### Basic definitions for continued fractions

A continued fraction is an expression of the form

 (1)

where , and are either real or complex values called the terms of continued fraction (1). The number of terms can be either finite () or infinite (). If continued fraction (1) is called finite, otherwise infinite. The ’s are called partial denominators and ’s partial numerators.

Thought the notation used for a continued fraction in (1) is intuitive and lucid, it takes up a lot of space and is not easy to typeset. So several alternative notations were developed. For instance,  A.I. Pringsheim introduced the notation

while C.F.Gauß used

where K stands for the initial of German mane Kettenbrüche for continued fraction.

The rational value

 (2)

or

or

which we get by truncation of the original continued fraction  after the th partial denominator , , is called the th convergent of the given continued fraction.

The first four convergents are

 (3)

Although a given convergent may naturally  be worked out “from the bottom” as it was done above, it is more practical to generate the sequence of convergents “from the top” . Let , denote the numerator and the denominator of the th convergent. J.Wallis  [1] ,  [2]  and L.Euler  [3]  found that the values of , and can be determined recursively

 (4)

 (5)

for and initial conditions  , , , . It is sometimes useful to start with , , , , or with  , , , .

Recurrence relations (4) and (5) are called fundamental recurrence formulas and the fraction

 (6)

is the th convergent of (1).

For the sake of simplicity it is usually supposed that all are non-vanishing.

We say that the continued fraction (1) converges or diverges (or oscillates) if the sequence converges or diverges (or oscillates) for . If the continued fraction (1) converges, that

 (7)

then is assigned as the value of (1).

Theorem (  [4]  , Satz 1): Let every , , be non-vanishing.  It two of three equalities

 (8)

 (9)

 (10)

are valid, then also the third one is true, no matter whether the continued fractions  (9),  (10) are infinite or finite but ending with the same term.

Theorem: We have:

 (11)

 (12)

 (13)

Two continued fractions and with convergents and , , respectively, are called equivalent if

Theorem ( [5] , Theorem 2.2.4): Two continued fractions and are equivalent if and only if  there exists a sequence of non-vanishing numbers , with such that

 (14)

Consequently

Theorem: If  , , is an infinite sequence of non-zero complex numbers, then the continued fractions

 (15)

 (16)

have the same successive convergents, that is the successive convergents of continued fraction (15) are exactly the same as the convergents of the fraction (16).

Theorem ( [5]  , (2.3.19a)): Let for every , and

 (17)

then and are equivalent

If we use the formula (22) for then we get

 (18)

#### Simple continued fractions

If for every , then the continued fraction (1) is called simple. Simple continued fractions

 (19)

are usually written using a more compact abbreviated notation

 (20)

The standard textbooks usually works only with simple continued fractions, but simple continued fractions are not necessarily “simple”. For instance, presently no regularity in the sequence of partial denominators of the simple continued fraction of π is known

 (21)

On the other hand compare with a regular “non-simple” expansion

 (22)

which we get from (25).

Theorem: Every continued fraction (1) can be transformed into a simple continued fraction.

To see this take

 (23)

in (16) to prove the required transformation.

#### Functional continued fractions

The terms of a continued fraction (1) can also be functions, say , , and , . For instance, to J.H.Lambert  [6] ,  [7]  attributed relation

 (24)

appears already in L.Euler   [3]  .  Lambert used this continued fraction expansion to prove that is irrational. He showed by an infinite descent argument that if is rational, then the right hand side of (24) is irrational. Since is rational, the conclusion follows (Some authors claim that Lambert's proofs is not complete. Pringsheim [8]   proved that this is not true.) Lambert gave a list of the first 27 convergents of the simple continued fraction expansion of from which the first 25 were correct, but the last two not.

For the arc tangent we have

 (25)

#### Ascending continued fractions.

Continued fractions of type (1) are called  descending continued fractions. However there are possible also other types of continued fractions. One of them are  so-called the ascending continued fractions. They were already developed by Fibonacci, Lambert or Lagrange.

For instance, following Arabic sources Fibonacci writes (in our notation)

 (26)

which stands for plus of plus of of .

Ascending continued fractions are of the form

 (27)

It is equivalent with descending continued fraction

 (28)

### References

 [1] Wallis, J. (1655). Arithmetica infinitorum .

 [2] Wallis, J. (1695). Opera mathematica I. Osord.

 [3] Euler, L. (1744). De fractionibus continuis dissertatio. Commentarii academiae scientiarum Petropolitanae, 9, 98-137 (=Opera Omnia: Series 1, Volume 14, pp. 187 - 216)   .

 [4] Perron, O. (1913). Die Lehre von den Kettenbrüchen. (German). Leipzig u. Berlin: B.G.Teubner.

 [5] Jones, W. B., & Thron, W. J. (1980). Continued fractions. Analytic theory and applications. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company.

 [6] Lambert, J. H. (1761). Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques. Histoire de l'Académie Royale des Sciences et des Belles-Lettres der Berlin, 265-276.

 [7] Lambert, J. H. (1770). Vorläufige Kenntnisse für die, so die Quadratur und die Rectification des Circuls suchen.

 [8] Pringsheim, A. I. (1898). Ueber die ersten Beweise der Irrationalität von e und π. Sitzungsberichte der Bayerischen Akademie der Wissenschaften Mathematisch-Physikalische Klasse, 28, 325-337.

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Štefan Porubský: Basic definitions for continued fractions.

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