  Previous studies have found that the width of gamma-ray burst (GRB) pulse is
energy dependent and that it decreases as a power-law function with increasing
photon energy. In this work we have investigated the relation between the
energy dependence of pulse and the so-called Band spectrum by using a sample
including 51 well-separated fast rise and exponential decay long-duration GRB
pulses observed by BATSE (Burst and Transient Source Experiment on the Compton
Gamma Ray Observatory). We first decompose these pulses into rise, and decay
phases and find the rise widths, and the decay widths also behavior as a
power-law function with photon energy. Then we investigate statistically the
relations between the three power-law indices of the rise, decay and total
width of pulse (denoted as $\delta_r$, $\delta_d$ and $\delta_w$, respectively)
and the three Band spectral parameters, high-energy index ($\alpha$),
low-energy index ($\beta$) and peak energy ($E_p$). It is found that
(1)$\alpha$ is strongly correlated with $\delta_w$ and $\delta_d$ but seems
uncorrelated with $\delta_r$; (2)$\beta$ is weakly correlated with the three
power-law indices and (3)$E_p$ does not show evident correlations with the
three power-law indices. We further investigate the origin of $\delta_d-\alpha$
and $\delta_w-\alpha$. We show that the curvature effect and the intrinsic Band
spectrum could naturally lead to the energy dependence of GRB pulse width and
also the $\delta_d-\alpha$ and $\delta_w-\alpha$ correlations. Our results
would hold so long as the shell emitting gamma rays has a curve surface and the
intrinsic spectrum is a Band spectrum or broken power law. The strong
$\delta_d-\alpha$ correlation and inapparent correlations between $\delta_r$
and three Band spectral parameters also suggest that the rise and decay phases
of GRB pulses have different origins.
