A continued fraction of the form

is called aStieltjes fraction ($S$-fraction). We say that itcorresponds 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 inany case the coefficients$a_n$ can always be calculated from the power-seriescoefficients by means of thequotient-difference algorithm; seeTable3.10.1.

The first two columns in this table are defined by

where the$c_n$ ($\ne 0$) appear in (3.10.7). We continue bymeans of therhombus rule

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

The quotient-difference algorithm is frequently unstable and may requirehigh-precision arithmetic or exact arithmetic. A more stable version of thealgorithm is discussed inStokes (1980). For applications to Besselfunctions and Whittaker functions (Chapters 10 and13), seeGargantini and Henrici (1967).

A continued fraction of the form

is called aJacobi fraction ($J$-fraction). We say that it isassociated with the formal power series$f(z)$ in(3.10.7) if the expansion of its$n$th convergent$C_n$ inascending powers of$z$, agrees with (3.10.7) up to andincluding 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 and4.35.

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 and10.33 (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 seeLorentzen and Waadeland (1992, pp. 292–330, 560–599) andCuyt et al. (2008).

The$A_n$ and$B_n$ of (3.10.2) can be computed by means ofthree-term recurrence relations (1.12.5). However, this may beunstable; 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, eithera prioriknowledge is needed of the value of$n$, or$n$ is determined by trial anderror. In general this algorithm is more stable than the forward algorithm; seeJones 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 convergentof (3.10.13),$n=1,2,3,\mathrm{\dots }$. In contrast to the precedingalgorithms in this subsection no scaling problems arise and noa prioriinformation is needed.

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

This forward algorithm achieves efficiency and stability in the computation ofthe convergents$C_n=A_n/B_n$, and is related to the forward seriesrecurrence algorithm. Again, no scaling problems arise and noa prioriinformation is needed.

Let

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

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

Alternatives to Steed’s algorithm are theLentz algorithmLentz (1976)and themodified Lentz algorithmThompson and Barnett (1986).

For further information on the preceding algorithms, including convergence inthe complex plane and methods for accelerating convergence, seeBlanch (1964) andLorentzen and Waadeland (1992, Chapter 3). For theevaluation of special functions by using continued fractions seeCuyt et al. (2008),Gautschi (1967, §1),Gil et al. (2007a, Chapter 6), andWimp (1984, Chapter 4, §5). Seealso §§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).