d'Ocagne的斐波那契數字身份歷史


5

d'Ocagne的身份通常表示為$(-1)^ n F_ {m-n} = F_m F_ {n + 1} -F_n F_ {m + 1} $。

每本有關斐波那契數的書中都有此公式,但是我找不到關於它的任何上下文。首次發佈時,為什麼要加上名稱,即使最初$ m,n $被認為是任何整數也是如此。

3

A special case for $m=n-1$ was discovered by Cassini in 1680, it was rediscovered by Simson in 1753. Cassini's identity $(-1)^{n+1} = F_{n-1} F_{n+1}-F_n^2$ was generalized in two ways: one was found by Catalan in 1879, and reads $(-1)^{n+k} = F_{n-k} F_{n+k}-F_n^2$, the other by d'Ocagne (1862-1938). It is unclear when exactly, but in 1889 he studied recurrence relations of Fibonacci and Lucas type, in connection with periodic continued fractions.

As for context, both identities are examples of determinantal identities for Fibonacci numbers, and show close relation between recurrence relations and matrix algebra. Koshy's survey book Fibonacci and Lucas Numbers with Applications has a chapter on determinantal identities. Cassini's identity can be used to generate Curry's paradox invented by Curry in 1953, a dissection of $8×8$ square "rearranged" into a $5×13$ rectangle despite $64\neq65$. Other cases of d'Ocagne's identity can be used to generate similar paradoxes. Recently, Spivey used determinants to derive an identity that generalizes both d'Ocagne's and Catalan's cases.

Interestingly, d'Ocagne's Wikipedia entry does not even mention this identity. He served as the president of the Mathematical Society of France in 1901 and the chair of geometry at École Polytechnique since 1912. But he is most remembered for inventing nomography, a method of graphic calculation popular with engineers before the onset of pocket calculators.