On the convergence behavior of continued fractions with real elements

Author:
Walter Gautschi

Journal:
Math. Comp. **40** (1983), 337-342

MSC:
Primary 40A15; Secondary 33A15

MathSciNet review:
679450

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Abstract: We define the notion of transient (geometric) convergence rate for infinite series and continued fractions. For a class of continued fractions with real elements we prove a monotonicity property for such convergence rates which helps explain the effectiveness of certain continued fractions known to converge "only" sublinearly. This is illustrated in the case of Legendre's continued fraction for the incomplete gamma function.

**[1]**Walter Gautschi,*A computational procedure for incomplete gamma functions*, Rend. Sem. Mat. Univ. Politec. Torino**37**(1979), no. 1, 1–9 (Italian). MR**547763****[2]**W. Gautschi, "Algorithm 542--Incomplete gamma function,"*ACM Trans. Math. Software*, v. 5, 1979, pp. 482-489.**[3]**Peter Henrici,*Applied and computational complex analysis. Vol. 2*, Wiley Interscience [John Wiley & Sons], New York-London-Sydney, 1977. Special functions—integral transforms—asymptotics—continued fractions. MR**0453984****[4]**E. P. Merkes,*On truncation errors for continued fraction computations*, SIAM J. Numer. Anal.**3**(1966), 486–496. MR**0202283**

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DOI:
https://doi.org/10.1090/S0025-5718-1983-0679450-2

Keywords:
Convergence of real continued fractions,
Legendre's continued fraction for the incomplete gamma function

Article copyright:
© Copyright 1983
American Mathematical Society