Recursive Patterns and Polynomials

Alyssa Farrell

Iowa State University

MSM Creative Component

Summer 2006

Heather Thompson, Major Professor

Irvin Hentzel, Major Professor

Patricia Leigh, Committee Member

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Patterns have been studied for centuries. Some of these patterns have been graced with names of the people who are most remembered for their work with them or related topics. Among these are the Fibonacci numbers, Lucas numbers, Pascal’s Triangle,

Chebyshev’s Polynomials, and Cardan Polynomials.

Patterns are used as building blocks in education from newborn babies and young children all the way through high school and beyond into post-secondary work. We were introduced as early as infancy to learning to recognize patterns, meaning both similarities

and differences, when learning colors, shapes, the alphabet, and numbers. As we grew,

we were introduced to new mathematical patterns like learning to count by 2’s, or 5’s.

As we were more formally introduced to shapes, we learned to recognize that a square has four sides the same length, but unlike a kite, the square has four angles that are 90 degrees. Then we learned to add, subtract, multiply, and divide. All of these necessary operations were taught using a pattern-approach. Do you remember learning to read? We were taught to recognize patterns of letters which made certain sounds and that when certain sounds were strung together they made words.

These patterns continued to be used as we developed higher in our math education. Teachers in junior high or middle school introduced us to an abstract idea of

letting letters represent numbers as variables. We found out that adding “letters” was

similar to adding numbers. In high school we elaborated on algebraic thought processes.

Learning to multiply two binomials could be described using a pattern.

Mathematics itself has been described as the “science of patterns” (Smith, 137).

This description places a direct emphasis on the ability of math student to recognize,

process, and discuss patterns.

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Recursive thinking, or thinking with respect to patterns, has been a topic of many documents. The National Council of Teachers of Mathematics recommends in their 2000 publication, Principles and Standards for School Mathematics, that patterns are

introduced as an integral part of math curriculum as early as preschool. They recommend

that students continue being exposed to patterns through college. The importance of

understanding patterns is addressed as an integral piece of knowledge for students to be

able to use the process standards: problem solving, reasoning and proof, communication,

connections, and representation.

Patterns are an important part of mathematics curricula and also have more impact

on a student’s life than just in their mathematics career. “…mathematical habits of mind

are the most important things students can take away from their mathematics

education…the real utility of mathematics is that it provides you with the intellectual

schemata necessary to make sense of a world in which the products of mathematical

thinking are increasingly pervasive in almost every walk of life… Recursive thinking also

gives students genuine intellectual power.” (Cuoco).

Patterns continue to be building blocks in learning new math. Even as I continue

my education with my master’s degree, I use patterns to help me describe what happened;

learning to relate patterns to new material made this learning easier. This is why I chose

to look deeper into Thomas Osler’s article “Variations on a Theme from Pascal’s

Triangle”. At first, I was drawn to the patterns and generating functions of any variation

of Pascal’s Triangle, but as I delved into these, I realized there was more out there to

explore than just the patterns of Pascal’s Triangle.

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I started by looking at patterns developed when binomials were expanded and

2 i multiplied by a trinomial (a2 x + a1x + a0 )(b1x + b0 ) . This was interesting to me because of the ties it had within Algebra I and Algebra II curriculum. What I found was that there

was a formula that could be used to find the coefficients of the polynomial for any value

of i .

i−c+2 c−2 i−c+1 c−1 i−c A()i,c = C(i,c − 2)⋅b1 ⋅b0 ⋅ a0 + C(i,c −1)⋅b1 ⋅b0 ⋅ a1 + C(i,c)⋅b1 ⋅b0 ⋅ a2 , where A(i,c) refers to the coefficient of the cth term in the polynomial expansion of

2 i (a2 x + a1 x + a0 )()b1 x + b0 and C(n,r)represents the combination of n things taken r at a time.

This follows from the binomial expansion, which states that given a binomial,

i (b1 + b0 )can be expanded using the formula:

i i i−1 i−2 2 i ()b1 + b0 = b1 + C(i,1)⋅b1 ⋅b0 + C(i,2)⋅b1 ⋅b0 + ...+ b0 . To find a specific term, say

i−c+1 c−1 the cth term, one can use the formula: C(i,c −1)⋅b1 ⋅b0

After working with the above idea and then moving on to work with the expansion of trinomials, I realized that this work had already been done. I went back to

Osler’s article again and looked at other patterns he had unmasked during his explorations. He writes about the Fibonacci numbers and polynomials, the Lucas numbers and polynomials, and also the Cardan polynomials and their relation to the

Lucas polynomials (216-23).

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Fibonacci Numbers

One variation that Osler looks at is the following (218):

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1

This variation of Pascal’s Triangle is formed by the expansion of the polynomial

xm (1+ x)m . If one adds the columns of the arrangement, the sums are 1, 1, 2, 3, 5, 8,

13… This series is the well known Fibonacci numbers. The Fibonacci numbers are a

series of naturally occurring numbers. The most familiar place where the Fibonacci

numbers are represented is in the solution to the “rabbit problem” and in the family tree

of a honeybee (Anderson, Frazier, and Popendorf). Fibonacci numbers also have a very

close relation to the Golden Ratio (Knott, Fibonacci Number).

Leonardo Pisano, better known by his nickname “Fibonacci”, meaning son of

Bonacci, (1170-1250) is credited with solving the infamous rabbit problem, whose solution represents the series of numbers known by his name(O’Connor and Robertson,

Fibonacci). The rabbit problem is stated as “A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?” (Anderson, Frazier, and

Popendorf).

The solution is represented by the table below, where M represents the number of the month and P represents the number of rabbit pairs in the population at the end of that month.

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M 0 1 2 3 4 5 6 7 8 9 10 11 12 P 1 1 2 3 5 8 13 21 34 55 89 144 233

So, at the end of the 12th month, there are 233 pairs of rabbits in the population.

This, however, is a very unrealistic problem, since we know that every birth does not necessarily produce a pair, that the population would have to be reproducing within itself and mutations are bound to occur, and that the food supply is unlimited. What are the chances that no deaths occur in this population for an entire year?

Another example where the Fibonacci numbers are displayed is in the family tree of a honeybee. Since honeybees have a strange parentage, their family trees are very interesting. Some bees have two parents, some have only one. In the honeybee colony there are three types of bees: drones (males), workers (females) and a queen (special female). A queen is the only female that reproduces and her offspring can be male, produced from a non-fertilized egg, or female, from a fertilized egg. Thus a male honeybee has only one parent, the queen bee, while a female honeybee has two parents, both a male and a female. If we look at the family tree of a male honeybee (1 bee in generation 1), we see that he has only one parent, a female (1 bee in generation 2). This female honeybee has two parents, a male and female (2 bees in generation 3). Of the bees in generation 3, the male has one parent, a female, the female has two parents, male and female (3 bees in generation 4). The diagram and table below shows the number of bees in each generation.

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M Generation 1 F Generation 2 M F Generation 3 F M F Generation 4

Generation 5 M F F M F

Generation 1 234567 8 9 1011 12 Number of bees 1 123581321345589 144

This series of numbers, Fn ,can be defined as the recursive function

Fn = Fn−1 + Fn−2 with F1 = 1and F2 = 1 (Knott, General Fibonacci). Any specific

Fibonacci number, Fn , can be found by using the closed formula

n n 1 ⎡⎛1+ 5 ⎞ ⎛1− 5 ⎞ ⎤ F = ⎢⎜ ⎟ − ⎜ ⎟ ⎥ . Simplified, this is the same as Binet’s Fibonacci n 5 ⎢⎜ 2 ⎟ ⎜ 2 ⎟ ⎥ ⎣⎝ ⎠ ⎝ ⎠ ⎦

n n 1+ 5 − 1− 5 Number Formula F = ()( ) (Knott, Fibonacci Number). n 2n 5

It can easily be proven that Binet’s formula satisfies the recursion formula with a proof by induction. To start with, we need to show that Binet’s formula works for n =1, n = 2 and n = 3.

1 1 ()1+ 5 − (1− 5) 1+ 5 −1+ 5 2 5 F1 = = = = 1 21 5 2 5 2 5

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2 2 1+ 5 − 1− 5 1+ 2 5 + 5 − 1− 2 5 + 5 4 5 F = ()( ) = ( ) = =1 2 22 5 4 5 4 5

3 3 1+ 5 − 1− 5 1+ 3 5 +15 + 5 5 − 1− 3 5 +15 − 5 5 16 5 F = ()()= ( ) = = 2 3 23 5 8 5 8 5

Assume that Binet’s formula works for n = k . We will now show that Binet’s formula holds for n = k −1.

Show Fk +1 = Fk + Fk −1 is true for Binet’s Formula.

k k k −1 k −1 1+ 5 − 1− 5 1+ 5 − 1− 5 F = ()( ) + ( ) ( ) k +1 2k 5 2k −1 5

k k k −1 k −1 1+ 5 − 1− 5 2 1+ 5 − 2 1− 5 = ()( ) + ( ) ( ) 2k 5 2k 5

k k −1 k k −1 1+ 5 + 2 1+ 5 − 1− 5 − 2 1− 5 = ()( ) ( ) ( ) 2k 5

k −1 k −1 1+ 5 1+ 5 + 2 − 1− 5 1− 5 + 2 = ()( ) ( ) ( ) 2k 5

k −1 k −1 1+ 5 3+ 5 − 1− 5 3− 5 = ()( ) ( ) ( ) . 2k 5

2 2 (1+ 5) (1− 5) Substitute ()3+ 5 = and ()3− 5 = 2 2

⎛ 2 ⎞ ⎛ 2 ⎞ k −1 (1+ 5) k −1 (1− 5) ()1+ 5 ⎜ ⎟ − ()1− 5 ⎜ ⎟ ⎜ 2 ⎟ ⎜ 2 ⎟ F = ⎝ ⎠ ⎝ ⎠ k +1 2k 5

⎛ k +1 k +1 ⎞ ⎜ (1+ 5) − (1− 5) ⎟ ⎜ 2 ⎟ = ⎝ ⎠ 2k 5

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k +1 k +1 1+ 5 − 1− 5 F = ( ) ( ) . k +1 2k +1 5

Binet’s formula works for all n =1,2,3,...

1+ 5 The Fibonacci numbers are also related to the Golden Ratio, φ = or phi, by 2

F the relationship φ = lim n . The Golden Ratio can also be used to find a specific n→∞ Fn−1

φ n − (ϕ)n 1− 5 Fibonacci number with the formulas Fn = , where ϕ = is the reciprocal 5 2 of φ . This is the same as Binet’s Formula using the variables φ and ϕ for the Golden

Ratio instead of the numerical representation (Knott, Fibonacci Number).

Fibonacci Polynomials

By expanding the polynomial xm (x + y)m , the Fibonacci polynomials can be generated:

1 xy + x2 x2 y 2 + 2yx3 + x4 y3 x3 + 3y 2 x4 + 3yx5 + x6 y 4 x4 + 4y3 x5 + 6y3 x6 + 4yx7 + x8

If the columns are added we get the polynomial:

1+ yx + (y2 +1)x2 + (y3 + 2y)x3 + (y4 + 3y2 +1)x4 +..., where the coefficients of xn are the

Fibonacci polynomials (Osler, Variations).

The first seven Fibonacci Polynomials are:

F1(x) 1

F2 (x) x

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2 F3 (x) x +1 F (x) x3 + 2x 4 4 2 F5 (x) x + 3x +1 5 3 F6 (x) x + 4x + 3x F (x) x6 + 5x4 + 6x2 +1 7

The Fibonacci polynomials follow the recursive relationship

Fn ()x = x ⋅ Fn−1(x)+ Fn−2 (x) where F1(x) =1 and F2 (x) = x . The Fibonacci polynomials are interesting for more than their recursive patterns, because as one substitutes x =1into

any Fn (x), the nth Fibonacci number results (Weisstein, Fibonacci Polynomial).

Lucas Numbers

Another variation of Pascal’s triangle that Osler demonstrates is as follows:

1 2 1 3 2 1 4 5 2 1 5 9 7 2 1 6 14 16 9 2 1 7 20 30 25 11 2

This array is developed by the coefficients of expansion of the polynomial ()1+ 2x xm (1+ x)m (219).

If the columns of this array are added, the series 1, 3, 4, 7, 11, 18… emerges. This is another famous series of numbers, the Lucas Numbers. The Lucas numbers are named after the 19th century mathematician Eduard Lucas. Lucas is known for naming the sequence of numbers that Fibonacci found while counting rabbit pairs and honeybee ancestors, the Fibonacci numbers. This sequence of numbers that he studied extensively

Farrell 11 was named the Lucas numbers in his honor. He is also famous for devising the test that is still in use today for identifying Mersenne primes (O’Connor and Robertson, Lucas).

The Lucas numbers are a series of numbers similar to the Fibonacci numbers. The same recursive relation applies to the Lucas numbers as with the Fibonacci numbers,

Ln = Ln−1 + Ln−2 . The difference between the two sequences of numbers is in their starting

values. The Fibonacci sequence starts with F1 =1 and F2 =1, where the Lucas sequence

starts with L1 =1 and L2 = 3 (Knott, Lucas Numbers).

There are other series which use the Fibonacci recursion formula, but do not use the starting values above. These are called Generalized Fibonacci Series, orG . A generalized Fibonacci series is denoted by G(a,b,i) where a andb are the starting terms and i represents the index of the term (starting with zero). One example of a G-series is

G()4,5,i = 4,5,9,14,23,37,.... There are two special cases of Generalized Fibonacci Series.

One is the seriesG(0,1,n), the Fibonacci (Numbers) Series. The other is G(2,1,n), Lucas

Numbers (Knott, General Fibonacci Series).

An interesting relationship between the Fibonacci series and any Generalized

Fibonacci series is the pattern G(a,b,n) = a ⋅ F(n −1)+ b⋅ F(n) where F(n) is the nth term in the Fibonacci series (Knott, General Fibonacci Series).

There are also many direct connections between the Lucas numbers and the

Fibonacci numbers. The Lucas numbers can be used to find or identify Fibonacci numbers using a multitude of formulas, some of which I have listed here (Weisstien,

Lucas Number).

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1. Ln = Fn−1 + Fn+1

L + L 2. n−1 n+1 = F 5 n

3. Ln = Fn+2 − Fn−2

4. 2⋅ Ln = Fn−3 + Fn+3

5. F2n = Fn ⋅ Ln

The Lucas numbers are also connected to the Golden Ratio by the formula

1+ 5 1− 5 L = φ n − (ϕ)n , where φ = is the Golden Ratio, and ϕ = is the reciprocal of n 2 2

n n ⎛1+ 5 ⎞ ⎛1− 5 ⎞ the Golden Ratio. This formula can be written as L = ⎜ ⎟ + ⎜ ⎟ (Weisstein, n ⎜ ⎟ ⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠

Golden Ratio).

Lucas Polynomials

The expansions of ()y + 2x xm (y + x)m for m = 0,1,2,3... are used to develop the

Lucas polynomials.

y + 2x y 2 x + 3yx2 + 2x3 y3 x2 + 4y 2 x3 + 5yx4 + 2x5 y 4 x3 + 5y3 x4 + 9y 2 x5 + 7yx6 + 2x7

When the columns are added we get: y + (y2 + 2)x + (y3 + 3y)x2 + (y4 + 4y 2 + 2)x3 +... where the coefficients of xn are the Lucas polynomials (Osler, Variations, 221).

The first seven Lucas Polynomials:

L1(x) x 2 L2 ()x x + 2 3 L3 (x) x + 3x

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4 2 L4 ()x x + 4x + 2 5 3 L5 (x) x + 5x + 5x 6 4 2 L6 (x) x + 6x + 9x + 2 7 5 3 L7 ()x x + 7x +14x + 7x

Just as with the Lucas numbers, the Lucas polynomials also behave similarly to their Fibonacci counterparts. The Lucas polynomials follow the same recursive pattern

only with different starting values. The relationship is Ln (x) = x ⋅ Ln−1(x)+ Ln−2 (x) with

2 L1()x = x and L2 ()x = x + 2 (Weisstein, Lucas Polynomial).

The “[Lucas polynomials] are closely related to the Cardan polynomials” (Osler,

Variations, 221), and the Cardan polynomials are closely related to the Chebyshev polynomials. Prior to examining the Cardan polynomials, Chebyshev polynomials will be discussed. Chebyshev polynomials are used in the definition of the Cardan polynomials, thus making the understanding of Chebyshev polynomials instrumental in understanding the Cardan polynomials.

Chebyshev Polynomials

Pafnuty Lvovich Chebyshev (1821-1894) was a Russian mathematician that is best known for his investigations with number theory. Chebyshev also worked with probability and made contributions in this area of mathematics as well. In 1854 he published a paper, Théorie des mécanismes connus sous le nom de parallélogrammes

(Theory of mechanisms known under the name parallelograms) which introduced the world to the Chebyshev polynomials for the first time. In this paper, he first explored the concept of orthogonal polynomials. He later continued his work with orthogonal polynomials and developed a theory describing them (O’Connor and Robertson,

Chebyshev).

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“A polynomial sequence is an orthogonal sequence with respect to the weight function, W , when any two different polynomials in the sequence are orthogonal, using

x2 that weight function i.e. pm , pn = pm ()x pn ()x W ()x dx where m ≠ n ” (Orthogonal ∫x 1

Polynomials). Chebyshev polynomials are a set of orthogonal polynomials with respect to

1 the weight function (Chebyshev Polynomials). The Chebyshev polynomials are 1− x2

⎧0:nm≠ 1 dx ⎪ defined orthogonal on the interval [−1,1] as Txnm()T()x ==⎨π :0nm=, ∫−1 2 1− x ⎪ π :0nm=≠ ⎩⎪ 2

where Tn is the nth Chebyshev Polynomial.

To show that this works, let n = 2 and m = 3 :

1 1 1 dx 2 3 dx 5 3 dx T2 ()x T3 (x) = ()2x −1 (4x − 3x) = (8x −10x + 3x) ∫−1 2 ∫−1 2 ∫−1 2 1− x 1− x 1− x

1 4 2 ⎛ −8x + 6x − 3⎞ 2 ⎤ ⎡⎛ −8 + 6 − 3⎞ ⎤ ⎡⎛ −8 + 6 − 3⎞ ⎤ = ⎜ ⎟⋅ 1− x ⎥ ⎜ ⎟⋅0 − ⎜ ⎟⋅0 = 0 . ⎜ 5 ⎟ ⎢ 5 ⎥ ⎢ 5 ⎥ ⎝ ⎠ ⎦ −1 ⎣⎝ ⎠ ⎦ ⎣⎝ ⎠ ⎦

The Chebyshev polynomials, referring to the Chebyshev polynomials of the first kind, are, by one definition, the set of polynomials which are the solutions to the

2 d ⎡ 2 dy ⎤ n Chebyshev differential equation: ⎢ 1− x ⎥ + y = 0 (Sinwell, 22). This dx ⎣ dx⎦ 1− x2 differential equation is a special case of the Sturm-Liouville Boundary Value Problem:

d ⎡ dy ⎤ p()x + []q()x + λr(x)y = 0 ; dx ⎣⎢ dx⎦⎥

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⎧a1 y()a + a2 y′()a = 0 ⎨ where a ≤ x ≤ b . ⎩b1 y()b + b2 y′()b = 0

1 If a = −1, b =1, p()x = 1− x2 , q(x) = 0 , r()x = , and λ = n2 , one gets the 1− x2

Chebyshev differential equation (Special Cases).

Written in a slightly different manner, the Chebyshev differential equation

d 2 y dy becomes ()1− x2 − x + n2 y = 0 where n is a real constant. The solutions of the dx2 dx

∞ m differential equation can be obtained by the power series y = ∑ am x where the m=0

()m + n (m − n) coefficient is given by a = a . m+2 ()m +1 (m + 2)m

When a0 =1 and a1 = 0 the equation

n2 ()n − 2 n2 (n + 2) (n − 4)(n − 2)n2 (n + 2)(n + 4) y ()x =1− x2 + x4 − x6 +... emerges. 1 2! 4! 6!

When a0 = 0 and a1 =1 the equation

()n −1 (n +1) ()n − 3 (n −1)(n +1)(n + 3) y ()x = x − x3 + x5 −... emerges (Clase). 2 3! 5!

These two equations, y1(x) and y2 (x), become the Chebyshev polynomials, up to

multiplication by a constant, where n is a constant. When n is even, y1(x) terminates at

n n x ; when n is odd, y2 (x) terminates at x .

1 If n =1, then y2 ()x = x giving T1(x) = x .

22 If n = 2 , then y ()x =1− x2 =1− 2x2 multiplying by −1, we get T ()x = 2x2 −1. 1 2! 2

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(3−1)(3+1) 4 If n = 3, then y ()x = x − x3 = x − x3 . Multiplying by − 3 , we get 2 3! 3

3 T3 ()x = 4x − 3x (Clase).

The Chebyshev polynomials are also defined as the set of polynomials that satisfy

the equationTn ()cosθ = cos(nθ ). It can also be defined by the recursive relationship:

⎧1 if n = 0 ⎪ Tn ()x = ⎨x if n = 1 (Sinwell, 22). ⎪ ⎩2x ⋅Tn−1 ()x − Tn−2 (x) if n ≥ 2

The first seven Chebyshev polynomials

T0 (x) 1

T1()x x 2 T2 (x) 2x −1 3 T3 (x) 4x − 3x 4 2 T4 (x) 8x −8x +1 5 3 T5 (x) 16x − 20x + 5x 6 4 2 T6 (x) 32x − 48x +18x −1

The closed formula for the kth polynomial of the Chebyshev polynomial can be

k −k eikθ + e −ikθ (eiθ ) + (eiθ ) developed using cos()kθ = = and the two properties 2 2 eiθ = cosθ + isinθ andsinθ = 1− cos2 θ . This gives

k −k cosθ + cos 2 θ −1 + cosθ + cos 2 θ −1 cos()kθ = ( ) ( ) . Recall that cos()nθ = T (cosθ ) 2 n

k −k x + x2 −1 + x + x2 −1 and substitute cosθ = x . We have T ()x = ( ) ( ) (Sinwell, 22-25). k 2

Proof of Tn ()cosθ = cos(nθ )

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Proof by induction:

When n =1: T1()cosθ = cos(1⋅θ )

cos()θ = cos(θ )

When n = 2 : T2 (cosθ ) = cos(2⋅θ )

−1+ 2 ⋅ cos2 ()θ = −1+ 2 ⋅ cos 2 (θ )

Assume true for n = k : Tk (cosθ ) = cos(k ⋅θ )

Show Tk +1 ()cosθ = cos((k +1)⋅θ )

cos()(k +1)⋅θ = cos(kθ +θ ) = cos kθ cosθ − sin kθ sinθ

1 = cosθ ⋅T ()cosθ − []cos()kθ −θ − cos(kθ +θ ) k 2

1 1 = cosθ ⋅T ()cosθ − cos()()k −1 θ + cos()()k +1 θ . k 2 2

1 1 We have: cos()()k +1 ⋅θ = cosθ ⋅Tk ()cosθ − Tk−1()cosθ + cos()()k +1 θ 2 2

1 1 Collecting like terms: cos()()k +1 θ = cos(θ )⋅T (cosθ )− T ()cosθ 2 k 2 k−1

Multiply both sides by 2.

Tk+1 ()cosθ = cos()(k +1)θ = 2⋅cos(θ )⋅Tk (cosθ )−Tk−1(cosθ ).

If we let cosθ = x we have Txkk+−11()=⋅2x⋅T( x) −Tk( x) , which is the

definition of Chebyshev polynomials.

The Chebyshev polynomials are used in many areas of mathematics. One such area is in polynomial interpolation. Due to their use in interpolation, the roots of

Chebyshev polynomials are sometimes called Chebyshev nodes. There are n simple roots

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⎛ 2i −1 ⎞ in an nth degree Chebyshev polynomial. These roots are found as xi = cos⎜ π ⎟ ⎝ 2n ⎠ where i =1,...,n (Chebyshev Polynomials).

Chebyshev polynomials are also used in the definition of another type of recursive polynomials, the Cardan polynomials.

Cardan Polynomials

The Cardan polynomials are so named for their connection to Cardan’s solution to cubic equations. Girolamo Cardan published this method to find the solution to the cubic equation x3 − 3cx = 2a in his work Ars Magna in 1545. Although he published this method, Cardan does acknowledge the previous work on this problem by another mathematician, Tartaglia of Pisa. It is in this work that Cardan acknowledges the existence of complex numbers, since they are an integral part of the solutions to the cubic equations (O’Connor and Robertson, Cardan).

Osler uses the Cardan polynomials to help solve problems, however, not necessarily just for cubic equations. Osler uses these polynomials in reference to solving radicals of the form n a ± b (Cardan Polynomials).

The first seven Cardan Polynomials:

C1()c, x x 2 C2 ()c, x x − 2c 3 C3 ()c, x x − 3cx 4 2 2 C4 ()c, x x − 4cx + 2c 5 3 2 C5 ()c, x x − 5cx + 5c x 6 4 2 2 3 C6 ()c, x x − 6cx + 9c x − 2c 7 5 2 3 4 C7 ()c, x x − 7cx +14c x − 9c x

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Osler uses two methods to derive these polynomials. The first is a method of symbolic manipulations. The second shows the relationship between Chebyshev polynomials and the Cardan polynomials (Cardan Polynomials).

One approach to deriving the Cardan polynomials makes use of the expression x =+nab+na−b. By defining c = n a + b n a − b = n a2 − b , we can raise x to any power of n and manipulate our result to an expression equal to 2a . This

expression happens to be the Cardan polynomial,Cn (c, x). All Cardan polynomials

satisfy the equation Cn ()c, x = 2a (Osler, Cardan Polynomials).

3 For example, when n = 3, x3 = ⎜⎛ 3 a + b + 3 a − b ⎟⎞ ⎝ ⎠

3 2 2 3 = ⎜⎛ 3 a + b ⎟⎞ + 3⎜⎛ 3 a + b ⎟⎞ ⎜⎛ 3 a − b ⎟⎞ + 3⎜⎛ 3 a + b ⎟⎞⎜⎛ 3 a − b ⎟⎞ + ⎜⎛ 3 a − b ⎟⎞ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

3 3 = ⎜⎛ 3 a + b ⎟⎞ + 3⎜⎛ 3 a + b ⎟⎞⎜⎛ 3 a − b ⎟⎞⎜⎛ 3 a + b + 3 a − b ⎟⎞ + ⎜⎛ 3 a − b ⎟⎞ . ⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠ ⎝ ⎠

Using the definitions of x and c from above, we have x3 = (a + b )+ 3cx + (a − b ) .

Simplifying this statement yields x3 = 2a + 3cx , solving for 2a : 2a = x3 − 3cx . Thus,

3 C3 ()c, x = x − 3cx . (Osler, Cardan Polynomials).

A second approach to deriving the Cardan polynomials is to connect them to the

Chebyshev polynomials. The substitution sum

nnab++a−b=rexp()iθ +rexp(−iθ ) which gives

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()ab++()a−b=rnexp()inθ +rnexp(−inθ ). If we rewrite x (from above)

with this we get: x = n a + b + n a − b = reiθ + re −iθ = 2r cosθ .

r neinθ + r ne−inθ = (a + b )+ (a − b )= 2a .

a Since r neinθ + r n e −inθ = 2r cos(nθ ) , we get cos()nθ = . r n

Recall from that the Chebyshev polynomials that they are defined

⎛ x ⎞ a byTn ()cosθ = cos(nθ ). Also, recall that x = 2r cosθ . We have thatTn ⎜ ⎟ = , solving ⎝ 2r ⎠ r n

n ⎛ x ⎞ n ⎛ x ⎞ for a gives a = r Tn ⎜ ⎟ . SinceCn (c, x) = 2a , we can say thatCn ()c,.x = 2r Tn ⎜ ⎟ To ⎝ 2r ⎠ ⎝ 2r ⎠ eliminate r , we will use the equations c = n a + b n a − b and

nnab++a−b=rexp()iθ +rexp(−iθ ). Combining these two equations we get that c = reiθ re−iθ = r 2 . The result is the Cardan polynomials using their connection with

n 2 ⎛ x ⎞ the Chebyshev polynomials: Cn ()c, x = 2c Tn ⎜ ⎟ (Osler, Cardan Polynomials). ⎝ 2 c ⎠

Cardan polynomials can be used to reduce radicals of the form n a ± b . The first step is to solve the equation b = a2 − cn for c . Then, using the table of Cardan

polynomials, find Cn (c,x). Set Cn (c, x) = 2a and solve for x . This is the reduced form of the radical. If an integer or rational root can be found, then the radical n a ± b can be

x ± x2 − 4c written as (Osler, Cardan Polynomials). 2

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To demonstrate this, we’ll use the radical 3 2 + 5 + 3 2 − 5 . Notice that a = 2 ,b = 5 and n = 3; substituting these values into b = a2 − cn gives5 = 22 − c3 . So,

3 c = −1. The next step is to look up C3 (c, x) = x − 3cx . We need to solve for x by taking x3 − 3cx = 2a ; x3 + 3x = 4 , where x =1. This means that 3 2 + 5 + 3 2 − 5 = 1.

Another example is to simplify 938 ++17 5 938 −17 5 , where a = 38 , b =1445 , and n = 9 . We need to find c : 1445 = 382 − c9 where −1 = c . Substituting

9 7 5 3 a and c into C9 ()c, x = 2a yields the equation x + 9x + 27x + 30x + 9x = 76. The real solution to this equation is x =1. Which means that the radical

938 ++17 5 938 −17 5 can be simplified to 1 (Osler, Cardan Polynomials, 31).

Suppose the process above does not yield a solution. In this case, use a divisor, d of n where n = d ⋅e . Then examine the radical e d a ± b with the largest d first, then with smaller divisors of n . Continue until the expression simplifies or all divisors of n are used (Osler, Cardan Polynomials).

Lucas Polynomials and Cardan Polynomials

Below is a table comparing the Lucas polynomials (left) and the Cardan

Polynomials (right). The similarities and differences can be seen easily.

Lucas Polynomials: Ln (x) n Cardan Polynomials:Cn ()c,x

Ln ()x = x ⋅ Ln−1()x + Ln−2 ()x n ≥ 2 Cn (c, x) = x ⋅Cn−1(c, x)(− Cn−2 c, x) x 1 x x2 + 2 2 x2 − 2c x3 + 3x 3 x3 − 3cx x4 + 4x + 2 4 x4 − 4cx2 + 2c2 x5 + 5x3 + 5x 5 x5 − 5cx3 + 5c2 x x6 + 6x4 + 9x2 + 2 6 x6 − 6cx4 + 9c2 x2 − 2c3

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x7 + 7x5 +14x3 + 7x 7 x7 − 7cx5 +14c2 x3 − 7c3 x x8 + 8x6 + 20x4 +16x2 + 2 8 x8 −8cx6 + 20c2 x4 −16c3 x2 + 2c4

Similarities:

The numerical coefficient and powers of x for each corresponding polynomial are the same. The powers of x are the same because both recursion functions multiply the previous polynomial by x before continuing.

Differences:

Lucas polynomials have only positive coefficients, while the Cardan Polynomials have alternating positive and negative coefficients. If one looks closely, the negative terms in Cardan sequence have factors of c with odd powers, while the terms that have even powers of c are positive. This indicates that there is a link between the factor of c and the sign of the term.

It is easy to predict the power of c for the Cardan polynomial.

Since we know what the absolute value of the coefficient and the power of x for each term (from the Lucas polynomial), the”c ” portion is easy to find. The last term in the Cardan polynomial has the highest power of c within that polynomial. This highest

n n −1 power can be found by taking if n is even, or if n is odd, where n is the degree 2 2 of the Lucas polynomial. The preceding powers of c are determined by subtracting one for each term. The sign of the term (positive or negative) of the term can be determined by the power of c . If the power is even then the sign will be positive, if the power is odd then the term will be negative.

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Take for instance the fifth Lucas Polynomial. To convert this to a Cardan polynomial, we need to include thec ’s and change some of the terms to negative.

Lucas: x5 + 5x3 + 5x Cardan: x5 − 5cx3 + 5c2 x

Convert to Cardan:

Since our polynomial has degree 5, the last term should include c with a power

n −1 of , or 2. Since the power of c is even, this term will be positive ( + 5c2 x ). The 2 preceding term will have c with a power of 2-1. Since 1 is odd, this term will be negative

( − 5cx2 ). If we take 1-1 (the power the second term minus one) we get zero, therefore, there is no power of c ( c does not exist in this term). Our Cardan polynomial becomes x5 − 5cx3 + 5c2 x .

Classroom Applications

Gearing Up

Using patterns and recursive functions in the classroom can be a good way to introduce or reinforce topics in mathematics. Pattern exploration can help students develop understanding of hands-on problems or activities and applications of math skills that they already know or are learning at the time. Exploring recursive functions is just one way that students can explore patterns in algebra.

There are many different recursive functions. I’ve talked about some in my paper, but there are many more to be explored. In an Algebra II classroom recursive functions such as the Cardan Polynomials or Chebyshev Polynomials may be a place to explore, but I personally would use these two functions only as extensions or examples of what recursion looks like beyond the high school classroom. The Fibonacci Polynomials and Lucas Polynomials are more likely what I would use to explore with my students.

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Assuming that students know what a function is, in order to discuss the recursive polynomials, students must understand how to evaluate a composite function. Since we are no longer just substituting numbers into our functions, students tend to lose track of where the variables are supposed to go. For instance, with the function h()x = g()f (x), students can visualize that the f (x) is “inside” the function g(x). However, when we add the information that f ()x = 2x and g(x) = x +1, it gets more complicated to see

WHERE the f (x)goes “inside” g(x). Unfortunately, this concept that one can substitute a new function into an existing function is a very important piece and we spend at least one entire, sometimes two classes working on this alone.

In order to work with recursive functions, students must have already worked with recursive series or sequences. Generally, an Algebra II book dedicates a short chapter or unit to sequences and series. This chapter is usually found toward the end of the book, and most of the time is one of the first pieces of curriculum that is “thrown out” due to the time constraints in a classroom. However, sequences have a very important place in math and I would enjoy having an opportunity to spend more time on them in the classroom.

Most books at this level (Algebra II) devote their pages to only arithmetic and geometric series and sequences. There are times when other sequences are introduced, but these occur in challenge problems in the homework. Students are introduced to sequences with notation, rule writing and having to find a certain number of terms of the sequence. Series notation is also introduced at the beginning of the chapter.

The next things that students are introduced to are the arithmetic and geometric sequences and series. Students are asked to find the common difference (arithmetic) or

Farrell 25 ratio (geometric) of a sequence. They are to use this common difference or ratio to write a rule that can be used to find the n th term in the sequence. They are also shown how to use the common difference or ratio in order to find the sum of the first n terms of the series.

Students are then introduced to recursive functions that may or may not be arithmetic or geometric in nature. It’s generally in this section that students are shown that there are ways to write formulas both recursively (using previous terms to find another term.) and explicitly (using only the number of the term to find the value of the term).

Introducing Fibonacci

One way to introduce students to the Fibonacci series is to start them down the same path that Fibonacci himself took. Let’s start with the Rabbit Problem and the

Honeybee’s Genealogy.

The Rabbit Problem is the classic problem of “how many rabbits do we have after a year” provided that the ideal conditions are met. Students will be asked to walk through each month (0-12) of the one year period recording their trek in two ways. One way will be a pictorial representation, similar to a genealogy chart, for our young rabbit pair and their offspring. The second way to track their data is to use a table to record the number of the month and the number of rabbit pairs. (See handouts 1, 2 and 3)

The Honeybee’s Genealogy activity examines the genealogy of one male honeybee in a colony. Honeybees, as described earlier, have a very interesting family.

There are three types of honeybees. The queen bee, the only female to reproduce, has two parents. The queen bee’s offspring are either male, from an unfertilized egg, or a female,

Farrell 26 from an egg that has been fertilized by a male. Some of the young female bees are fed

“royal jelly”, thus causing them to grow into new queen bees. This strange family history of honeybees makes the family tree of a male honeybee very interesting. The activity is set up to uncover the interesting links between the number of bees in a generation and the

Fibonacci numbers. This activity will be set up in a similar manner as the Rabbit

Problem. (See handouts 4, 5, and 6)

During the activity, the students will be working in groups of three. One person will be assigned to each of the three tasks above. One person will be designated as responsible for reporting back to the class after the activity is finished. Half of the groups will work on the Rabbit Problem, the other half of the groups will work on the

Honeybee’s Genealogy problem.

While the groups are working, some will figure out that a pattern exists in the number of rabbits per month (or honeybees in a generation). Others may not catch the pattern until the full group discussion after the activity is over.

The goal for the activities is to be able to have two completely separate, seemingly unrelated problems, and yet the solutions are completely identical. The pattern itself, the Fibonacci series, is being discovered. The timeline for an activity like this would be one 45 minute class period for the groups to work. A second 45 minute class period would be needed to have one group that worked Rabbit Problem and one group that worked on the Honeybee’s Genealogy present their findings to the class (these groups would be selected randomly). A full group discussion would follow. (See handout

7)

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After the group discussion of the three questions (found on Handout 7), I will go over the significance of this set of numbers, the Fibonacci numbers. I would then discuss that other sequences of numbers, like the Lucas numbers, can be formed by using the same recursive function.

The other activity that I would do with the students deals directly with Fibonacci again. This activity introduces the idea that recursive patterns do not occur in just numbers, but can also occur within polynomials. The Fibonacci polynomials are a set of polynomials that follow a recursive pattern.

The students will be given a list of polynomials (see handout 8) and asked to find a recursive function (written in the correct form) that will find the next polynomial in the sequence. This activity will only take a partial class period. Students should be able to determine the function within a 40 minute class period.

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Works Cited

Anderson, Matt, Jeffrey Frazier, and Kris Popendorf. The Fibonacci Series – The Series

– The Rabbit Problem. 1999. ThinkQuest.com. 16 June 2006.

.

Chebyshev Polynomials. 2006. Wikipedia. 16 June 2006.

.

Clase, Matt. Chebyshev Equation. 2002. PlanetMath.org. 14 June 2006.

.

Cuoco, Albert. “Mathematics as a Way of Thinking about Things.” High School

Mathematics at Work: Essays and Examples for the Education of All Students. 1

Dec. 2006 .

Knott, Ron. The Fibonacci Number and Golden Section in Nature – 1. 2005. University

of Surrey. 20 May 2006.

>.

---. General Fibonacci Series. 2006. University of Surrey. 20 May 2006.

.

---. The Lucas Numbers. 2006. University of Surrey. 20 May 2006.

.

O’Connor, J.J. and E.F. Robertson. Cardan biography. 1998. St. Andrew’s University. 5

May 2006. .

---. Fibonacci biography. 1999. St. Andrew’s University. 5 May 2006.

history.mcs.st-and.ac.uk/Biographies/Fibonacci.html>.

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---. Lucas Biography. 2006. St. Andrew’s University. 5 May 2006.

groups.dcs.st-and.ac.uk/~history/Biographies/Lucas.html>.

Orthogonal polynomials. 2006. Wikipedia.16 June 2006.

.

Osler, Thomas J. “Cardan Polynomials and the Reduction of Radicals.” Mathematics

Magazine. 47 (2001): p26-32.

---. “Variations on a Theme from Pascal’s Triangle.” The College Mathematics Journal.

34 (May 2003): p216-23.

Principles and Standards for School Mathematics. NCTM, 2000.

Sinwell, Benjamin. “The Chebyshev Polynomials: Patterns and Derivation.” Mathematics

Teacher 98 (August 2004): p 20-5.

Smith, Erika. “Stasis and Change: Integrating Patterns, Functions, and Algebra

Throughout the K-12 Curriculum.” A Research Companion to Principles and

Standards for School Mathematics. Ed. Jeremy Kilpatrick, W. Gary Martin, and

Deborah Schifter. NCTM, 2003. 136-50.

Special Cases of Sturm-Liouville Boundary Value Problem. Efunda: Engineering

Fundamentals. 16 June 2006.

.

Weisstein, Eric W. Chebyshev Polynomials of the First Kind. Wolfram Web Resource.

16 June 2006.

.

---. Fibonacci Polynomial. Wolfram Web Resource. 16 June 2006.

.

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---. Golden Ratio.Wolfram Web Resource. 16 June 2006.

.

---. Lucas Number.Wolfram Web Resource. 16 June 2006.

.

---. Lucas Polynomial Sequence. Wolfram Web Resource. 16 June 2006.

.

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Handouts

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Rabbit Problem Handout #1: Rabbit Problem Explanation Sheet

Here’s a question for your group to answer. Along with this explanation sheet, you also have a sheet with a table to be filled in and a sheet that you will fill out with a family-tree for your rabbits.

The Setting:

Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was...

The Question:

How many pairs will there be in one year?

Getting Started:

1. At the end of the first month, they mate, but there is still one only 1 pair.

2. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.

3. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.

4. At the end of the fourth month, the original female produces another pair and the female born two months ago produces her first pair. This makes a total of 5 pairs in the field.

5. At the end of the fifth month, the original female produces another pair…

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Rabbit Problem Table Handout #2: Rabbit Problem Table end of 0 1 2 3 4 month Pairs of 1 1 2 3 5 rabbits

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Rabbit Problem Family Tree Handout #3: Rabbit Problem Family Tree

List pairs of rabbits by the month they were born in.

Month 0 0 Month 1 0 Month 2 2 0 Month 3 2 0 3 Month 4 4 2 0 4 3 Month 5 Month 6 Month 7 Month 8 Month 9 Month 10 Month 11 Month 12

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Handout #4: Honeybee’s Genealogy Explanation Sheet Honeybee’s Genealogy

Here’s a question for your group to answer. Along with this explanation sheet, you also have a sheet with a table to be filled in and a sheet that you will fill out with a family-tree for your honeybees.

The Setting:

Honeybees have different families. Some honeybees have two parents, some have only one. There are three types of honeybees:

1. Drones: male bees that are produced from a Queen’s unfertilized egg. Male honeybees only have ONE parent, a female (the queen bee).

2. Workers: female bees that cannot reproduce, but do all other work for the hive. Female honeybees have TWO parents, a female (a queen bee) and a male.

3. The Queen Bee: female bee that has been fed a special “royal jelly” which produces a queen bee. The queen bee has TWO parents, a female (another queen bee) and a male.

The Question:

If we look at the family tree for a male (drone) honeybee, how many bees are in his family tree 13 generations ago?

Getting Started:

We will look at generations of honeybees.

Generation 1: our male honeybee

Generation 2: He had 1 parent, a female.

Generation 3: He has 2 grand-parents, since his mother had two parents, a male and a female.

Generation 4: He has 3 great-grand-parents: his grand-mother had two parents but his grand-father had only one.

Generation 5: He has 5 great-great-grand-parents…

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Handout #5: Honeybee’s Genealogy Table Honeybee’s Genealogy Table

Generation 1 2 3 4 5

Number of 1 1 2 3 5 Honeybees

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Handout #6: Honeybee’s Genealogy Family Tree Honeybee’s Genealogy Family Tree

Generation M 1 Generation F 2 Generation M F 3 Generation F M F 4 Generation M F F M F 5 Generation

6 Generation

7 Generation

8 Generation

8 Generation

9 Generation

10 Generation

11 Generation

12 Generation

13

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Handout #7: Questions for discussion

Questions for discussion after the Rabbit Problem, Honeybee’s Genealogy Activities

Discussion questions to ask, not necessarily in this order:

1. What do you notice about the tables for these two activities?

2. Is there a pattern? Could we predict the next number?

3. If there is a pattern in either set of numbers, can we write a function that describes

how to find the next number?

The discussion up to this point would be lead by the students.

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Handout #8: Fibonacci Polynomials Activity Polynomial Exploration

Below is a list of polynomials. Your task is to figure out if you can develop a recursive formula to find the next polynomial in the table.

F1(x) 1 x F2 (x) 2 F3 (x) x +1 3 F4 (x) x + 2x 4 2 F5 (x) x + 3x +1 5 3 F6 (x) x + 4x + 3x 6 4 2 F7 (x) x + 5x + 6x +1

F8 (x)