for the Hermite-Tricomi functions are given and differential equations satisfied them is presented. A new expansions of the exponential for a wide class of functions, whose properties, including recurrences, addition theorems, generating functions ...

We consider the problem of counting the occurrences of patterns of the form xy-z within flattened permutations of a given length. Using symmetric functions, we find recurrence relations satisfied by the distributions on Sn for the patterns 12-3, 21-3, 23-1 and 32-1, and develop a unified approach to obtain explicit formulas. By these recurrences, we are able to determine simple closed form expr...

It is well known that members of families of polynomials, that are orthogonal with respect to an inner product determined by a nonnegative measure on the real axis, satisfy a three-term recursion relation. Analogous recursion formulas are available for orthogonal Laurent polynomials with a pole at the origin. This paper investigates recursion relations for orthogonal rational functions with arb...