Discrete Harmonic Analysis Associated with Ultraspherical Expansions

In this paper we study discrete harmonic analysis associated with ultraspherical orthogonal functions. We establish weighted l ^p -boundedness properties of maximal operators and Littlewood-Paley g-functions defined by Poisson and heat semigroups generated by the difference operator ??f(n):=an?f(n+1)-2f(n)+an-1?f(n-1),n?N,?>0,where an?:={(2?+n)(n+1)/[(n+?)(n+1+?)]}1/2, n? N, and a-1?:=0. We also prove weighted l ^p -boundedness properties of transplantation operators associated with the system {fn?}n?N of ultraspherical functions, a family of eigenfunctions of ? _? . In order to show our results we previously establish a vector-valued local Calderón-Zygmund theorem in our discrete setting.