  The backward differentiation formula (BDF) is a useful family of implicit
methods for the numerical integration of stiff differential equations. It is
well noticed that the stability and convergence of the $A$-stable BDF1 and BDF2
schemes for parabolic equations can be directly established by using the
standard discrete energy analysis. However, such classical analysis technique
seems not directly applicable to the BDF-$\mathbf{k}$ schemes for $3\leq
\mathbf{k}\leq 5$. To overcome the difficulty, a powerful analysis tool based
on the Nevanlinna-Odeh multiplier technique [Numer. Funct. Anal. Optim.,
3:377-423, 1981] was developed by Lubich et al. [IMA J. Numer. Anal.,
33:1365-1385, 2013]. In this work, by using the so-called discrete orthogonal
convolution kernels technique, we will recover the classical energy analysis so
that the stability and convergence of the BDF-$\mathbf{k}$ schemes for $3\leq
\mathbf{k}\leq 5$ can be established. One of the theoretical advantages of our
analysis technique is that less spacial regularity requirement is needed on the
initial data.
