On the margin of continued fraction blogging I would like to present simple result as regards to continuant polynomials. Continued fraction may be expressed as quotient of two polynomials of , named continuant (see continuants on Wikipedia)
Continuants has many interesting recurrence relations some of which You may find in Graham, Knuth, Patashnik book „Concrete mathematics”. The most important of this recurrences is as follows:
During my plays with matrix representation of continued fractions I found interesting relation:
that is – between variables and You put two of „1” in the first and one „1” in the second term. You may consider this as generalisation of Fibonacci recurrence – because if You put all You obtain Fibonacci numbers.
And it is true that:
How to prove it? W have:
where is trace of a matrix, and maybe You know that I define M as follows (see here ):
So we have the following formulas true:
Please note that upper formula has different arguments than the lower so they agree.
And there is true that and which we may use to produce „unity decomposition” as below:
Then You may insert it in any place between which gives expression above and many more if You consider for etc.
Remark 1: proved here formula should be possible also to prove from recursion relation directly – so if You would like to try this way – please let me know.
Remark 2: I post this on mathoverflow, and someone provide the sketch of proof by induction – but it seems to be only a sketch – and rather incomplete.