On a power series distribution with mean parameterization

The article examines the distribution of the power series of the function $w(y)={(1+\sqrt{1-y})}^{-\frac{1}{2}}\mathrm{.}$ The distribution of the considered function into a power series is obtained ${(1+\sqrt{1-y})}^{-\frac{1}{2}}={\sum }_{m=0}^{\mathrm{\infty }}\frac{(4m)!{16}^{-m}}{(2m)!(2m+1)!\sqrt{2}}{y}^m\mathrm{.}$ The dispersion function is found $\nu (x)=x(2x+1)(4x+1),x>0.$ A distribution with mean parameterization is constructed $\mathrm{Pr}(\xi =k)=\left(\genfrac{}{}{0px}{}{4k+1}{2k}\right)2^{-k}{x}^k(2k+1{)}^{k+\frac{1}{2}}(4k+1{)}^{-2k-\frac{3}{2}},x>0.$ It is proved that the raw moments ${\alpha }_m$, central moments ${\mu }_m$, cumulants ${\chi }_m,m=1,2,\dots$ satisfy the following recurrence relations: ${\alpha }_{m+1}=x{\alpha }_m+\nu (x)\frac{d{\alpha }_m}{dx},{\alpha }_0=1,{\alpha }_1=x;{\mu }_{m+1}=m{\mu }_{m-1}+\nu (x)\frac{d{\mu }_m}{dx},{\mu }_0=1,{\mu }_1=0;{\chi }_{m+1}=\nu (x)\frac{d{\chi }_m}{dx},{\chi }_1=x\mathrm{.}$