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Balancing polynomials, Fibonacci numbers and some new series for π

07/13/2022
by   Robert Frontczak, et al.
0

We evaluate some types of infinite series with balancing and Lucas-balancing polynomials in closed form. These evaluations will lead to some new curious series for π involving Fibonacci and Lucas numbers. Our findings complement those of Castellanos from 1986 and 1989.

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1 Motivation and Preliminaries

Castellanos [1, 2] has found, among other things, the following curious series for :

(1)

When the expression simplifies to ([1, Eq. (46)] and [2, Eq. (3.4)])

which has the equivalent form

(2)

Another series involving the squares of is ([2, Eqs. (3.2) and (3.3)])

(3)

with

which, for , simplifies to

(4)

In the identities above, (respectively ) are the famous Fibonacci (Lucas) numbers, defined for by the recursion () with initial conditions ( and ), and is the golden ratio. Castellanos’ series pose the question, whether more series of these types exist? Here, we give a positive answer to this question. Working with balancing and Lucas-balancing polynomials we derive more series for involving Fibonacci and Lucas numbers and exhibiting such a structure.

Recall that, for any integer and , balancing polynomials and Lucas-balancing polynomials are defined by the second-order homogeneous linear recurrence [4]

(5)

but with different initial terms. Balancing polynomials start with and , while for Lucas-balancing polynomials we set , . These polynomials have been introduced as a natural extension of the popular balancing and Lucas-balancing numbers and , respectively, and must be seen as a special member of the Horadam sequence. Obviously, and . The first few polynomials are

and

The closed forms, known as Binet’s formulas, for balancing and Lucas-balancing polynomials are given by

(6)

where and . We note the following properties: and for . The polynomials possess a simple connection to Chebyshev polynomials [4] via

where and are Chebyshev polynomials of the first and second kind, respectively. Some other interesting properties have been discovered in the articles [5, 6, 7, 8, 9]. Balancing and Lucas-balancing polynomials are also linked in various ways to Fibonacci and Lucas numbers. Two such connections, which will be exploited in the text, are

(7)

and

(8)

These relations can be deduced from the corresponding relations between Fibonacci numbers and Chebyshev polynomials (see [2] and [4]).

In the present article, we study special infinite series involving and , respectively. We evaluate these series in closed form. Based on these results, new infinite series for with Fibonacci and Lucas numbers will be stated as an immediate consequence. These new and curious series must be seen as complements of Castellanos’ results from 1986 and 1989.

2 Complementing the work of Castellanos (Part 1)

The next theorem is our starting point.

Theorem 1.

For each real with and with , we have

(9)

and

(10)
Proof.

Recall the Taylor series for the arctangent function

(11)

Hence, for all and with ,

where in the last step we applied the identity

(12)

This proves the first identity. The proof of the second identity is very similar and omitted. ∎

Corollary 2.

The following series representation for holds

(13)
Remark 3.

Observe the similarity of equation (13) to Castellanos’ series (4). To be particular, the two series provide an interesting example for series whose members’ denominators and the sum are the same but the numerators are different. Namely,

Such series seem to be rare. Another example given by Mező [10] (and involving Lucas numbers) are the series

2.1 Series with even Fibonacci (Lucas) coefficients

Setting , we first note that

with and where we have used that . We can now use (7) to obtain the relations, valid for all ,

(14)

and

(15)

Especially, for ,

(16)

and

(17)

Next, from we see that

Hence, inserting , we also get the relations

(18)

and

(19)

where we have used the Catalan identities

To derive Castellanos-like expressions for with Fibonacci (Lucas) coefficients, we use (14) and (15), and relate them to special arctan arguments. We start with

Solving gives

But it can be checked easily that for all

with and as . Similarly, solving the equation gives

and

Note also that while as . Hence, inserting in (15) and simplifying proves the next result.

Theorem 4.

For each integer the following expression for is valid

(20)

Equation (20) is the first Lucas number counterpart of (1). It is worth to note, that the case yields

which can be obtained directly from (11) with and using the arctan identity

When , then

In a similar manner, working with other arguments of the arctan function, we can derive the following presumably new series for involving even Lucas numbers.

Theorem 5.

For each integer the following expressions for are valid

(21)
(22)

and

(23)

2.2 Series with odd Fibonacci (Lucas) coefficients

Setting , we observe that

where again we have used . Proceeding as before, inserting the value of into (9) and (10) we obtain the following expressions via (8), valid for all ,

(24)

and

(25)

Especially, for ,

(26)

and

(27)

Once more, deriving Castellanos-like expressions for is fairly simple. We start with

Solving gives

But it can be checked easily that for all . Inserting the value in (24) gives Castellanos’ series (1). Similarly, solving the equation gives

We note that, for the zeros are complex numbers with . For , the zeros are real and . Hence, no such series for will exist.

In the same manner other arctan arguments can be analyzed. Interestingly, the careful analysis shows that no series involving odd Lucas numbers will come to appearance. The next theorem contains additional odd-indexed Fibonacci series for

that we found.

Theorem 6.

For each integer the following expressions for are valid

(28)
(29)

and

(30)

The special case of (28) when appears in one of Castellanos’ papers [1, Eq. (48)]. The generalization and the other two series seem to be new.

3 Complementing the work of Castellanos (Part 2)

Replacing in (9) by and , respectively, and combining the terms according to the Binet form gives

Now, we can apply (12) and simplify to end with the following identity involving squares of balancing polynomials

(31)

Similarly, the same replacement in (10) gives

(32)

To get a series for, say, we set in (31) and are left with the quadric

(33)

The roots are

None of the roots satisfies the condition for convergence and therefore we can conclude that no such series with exists. Setting in the above Lucas-balancing identity we obtain

where is a positive root of the polynomial equation

(34)

Here, also all four roots of equation (34) are real

and the only (biggest) root that satisfies the condition is

Hence,

(35)

From (31) and (32), with , we arrive at

(36)

and

(37)

To get a new Castellanos’ type series for it is necessary to study the equations

(38)

and

(39)

Note that, if in (38) we set then the quartic takes the form

(40)

The four roots are given by

(41)

and

(42)

Finally, if we set and then

Among the four roots we choose the biggest root that satisfies the condition for convergence and arrive at the main theorem.

Theorem 7.

For each we have the following series for involving squared even-indexed Lucas numbers

(43)

with

(44)

Theorem 7 is a wonderful Lucas analogue of Castellanos’ equation (3). For , and

(45)

For , and

(46)

The analysis of the quartic (39) is very similar. With the relevant equation becomes

(47)

The four roots are given by

and

These roots can be expressed as

with and . For and we have only complex roots: