Consider an exponential family such that the variance function is given by the power of the mean function. This family is denoted by ED (~) if the variance function is given by #(2-~)/0-~), where # is the mean function. When 0 < a < 1, it is known that the transformation of ED (~) to normality is given by the power transformation X (1-2~)/(3-3a), and conversely, the power transformation characterizes ED (~). Our principal concern will be to show that this power transformation has an another merit, i.e., the density of the transformed variate has an absolutely convergent Gram-Charier expansion. In statistical theory, the study of the exponential family has a long history. The family has nice statistical properties, and many important distributions are members of the family (Barndorff-Nielsen (1978) gave an excellent review on this family). In particular, an exponential family whose variance function is given by a power function of the mean is very interesting. This family is called the exponential family with power variance function (PVF). All exponential families with PVF were found by Jorgensen (1987). He showed that these families are also exponential dispersion (ED) models. The exponential family with PVF is denoted by ED (a) when the variance function is given by #(2 -a)/(l-a), where # is the mean value. We may note that ED (a) coincides with Poisson, gamma, inverse Gaussian and normal distributions when c~ = -o% 0, 1/2 and 2 respectively. When a E (0, i) U (1, 2), the density of ED (a) is given by exponential tilting of the stable distribution. The density, however, is developed by an infinite series except the case c~ = i/2. It is well-known that the transformation of a chi-squared variate (or of a gamma variate) to normality is given by the cube root transformation. In general,