Recurrence relations for discrete hypergeometric functions

Abstract

Abstract. We present a general procedure for finding linear recurrence relations for the solutions of the second order difference equation of hypergeometric type. Applications to wave functions of certain discrete system are also given.

On the properties of special functions on the linear-type lattices

“…Therefore we will complete the work started in [16] where few recurrence relations where obtained. In fact we will prove, by using the q-analoge of the technique introduced in [4] for the discrete case (uniform lattice), that the solutions (not only the polynomial ones) of the difference equation on the q-linear lattice x(s) = c 1 q s + c 2 satisfy a very general recurrent-difference relation from where several well known relations (such as the three-term recurrence relation and the ladder-type relations) follow.…”

“…Proof. Since in [4] we have proved the case when x(s) = s (the uniform lattice) we will restrict here to the case of the q-linear lattice x(s) = c 1 q s + c 2 ). Moreover, we will give the proof for the case of functions of the form (11), the other case is completely equivalent.…”

“…In this section we will apply the previous results to the q-classical orthogonal polynomials [2,10,11] in order to show how the method works. We first notice that these polynomials are instances of the functions y ν on the lattice x(s) = q s defined in (4). In fact we have [13,16]…”

“…Our main aim in this paper is to present a constructive approach for generating recurrence relations and ladder-type operators for the difference analogues of special functions of hypergeometric type on the linear-type lattices. Here we will focus our attention on functions defined on the q-linear lattice (for the linear lattice x(s) = s see [4] and references therein, and for the continuous case see e.g. [18]).…”

Cooling ultracold bosons in optical lattices by spectral transform

“…where J 0 = In the vicinity of the lattice center i = (N − 1)/2 + j with j a small integer j ≪ N , the tunneling coefficients (28) are indeed approximately quadratic:…”