  Let $B^{a,b}$ be a weighted fractional Brownian motion with indices $a,b$
satisfying $a>-1,-1<b<0,|b|<1+a$. In this paper, motivated by the asymptotic
property $$ E[(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2]
=O(\varepsilon^{1+b})\not\sim
\varepsilon^{1+a+b}=E[(B^{a,b}_{\varepsilon})^2]\qquad (\varepsilon\to 0) $$
for all $s>0$, we consider the generalized quadratic covariation
$\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}$ defined by $$
\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t=\lim_{\varepsilon\downarrow
0}\frac{1+a+b}{\varepsilon^{1+b}}\int_\varepsilon^{t+\varepsilon}
\left\{f(B^{a,b}_{s+\varepsilon})
-f(B^{a,b}_s)\right\}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^{b}ds, $$ provided
the limit exists uniformly in probability. We construct a Banach space
${\mathscr H}$ of measurable functions such that the generalized quadratic
covariation exists in $L^2(\Omega)$ and the generalized Bouleau-Yor identity $$
[f(B^{a,b}),B^{a,b}]^{(a,b)}_t=-\frac1{(1+b){\mathbb B}(a+1,b+1)} \int_{\mathbb
R}f(x){\mathscr L}^{a,b}(dx,t) $$ holds for all $f\in {\mathscr H}$, where
${\mathscr L}^{a,b}(x,t)=\int_0^t\delta(B^{a,b}_s-x)ds^{1+a+b}$ is the weighted
local time of $B^{a,b}$ and ${\mathbb B}(\cdot,\cdot)$ is the Beta function.
