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Further results on the computation of incomplete gamma functions. (English) Zbl 0597.33002
Analytic theory of continued fractions II, Proc. Semin.-Workshop, Pitlochry and Aviemore/Scotl. 1985, Lect. Notes Math. 1199, 67-89 (1986).

[For the entire collection see Zbl 0583.00009.]

The complementary incomplete gamma function

${\Gamma }\left(a,z\right)={\int }_{z}^{\infty }{e}^{-t}{t}^{a-1}dt,\phantom{\rule{1.em}{0ex}}a\in ℂ,\phantom{\rule{1.em}{0ex}}|argz|<\pi ,$

can be expressed by means of a Stieltjes fraction $K\left({\alpha }_{n}{z}^{-1}/1\right),$ where ${\alpha }_{1}=1,$ ${\alpha }_{2n}=n-a$ and ${\alpha }_{2n+1}=n·$ We investigate convergence, truncation error bounds and speed of convergence for these and related continued fractions. Moreover, we suggest a modifying factor which is easy to compute and which accelerates the convergence of $K\left({\alpha }_{n}{z}^{-1}/1\right)$ considerably.

##### MSC:
 33B15 Gamma, beta and polygamma functions 30B70 Continued fractions (function-theoretic results) 40A15 Convergence and divergence of continued fractions 40D15 Convergence factors; summability factors 41A20 Approximation by rational functions 41A25 Rate of convergence, degree of approximation 65B99 Acceleration of convergence (numerical analysis) 65D20 Computation of special functions, construction of tables