4.6 Examples of Continued Fractions Which are Padé Approximants

We present here some examples of continued fractions which are also Padé approximants. The examples are either staircase or diagonal sequences in the Padé table, obtained from ./-fractions or S-fractions. We do not give continued fractions which are merely Euler's corresponding fractions whose

 

 

(6.4)

This converges for all z except — oo<z<l, unless v is integral, when the continued fraction terminates and the result is exact.

 

Inverse tangent

convergents are identical to truncated Maclaurin series. We quote a quite comprehensive set of formulas for the functions which are known to have useful continued-fraction expansions. We conclude the section with their formal algebraic derivation, which consists of showing that the Maclaurin series of each continued fraction is the same as that of the given function. The question of convergence is left to Section 4.7.

Exponential function

 

 

 

 

(6.1b)

 

 

(6.1c)

These expansions converge for all z. Tangent function

 

n integral.

 

 

This converges for all z except

 

Hyperbolic tangent

 

n integral.

 

 

This converges for all z except

 

 

 

 

This converges for all z in the z-plane cut from i to /oo and from — i to — i'oo.

Inverse hyperbolic tangent

 

This fraction converges in the whole z-plane cut b}

 

 

 

 

Natural logarithm

 

This fraction converges for all z except

 

 

 

 

Exponential integral

 

 

 

 

This is valid in the entire z-plane cut along — oo<z<0; the integral representation is only valid for Rez>0. See Exercise 1.

Complementary error function

 

 

 

(6.9)

 

 

 

 

This converges for

 

 

 

 

 

 

 

 

 

This is valid for all z except in — oo<z<0. If a is a positive integer, the fraction terminates and so the representation is valid for all z. The connec-

tion with (6.8) is that

 

 

 

 

 

This converges for all z. However, convergence is not fast for Re z > 2, and erfc(z) and its continued fraction (6.10) are more useful in such applica­tions. The relation with Dawson's integral, e ~z2/0z e'* dt, is given in (II. 1.1.44), and T-fraction representations are given by (II.1.1.45d).

 

 

 

 

 

 

Incomplete Gamma function

 

 

 

 

 

(6.12)

where a is not a negative integer or zero. If a is a strictly positive integer, this converges for all z. If a is not integral, the continued fraction in (6.12) converges, but y(a,z) is only defined in the z-plane cut by — oo <z<0. A T-fraction representation is given by (1.1.1.45a), and an error formula is given by Luke [1975].

Definition of hypergeometric functions. We use the hypergeometric func­tion pFq(ax, a2,..., ap, bx, b2,..., bq\ z) with p numerator parameters and q denominator parameters. The examples make the definition clear. The definitions are valid formally provided the denominator parameters are not negative integers or zero. Notice that oF{(a-, z) is an entire function, 2Fx{a, g, c; z) is analytic in thez-plane cut b\ l<z<oo, and 2F0{a,b-, z) has a purely formal definition, since the radius of convergence of the series is zero. The given expansion is a formal expansion of which is the proper definition, valid for Rea>0 and z not on the positive real axis.

 

oi7, hypergeometric-function relation

(6.13a)

 

This converges for all z not a zero of 0 Ft(a\ z). A relation for Bessel functions follows from the formula relating 0Fl hypergeometric functions to Bessel functions,

 

 

We deduce that

 

 

 

Confluent-hypergeometric-function relation

 

 

(6.14)

This converges for all z except for the zeros of xFx(a, b\ z).

2 F0 hypergeometric-function relation

 

 

(6.15)

 

 

This converges in the cut z plane except in the cutThis converges in the cut z-plane except on the cut 1 s£z< oo, and except for the zeros of 2F\{a, b, c; z).

Hypergeometric-function relation [Gauss, 1813]

 

 

Other continued-fraction developments which are Pade approximants for special functions are known [Wall, 1945, p. 369]: they mostly involve integrals of hyperbolic and elliptic functions.

 

 

Hence, (6.1c) follows from (6.3) by taking 2y=z and using the formula

 

Proof of (6.2).

 

 

and the result follows from (6.13).

The derivation of each of the preceding formulas (6.1)-(6.16) consists of both an algebraic and an analytical part. First we show that the Maclaurin series of the two sides of the equations agree term by term. Since the results (6.1)-(6.12) are corollaries of (6.13), (6.14), and (6.15), we discuss these cases first.

Proof of (6.1). Take a = 0 in (6.14). ,F,(0,6; z)=l by definition. This step is used in most of the corollaries. Accordingly, we find that

Taking b = 0, the left-hand side is exp(z), and (6.1a) follows. To prove (6.1b), write exp(z) = {exp(—z)}~' and use the representation (6.1a) for exp(—z). To prove (6.1c), use

 

Proof of (6.3).

 

 

 

and the result follows from (6.13). Proof of (6.4)-(6.7).

 and (6.4)-(6.7) follow from (6.16) with b = 0. Proof of (6.8). Use the representation

 

For

 

taking

and (6.8) follows from (6.15).

 

 

 

 Proof of (6.9).

 

 

and so (6.9) follows from (6.10).

 

 

Proof of (6.10). This is the same as for (6.8). Proof of (6.11).

and so (6.11) follows from (6.14).The continued-fraction expansion then follows from (6.14).

Proof of (6.13). Series expansion of the hypergeometric function shows that

 

 

Therefore

 

 

This formula is simple to iterate and is used to generate the continued- fraction expansion (6.13a).

Proof of (6.14). Series expansion of the confluent hypergeometric func­tion shows that

 

 

Therefore

 

 

Again, this formula is simple to iterate and is used to generate the continued- fraction expansion (6.14).

Proof of (6.12).

 

 

by expansion

 

by setting t = z~u

 

by expansion.

Proof of (6.15). Formal operations with the power series similar to the previous operations lead to the formula

 

 

 

 

 

However, we must use the representation

which is valid for Rea>0 and z not on the positive real axis, to establish the result (6.17). The identity

 

 

leads to equality of the integrands and provides the proof of (6.15) for Rea>0. Extension to complex values of a is by analytic continuation. Hence

(6.19)

Since 2cl, b; z) = 2Fo(b, o; z), it follows from (6.19) that

(6.20)

Equations (6.19) and (6.20) together provide a formula which may be iterated to yield (6.15).

 

Proof of (6.16). The expansion of the hypergeometric function leads to the identity

 

This may be rewritten as

 

 

 

 

 

 

Since we may rewrite (6.21), replacing c byTogether (6.21) and (6.22) yield a formula which connects ratios of hyperge­ometric functions in which the numerator parameters a and b are increased by 1 and the denominator parameter c is increased by 2. The result (6.16) follows by iteration.

To summarize this section, we observe that a variety of familiar functions have continued-fraction expansions given by (6.1)-(6.16). Using the alge­braic results (6.17), (6.19)-(6.22), we have proved that the Maclaurin expansion of each function is the same as that of the corresponding continued fraction. In this sense, the results (6.1)-(6.16) are formal equali­ties. In the next section, we find the domain of values of z for which (6.1)—(6.16) are true equalities.