A survey on the isometry of certain orthogonal polynomial systems in martingale spaces Edmundo J. Huertas 1 and Nuria Torrado 2 1 Centre for Mathematics, University of Coimbra (CMUC), Largo D. Dinis, Apartado 3008, EC Santa Cruz, 3001-501 Coimbra, Portugal (E-mail: ehuertasce@mat.uc.pt, ehuertasce@gmail.com) 2 Centre for Mathematics, University of Coimbra (CMUC), Largo D. Dinis, Apartado 3008, EC Santa Cruz, 3001-501 Coimbra, Portugal (E-mail: nuria.torrado@mat.uc.pt, nuria.torrado@gmail.com) Abstract. In this paper we survey how an inner product derived from an Uvarov transformation of the Laguerre weight function is used in the orthogonalization proce- dure of a sequence of martingales related to a L´ evy process. The orthogonalization is done by isometry and it is based in previous works of Nualart and Schoutens (see [18] and [19]), where the resulting set of pairwise strongly orthogonal martingales involved are used as integrators in the so-called chaotic representation property. Finally, we give an idea of how to generalize the above works. Keywords: Orthogonal polynomials; Laguerre-type polynomials; Krall-Laguerre polynomials; Inner products; L´ evy processes; Stochastic processes. 1 Introduction The Laguerre orthogonal polynomials are defined as the polynomials orthogonal with respect to the Gamma distribution. Therefore, they are orthogonal with respect to the inner product in the linear space P of polynomials with real coefficients (see [2]) 〈p, q〉 α =  ∞ 0 pqx α e −x dx, α> −1, p,q ∈ P. (1) From now on, {  L α n (x)} n≥0 stands for the sequence of monic Laguerre polyno- mials orthogonal with respect to (1). From the above inner product, let us introduce the modified inner product 〈p, q〉 =  ∞ 0 pqx α e −x dx + σ 2 p(c)q(c), α> −1, p,q ∈ P (2) where σ 2 ∈ R + , and c ∈ (−∞, 0]. Notice that 〈p, q〉 = 〈p, q〉 α + σ 2 p(c)q(c), so therefore (2) can be interpreted as a modification (or perturbation) of the 3 rd SMTDA Conference Proceedings, 11-14 June 2014, Lisbon Portugal C. H. Skiadas (Ed) c  2014 ISAST