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## Unverified Families

The following is a list of functions which appear to be families of continued fractions, because a computer program written by the author in the summer of 1999 showed that for small values for the variables in M, the period of grew linearly. These results would be highly unlikely if M where in fact not a family of continued fractions.

Let F = 4a + 1, let j = 1 when a = 1, and let j = 2 when a > 1.

Where E = 4a - 1,

Also, where p = 0 for a= 1 and p = 1 for a> 1,

Now let, , with d > 2; then,

Letting where d > 2, one finds,

Let with d > 2; then,

Where with d > 2, one has

Looking at G again,

Similarly with N,

Some more families involving G:

The sort of patterns above can also be seen in N:

Though the following continued fractions have only one variable, their growth rate is significant (for two of them), so it should be useful to mention them.

Next: The Fundamental Correspondence Up: Families of Continued Fractions Previous: Levesque and Rhin's Families
mcenter
2000-01-06