Fibonacci

Proof:

We want to show that
F_p+iF_q-i - F_pF_q = (-1)^k(F_p+i+kF_q-i+k - F_p+kF_q+k)

It is obvious that, if we can proof the theorem for all k≥0, then the theorem is automatically true for all k≤0 (for, change the left- and righthand side and replace p and q by respectively p-k and q-k).
The proof is by induction:
The hypothesis is true k = 0 (trivial).
Now, assume that the supposition is true for k=t. The proof is completed if we can show that
(-1)^t+1(F_p+i+t+1F_q-i+t+1 - F_p+t+1F_q+t+1) = (-1)^t(F_p+i+tF_q-i+t - F_p+tF_q+t) or that
-(F_p+i+t+1F_q-i+t+1 - F_p+t+1F_q+t+1) = F_p+i+tF_q-i+t - F_p+tF_q+t.
Now, for the sake of clearness, replace p+t by the character u and q+t by v. So, we have to show that
-(F_u+i+1F_v-i+1 - F_u+1F_v+1) = F_u+iF_v-i - F_uF_v.
We make use of a telescopic sum
F_u+iF_v-i - F_uF_v = (F_u+iF_v-i - F_u+i-1F_v-i+1) + (F_u+i-1F_v-i+1 - F_u+i-2F_v-i+2) + ... + (F_u+1F_v-1 - F_uF_v).

If we can show that for arbitrary x and y F_xF_y - F_x-1F_y+1 = -(F_x+1F_y+1 - F_xF_y+2), then we may increase all indices in the telescopic sum by 1 (and, of course, you have to put an extra minus sign if front of the expression). The result of that telescopic sum yields the result we are looking for..
The equation F_xF_y - F_x-1F_y+1 = -(F_x+1F_y+1 - F_xF_y+2) can also be written like:
F_x(F_y+2 - F_y) = F_y+1(F_x+1 - F_x-1)
Now F_y+2 - F_y = F_y+1 and F_x+1 - F_x-1 = F_x, thus the formulas in x and y are correct and the theorem has been proven.



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We want to show that
F_p+iF_q+iF_r+i - (-1)ⁱL_iF_pF_qF_r + (-1)ⁱF_p-iF_q-iF_r-i = F_iF_2iF_p+q+r
We make use of the formulas
F_p+iF_q+i - (-1)ⁱF_pF_q = F_iF_p+q+i    (1) and
F_p+iF_q+i - F_p-iF_q-i = F_2iF_p+q    (2)
F_iF_2iF_p+q+r =
{{ Replace in (2) p by p+q and q by r and multiply the result by F_i. }}
F_iF_p+q+iF_r+i - F_iF_p+q-iF_r-i =
{{ Use formula (1) twice,
the first time direct, the second time by replacing in (1) p and q by p-i and q-i. }}
(F_p+iF_q+i - (-1)ⁱF_pF_q)F_r+i - (F_pF_q - (-1)ⁱF_p-iF_q-i)F_r-i =
F_p+iF_q+iF_r+i - (-1)ⁱF_pF_qF_r+i - F_pF_qF_r-i + (-1)ⁱF_p-iF_q-iF_r-i =
F_p+iF_q+iF_r+i - (-1)ⁱF_pF_q{F_r+i + (-1)ⁱF_r-i) + (-1)ⁱF_p-iF_q-iF_r-i =
{{ F_r+i + (-1)ⁱ F_r-i = F_r L_i, see formula 1 }}
F_p+iF_q+iF_r+i - (-1)ⁱL_iF_pF_qF_r + (-1)ⁱF_p-iF_q-iF_r-i