Factorization of the Green's operator in the Dirichlet problem for $(-1)^m(d/d t)^{2m}$

In this article we propose a method for solving the Dirichlet boundary value problem $(-1)^{m}{u}^{(2m)}=f$, ${u}^{(k)}(\pm 1)= 0$, $k=0, \dots ,m-1$, which is based on the factorization of the Green's operator, $\mathbf{G}_{2m}=(-1)^m\mathbf{J}^m \, \overset{\infty}{\underset{m}{\mathbf{Proj}}}\, \mathbf{J}^m:L_2({\mathbb I}) \to H^{m}_{0}({\mathbb I}) \cap H^{2m}({\mathbb I}), {\mathbb I}=[-1,1]$. Here $\mathbf{J}^m$ is a Volterra operator of $m$-fold integration аnd $\overset{\infty}{\underset{m}{\mathbf{Proj}}}$ — operator of orthogonal projection in $L_2({\mathbb I})$. The polynomials $\widetilde{\mathbb P}^{[2m]}_{2m+N} =\mathbf{J}^{m}\overset{\infty}{\underset{m} { \mathbf{Proj}}}\, {\mathbb P}^{[m]}_{m+N}$ form the basis of the Sobolev space $H^{m}_{0}({\mathbb I}) \cap H^{2m}({\mathbb I})$, where ${\mathbb P}^{[m]}_{m+N}(t)= \mathbf{J}^{m}P_N(t) = \dfrac{(t-1)^m}{m!C^m_{m+N}} P^{(m,-m)}_{N}(t)$, $P_N$ are Legendre polynomials and ${P}^{(m,-m)}_{N}$ — non–classical Jacobi polynomials. The study of polynomials ${\mathbb P}^ {[m]}_{m+N}$ occupies the most part of this work including the problem of expanding ${\mathbb P}^{[m]}_{m+N}$ in Legendre polynomials. The formula for calculating the connection coefficients is obtained.