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We present a new fractional Taylor formula for singular functions whoseCaputo 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 expansioncoefficient of this type of singular functions, and further derive the optimal(weighted) $L^\infty$-estimates and $L^2$-estimates of the Legendre polynomialapproximations. This set of results can enrich the existing theory for $p$ and$hp$ methods for singular problems, and answer some open questions posed insome recent literature.

Optimal error estimates for Legendre expansions of singular functions with fractional derivatives of bounded variation