On quasi-orthogonal polynomials of order r

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ABSTRACTThe sequences of quasi-orthogonal polynomials of order r are defined for non-quasi-definite moment functionals. Properties concerning the existence of such sequences, and relations between a quasi-orthogonal polynomial of order r and a set of orthogonal polynomials are proved. Two determinantal expressions of quasi-orthogonal polynomials of order r are given. At last it is proved that three consecutive polynomials of a sequence of quasi-orthogonal polynomials of order r satisfy a three term recurrence relation.