The calculation of the probability density of a strictly stable law at large $X$

The article is devoted to the problem of calculating the probability density of a strictly stable law at $x\to\infty$. To solve this problem, it was proposed to use the expansion of the probability density in a power series. A representation of the probability density in the form of a power series and an estimate for the remainder term was obtained. This power series is convergent in the case $0<\alpha<1$ and asymptotic at $x\to\infty$ in the case $1<\alpha<2$. The case $\alpha=1$ was considered separately. It was shown that in the case $\alpha=1$ the obtained power series was convergent for any $|x|>1$ at $N\to\infty$. It was also shown that in this case it was convergent to the density of $g(x,1,\theta)$. An estimate of the threshold coordinate $x_\varepsilon^N$, was obtained which determines the range of applicability of the resulting expansion of the probability density in a power series. It was shown that in the domain $|x|\geqslant x_\varepsilon^N$ this power series could be used to calculate the probability density.