A continued fraction of the form

is called a Stieltjes fraction ($S$-fraction). We say that it corresponds to the formal power series

if the expansion of its $n$th convergent $C_n$ in ascending powers of $z$ agrees with (3.10.7) up to and including the term in $z^{n-1}$, $n=1,2,3,\mathrm{\dots }$.

For several special functions the $S$-fractions are known explicitly, but in any case the coefficients $a_n$ can always be calculated from the power-series coefficients by means of the quotient-difference algorithm; see Table 3.10.1.

The first two columns in this table are defined by

where the $c_n$ ($\ne 0$) appear in (3.10.7). We continue by means of the rhombus rule

Then the coefficients $a_n$ of the $S$-fraction (3.10.6) are given by

The quotient-difference algorithm is frequently unstable and may require high-precision arithmetic or exact arithmetic. A more stable version of the algorithm is discussed in Stokes (1980). For applications to Bessel functions and Whittaker functions (Chapters 10 and 13), see Gargantini and Henrici (1967).

A continued fraction of the form

is called a Jacobi fraction ($J$-fraction). We say that it is associated with the formal power series $f(z)$ in (3.10.7) if the expansion of its $n$th convergent $C_n$ in ascending powers of $z$, agrees with (3.10.7) up to and including the term in $z^{2n-1}$, $n=1,2,3,\mathrm{\dots }$. For the same function $f(z)$, the convergent $C_n$ of the Jacobi fraction (3.10.11) equals the convergent $C_{2n}$ of the Stieltjes fraction (3.10.6).

For elementary functions, see §§ 4.9 and 4.35.

For special functions see §5.10 (gamma function), §7.9 (error function), §8.9 (incomplete gamma functions), §8.17(v) (incomplete beta function), §8.19(vii) (generalized exponential integral), §§10.10 and 10.33 (quotients of Bessel functions), §13.6 (quotients of confluent hypergeometric functions), §13.19 (quotients of Whittaker functions), and §15.7 (quotients of hypergeometric functions).

For further information and examples see Lorentzen and Waadeland (1992, pp. 292–330, 560–599) and Cuyt et al. (2008).

The $A_n$ and $B_n$ of (3.10.2) can be computed by means of three-term recurrence relations (1.12.5). However, this may be unstable; also overflow and underflow may occur when evaluating $A_n$ and $B_n$ (making it necessary to re-scale from time to time).

To compute the $C_n$ of (3.10.2) we perform the iterated divisions

Then $u_0=C_n$. To achieve a prescribed accuracy, either a priori knowledge is needed of the value of $n$, or $n$ is determined by trial and error. In general this algorithm is more stable than the forward algorithm; see Jones and Thron (1974).

The continued fraction

can be written in the form

where

The $n$th partial sum $t_0+t_1+\mathrm{\cdots }+t_{n-1}$ equals the $n$th convergent of (3.10.13), $n=1,2,3,\mathrm{\dots }$. In contrast to the preceding algorithms in this subsection no scaling problems arise and no a priori information is needed.

In Gautschi (1979c) the forward series algorithm is used for the evaluation of a continued fraction of an incomplete gamma function (see §8.9).

This forward algorithm achieves efficiency and stability in the computation of the convergents $C_n=A_n/B_n$, and is related to the forward series recurrence algorithm. Again, no scaling problems arise and no a priori information is needed.

Let

($\nabla$ is the backward difference operator.) Then for $n\ge 2$,

The recurrences are continued until $(\nabla C_n)/C_n$ is within a prescribed relative precision.

Alternatives to Steed’s algorithm are the Lentz algorithm Lentz (1976) and the modified Lentz algorithm Thompson and Barnett (1986).

For further information on the preceding algorithms, including convergence in the complex plane and methods for accelerating convergence, see Blanch (1964) and Lorentzen and Waadeland (1992, Chapter 3). For the evaluation of special functions by using continued fractions see Cuyt et al. (2008), Gautschi (1967, §1), Gil et al. (2007a, Chapter 6), and Wimp (1984, Chapter 4, §5). See also §§6.18(i), 7.22(i), 8.25(iv), 10.74(v), 14.32, 28.34(ii), 29.20(i), 30.16(i), 33.23(v).