We investigate algebraic and analytic properties of sequences of polynomialsorthogonal with respect to the Sobolev type inner product(f.g)= ∫▒〖f(x)g(x)dμ(x)+∑_(k=1)^K▒∑_(i=0)^(N_k)▒M_█(k.i@) 〗 f^((i) ) (b_k ) g^i (b_k )where μ is a finite positive Borel measure belonging to the Nevai class, the masspoints bk are located outside the support of μ, and Mk,i are complex numbers suchthat Mk,Nk = 0. First, we study the existence as well as recurrence relations for such polynomials. When the values Mk,i are nonnegative real numbers, we can deduce thecoefficients of the recurrence relation in terms of the connection coefficients for thesequences of polynomials orthogonal with respect to the Sobolev type inner productand those orthogonal with respect to the measure μ. The matrix of a symmetric multiplicationoperator in terms of the above sequence of Sobolev type orthogonal polynomialsis obtained from the Jacobi matrix associated with the measure μ. Finally, wefocus our attention on some outer relative asymptotics of such polynomials, which arededuced by using the above connection formulas
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