 Given the starting point by formula_86, the orthogonality of formula_76 can be shown by induction.For formula_88 one has

Now if formula_90 are orthogonal, then also formula_91, because in

all scalar products vanish except for the first one and the one where formula_93 meets the same orthogonal polynomial.Therefore,

However, if the scalar product satisfies formula_95 (which is the case for Gaussian quadrature), the recurrence relation reduces to a three-term recurrence relation