The Legendre-type polynomials is the orthogonal polynomial system formed by applying the Gram-Schmidt orthogonalization process to the set \(\{x^ n : n \in \mathbb{N}_ 0 \}\) in the space \(L^ 2([-1,1]; \mu)\) where \(\mu\) is the measure generated by a function \(\widehat \mu\) of the form \(\widehat \mu (x) = - 1 - M\) if \(x \leq - 1\), \(=x\), if \(x \in (-1,1)\) and \(=1 + N\), if \(x \geq 1\), where \(M\), \(N\) are real numbers. It is known that these polynomials satisfy a Legendre-type differential equation. In this paper the case \(M>0\), \(N>0\) \(M \neq N\) is considered with the corresponding symmetric Lagrange differential form being defined by \(M[y](x): = - ((1 - x^ 2)^ 3y'''(x))''' + ((1 - x^ 2) (12 + \alpha (1 - x^ 2)) y''(x))'' - (\pi (x)y' (x))' + ky(x)\), \(x \in (-1,1)\), where \(k \geq 0\) and \(\alpha\), \(\pi (\cdot)\) depend on \(M\), \(N\). Let \(\Delta\) be the set of all functions \(f:(-1,1) \to \mathbb{C}\) with \(f \in AC^{(r)}_{\text{loc}} (-1,1)\) \((r=0,1,2,3,4,5)\) and \(f\), \(M[f] \in L^ 2(-1,1)\). Let \(D(T)\) be the domain of the self-adjoint operator \(T\) in \(L^ 2([-1,1], \mu)\) whose eigenvectors are the above Legendre-type polynomials. Then the authors prove the following properties: (i) if \(f \in \Delta\), then \(f' \in L^ 2 (-1,1)\), \(f \in AC [-1,1]\) and \(f'(x) = 0 (| \ln (1 - x^ 2) |)\) as \(x \to \pm 1\), (ii) if \(f \in D(T)\), then \(f'' \in L^ 2(-1,1)\) and \(f,f',f'' \in AC [-1,1]\), (iii) these results are the best possible in some sense.