Diagonal continued fraction
 |
in which the sequences 
and 
must satisfy the following conditions:
1) the numbers 
and 
are integers; 
; 
if 
; 
if 
;
2) 
for all 
; if 
, then 
for an infinite set of indices 
;
3) 
for all 
;
4) the partial fractions of the continued fraction are all irreducible fractions 
such that 
and 
, where 
is value of the continued fraction.
For each real number 
there exists one and only one diagonal continued fraction with 
as its value; this fraction is periodic if 
is a quadratic irrationality.
Comments
After truncation and evaluation one obtains
 |
with 
, 
, 
. These are the numbers 
alluded to in condition . The continued fraction as described above for a real number 
can be obtained by the nearest integer algorithm, that is, 
, 
, 
, 
, 
, etc., where 
denotes the nearest integer to 
. It is also possible to use the entier function 
instead of 
. One then has the continued fraction algorithm which is more commonly used.
The adjective "diagonal" stems from the fact that 
for all 
.
References
| [a1] | G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8 |
| [a2] | O. Perron, "Die Lehre von den Kettenbrüchen" , I , Teubner (1977) |
Diagonal continued fraction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diagonal_continued_fraction&oldid=46639