  We present a new fractional Taylor formula for singular functions whose
Caputo fractional derivatives are of bounded variation. It bridges and
``interpolates" the usual Taylor formulas with two consecutive integer orders.
This enables us to obtain an analogous formula for the Legendre expansion
coefficient of this type of singular functions, and further derive the optimal
(weighted) $L^\infty$-estimates and $L^2$-estimates of the Legendre polynomial
approximations. This set of results can enrich the existing theory for $p$ and
$hp$ methods for singular problems, and answer some open questions posed in
some recent literature.
