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...

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...

The problem of summing a set of mutual recurrence relations with constant coefficients is investigated. A method is presented for summing an order d system of the form A(n) = ∑d i=1 MiA(n − i) + G(n), where A,G : N → K and M1, . . . ,Md ∈ Mm(K) for some field K and natural number m. The procedure expresses the sum ∑n i=0 A(i) in terms of A(n), . . . , A(n− d), initial conditions and sums of the...

We obtain relations among boundary states in bosonic minimal open string theory using the boundary ground ring. We also obtain a difference equation that boundary correlators must satisfy. email: [email protected], Address after Sep 11, 2005: School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA email: [email protected]

We use the theory of vertex operator algebras and intertwining operators to obtain systems of q-difference equations satisfied by the graded dimensions of the principal subspaces of certain level k standard modules for ŝl(3). As a consequence we establish new formulas for the graded dimensions of the principal subspaces corresponding to the highest weights iΛ1 + (k − i)Λ2, where 1 ≤ i ≤ k and Λ...

and T (r, f), which is only true for finite order meromorphic functions. We have also obtained the proximity function and pointwise estimates of f(z+η)/f(z) which is a discrete version of the classical logarithmic derivative estimates of f(z). We apply these results to give new growth estimates of meromorphic solutions to higher order linear difference equations. This also allows us to solve an...