On a power series distribution with mean parameterization

The article examines the distribution of the power series of the function $w(y) = \left( 1 + \sqrt{1 - y} \right)^{-\frac{1}{2}}.$ The distribution of the considered function into a power series is obtained $\left(1 + \sqrt{1 - y}\right)^{-\frac{1}{2}} = \sum_{m=0}^{\infty} \frac{(4m)! \, 16^{-m}}{(2m)! \, (2m+1)! \, \sqrt{2}} \, y^m.$ The dispersion function is found $\nu(x) = x (2x + 1)(4x + 1), \; x > 0.$ A distribution with mean parameterization is constructed $\Pr(\xi = k) = \binom{4k + 1}{2k} \, 2^{-k} \, x^k \, (2k + 1)^{k + \frac{1}{2}} \, (4k + 1)^{-2k - \frac{3}{2}}, \; x > 0.$ It is proved that the raw moments $\alpha_m$, central moments $\mu_m$, cumulants $\chi_m, \; m = 1, 2, \ldots$ satisfy the following recurrence relations: $\alpha_{m+1} = x \alpha_m + \nu(x) \frac{d\alpha_m}{dx}, \; \alpha_0 = 1, \; \alpha_1 = x; \quad \mu_{m+1} = m \mu_{m-1} + \nu(x) \frac{d\mu_m}{dx}, \; \mu_0 = 1, \; \mu_1 = 0; \quad \chi_{m+1} = \nu(x) \frac{d\chi_m}{dx}, \; \chi_1 = x.$