Fibonomial coefficient explained

In mathematics, the Fibonomial coefficients or Fibonacci-binomial coefficients are defined as

\binom{n}{k} _{F} = (F _{nF} _{n-1} … F _{n-k+1})/(F _{kF} _{k-1} … F _{1}) = (n! _{F})/(k! _{F} (n - k)! _{F})

where n and k are non-negative integers, 0 ≤ k ≤ n, F_j is the j-th Fibonacci number and n!_F is the nth Fibonorial, i.e.

{n!} _{F} : = \prod ^{n} _{i =1} F _{i,}

where 0!_F, being the empty product, evaluates to 1.

Special values

The Fibonomial coefficients are all integers. Some special values are:

\binom{n}{ 0 } _{F} = \binom{n}{n} _{F} = 1 \binom{n}{ 1 } _{F} = \binom{n}{n -1 } _{F} = F _{n} \binom{n}{ 2 } _{F} = \binom{n}{n -2 } _{F} = (F _{n} F _{n-1})/(F _{2} F _{1}) = F _{n} F _{n-1} , \binom{n}{ 3 } _{F} = \binom{n}{n -3 } _{F} = (F _{n} F _{n-1} F _{n-2})/(F _{3} F _{2} F _{1}) = F _{n} F _{n-1} F _{n-2} / 2 , \binom{n}{k} _{F} = \binom{n}{n - k} _{F.}

Fibonomial triangle

The Fibonomial coefficients are similar to binomial coefficients and can be displayed in a triangle similar to Pascal's triangle. The first eight rows are shown below.

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n =0		1
n =1		1	1
n =2		1	1	1
n =3		1	2	2	1
n =4		1	3	6	3	1
n =5		1	5	15	15	5	1
n =6		1	8	40	60	40	8	1
n =7		1	13	104	260	260	104	13	1

The recurrence relation

\binom{n}{k} _{F} = F _{n-k+1} \binom{n -1 }{k -1 } _{F} + F _{k-1} \binom{n -1 }{k} _{F}

implies that the Fibonomial coefficients are always integers.

\varphi = (1+ \sqrt 5)/(2) :{\binom n k} _{F} = \varphi ^{k(n-k)} {\binom n k} _{-1 /\varphi ^{2}}

Applications

Dov Jarden proved that the Fibonomials appear as coefficients of an equation involving powers of consecutive Fibonacci numbers, namely Jarden proved that given any generalized Fibonacci sequence

G _{n} , that is, a sequence that satisfies G _{n} = G _{n-1} + G _{n-2} for every n, then\sum ^{k +1} _{j =0} ( -1 ) ^{j(j+1)/2} \binom{k +1 }{j} _{F} G ^{k} _{n - j} = 0 ,

for every integer

n , and every nonnegative integer k .

References

• • Ewa Krot, An introduction to finite fibonomial calculus, Institute of Computer Science, Bia lystok University, Poland.
  • Dov Jarden, Recurring Sequences (second edition 1966), pages 30–33.