Integrals of Legendre polynomials and solution of some partial differential equations (Q2706915)

The author solves the problem of determining all polynomials \(Q_n\) whose `inflection points all coincide with their interior roots'. This is formulated in the following explicit way: NEWLINE\[NEWLINE Q_n(x)=(1-x^2)q_{n-2}(x),\quad Q_n''(x)=-n(n-1)q_{n-2}(x),\tag{1}NEWLINE\]NEWLINE or equivalently NEWLINE\[NEWLINE (1-x^2)Q_n''(x)+n(n-1)Q_n(x)=0. \tag{2}NEWLINE\]NEWLINE It is obvious that the degree has to be at least \(2\). NEWLINENEWLINENEWLINEDifferentiating it becomes clear that \(Q_n'\) satisfies the differential equation of Legendre and thus NEWLINE\[NEWLINE Q_n(x)=-\int_x^1 L_{n-1}(t) dt NEWLINE\]NEWLINE is a solution with the nice normalizations \(Q_n(-1)=Q_n(1)=0\) (\(L_{n-1}\) is orthogonal to \(f\equiv 1\) as \(n\geq 2\)). NEWLINENEWLINENEWLINEAlthough the author does not explain why the constant \(-n(n-1)\) in the differential requirement in (1) has to be of that form, the reader can easily check that using a constant \(c\) and asking for a polynomial solution \(Q_n\) of degree \(n\) actually forces \(c=-n(n-1)\). NEWLINENEWLINENEWLINEUsing the connection between \(Q_n\) and the Legendre polynomials (or between \(q_{n-2}\) and Jacobi polynomials \(P^{(1,1)}\) or \(P^{(-1,-1)}\)), several properties of the \(Q_n\) are then derived. NEWLINENEWLINENEWLINEThey are orthogonal on \([-1,1]\) with weight function \(w(x)=1/(1-x^2)\) and generating function NEWLINE\[NEWLINE \sum_{n=2}^{\infty} Q_n(x)h^n=1-xh-\sqrt{1-2xh+h^2}. NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEFurthermore, some estimates are given and a theorem concerning convergence of the Fourier series in terms of the \(Q_n\) for functions \(f\) who are continuous on \([-1,1]\) with \(f(-1)=f(1)=0\), \(f'\) piecewise continuous and the curve \(y=f'(x)\) being rectifiable. As an appendix to the paper several graphs of functions compared with partial sums arising from this theorem are given. NEWLINENEWLINENEWLINEFinally five examples of partial differential equations are treated, which lead after separation to the differential equation for \(Q_n\).