Identities for the Classical Polynomials Through Sums of Liouville Type

Adama Diene and Mohamed El Bachraoui Department of Mathematical Sciences United Arab Emirates University P.O. Box 17551 Al-Ain United Arab Emirates

                       adiene@uaeu.ac.ae
                    melbachraoui@uaeu.ac.ae

Abstract

We prove identities involving Bernoulli, Euler, Fibonacci, Lucas, Chebychev, and Dickson polynomials by using sums of Liouville type.

1 Introduction

Polynomials defined recursively over the integers such as Dickson polynomials, Chebychev polynomials, Fibonacci polynomials, Lucas polynomials, Bernoulli polynomials, Euler polynomials, and many others have been extensively studied in the past. Most of these polynomials have some type of relationship between them and share a large number of interesting properties. They have been also found to be topics of interest in many different areas of pure and applied sciences. Most recently, some of these families of polynomials have been found to be useful in cryptography and related topics, which keep making them a very interesting area of research for many people in this era of communication. In this paper, we use a sum of Liouiville type to prove new properties concerning many of these families of polynomials. The following notations and definitions will be adopted throughout this paper.

The sets of positive integers, nonnegative integers, integers, real numbers, and complex numbers are respectively denoted by ℕ, ℕ_{0}, ℤ, ℝ, and ℂ. The sets of even positive integers and the set of odd positive integers will be written 2ℕ and 2ℕ − 1 respectively and the sets of even integers and odd integers will be written 2ℤ and 2ℤ − 1 respectively. The Euler phi function will be denoted by ϕ(n) and the sum of the kth powers of positive divisors of n will denoted by σ_{k}(n).

Definition 1. For n ∈ ℕ such that n > 1, let the sets B(n) and B′(n) be defined as follows:

B(n) = {(a, b, u, v) ∈ ℕ^{4} : au + bv = n},
B′(n) = {(a, b, u, v) ∈ ℕ^{4} : au + bv = n, gcd(a, b) = gcd(u, v) = 1},

and for n ∈ 2ℕ let the sets O(n) and O′(n) be defined as follows:

O(n) = {(a, b, u, v) ∈ (2ℕ − 1)^{4} : au + bv = n},
O′(n) = {(a, b, u, v) ∈ (2ℕ − 1)^{4} : au + bv = n, gcd(a, b) = gcd(u, v) = 1}.

Arithmetic sums of the types

∑_{(a,b,u,v)∈B(n)}(f(a − b) − f(a + b)) and ∑_{(a,b,u,v)∈O(n)}(f(a − b) − f(a + b))  (1)

were first investigated by Liouville in his seminal work on elementary methods in number theory, see for instance Liouville [8, 9]. A comprehensive treatment of Liouville’s methods has been given in a new book by Williams [12]. Adapting the proof of Williams [12] for the sums (1), El Bachraoui [3] found the following related identities for the sums over the sets B′(n) and O′(n).

Theorem 2. Let n > 1 be a positive integer.

(a) If f : ℤ → ℂ is an even function, then

∑_{(a,b,u,v)∈B′(n)} (f(a − b) − f(a + b)) = (f(0) + 2f(1) − f(n))ϕ(n) − 2 ∑_{\begin{matrix}1≤l<n \\ (l,n)=1 \end{matrix}} f(l).

(b) If moreover n ∈ 2ℕ, then

∑_{(a,b,u,v)∈O′(n)} (f(a − b) − f(a + b)) = (f(0) − f(n))ϕ(n).

In this note we shall apply Theorem 2 to deduce identities involving Bernoulli, Euler, Lucas, Fibonacci, Chebychev, and Dickson polynomials.

2 Preliminaries

Let (B_{n})_{n=0}^{∞} be the sequence of Bernoulli numbers for which we have B_{1} = −\frac{1}{2}, B_{2n+1} = 0 for any n ∈ ℕ, and the first few terms for even n are

B_{0} = 1, B_{2} = \frac{1}{6}, B_{4} = −\frac{1}{30}, B_{6} = \frac{1}{42}, B_{8} = −\frac{1}{30}, B_{10} = \frac{5}{66}.

Further we have the recursive formula

B_{n} = ∑_{l=0}^{n} \binom{n}{l}B_{l}, (n ≥ 2).  (2)

Definition 3. If n ∈ ℕ, then the Bernoulli polynomial is given by

B_{n}(x) = ∑_{l=0}^{n} \binom{n}{l}B_{l}x^{n−l} = nB_{1}x^{n−1} + ∑_{l=0}^{⌊ \frac{n}{2} ⌋} \binom{n}{2l}B_{2l}x^{n−2l}.

A well-known property of Bernoulli polynomial is

B_{n}(x + 1) − B_{n}(x) = nx^{n−1}, (n ≥ 1)

which by iteration yields

B_{n}(x + m) − B_{n}(x) = n ∑_{l=0}^{m−1}(x + l)^{n−1}, (m, n ≥ 1).  (3)

Let (E_{n})_{n=0}^{∞} be the sequence of Euler numbers. We have E_{2n−1} = 0 for are all n ∈ ℕ_{0} and the first few terms for even n are

E_{0} = 1, E_{2} = −1, E_{4} = 5, E_{6} = −61, E_{8} = 1385, E_{10} = −50521, E_{12} = 2702765.

Definition 4. For n ∈ ℕ_{0} the Euler polynomial is given by

E_{n}(x) = ∑_{l=0}^{n} \binom{n}{l}\frac{E_{l}}{2^{l}} · (x − \frac{1}{2})^{n−l} = ∑_{l=0}^{⌊ \frac{n}{2} ⌋} \binom{n}{2l}\frac{E_{2l}}{4^{l}} · (x − \frac{1}{2})^{n−2l} .

A well-known property of E_{n}(x) is

E_{n}(x + 1) − E_{n}(x) = 2(x^{n} − E_{n}(x)),

which by iteration yields

E_{n}(x + m) − E_{n}(x) = 2 ∑_{l=0}^{m−1}((x + l)^{n} − E_{n}(x + l)).  (4)

Definition 5. For n ∈ ℕ, the Dickson polynomials of the first kind and of the second kind of degree n with parameter a are respectively defined as follows:

D_{n}(x, a) = ∑_{j=1}^{⌊n/2⌋} \frac{k}{k − j} \binom{n − j}{j}(−a)^{j}x^{n−2j}, and D_{0}(x, a) = 2.

D_{n}^{(2)}(x, a) = ∑_{j=1}^{⌊n/2⌋} \binom{n − j}{j}(−a)^{j}x^{n−2j}, and D_{0}^{(2)}(x, a) = 1.

The first few Dickson polynomials of the first and of the second kind of degree n with parameter a are respectively:

D_{0}(x) = 2, D_{1}(x) = x, D_{2}(x) = x^{2} − 2a, D_{3}(x) = x^{3} − 3xa,

D_{4}(x) = x^{4} − 4x^{2}a + 2a^{2}, D_{5}(x) = x^{5} − 5x^{3}a + 5xa^{2},
D_{0}^{(2)}(x) = 1, D_{1}^{(2)}(x) = x, D_{2}^{(2)}(x) = x^{2} − a, D_{3}^{(2)}(x) = x^{3} − 2xa,
D_{4}^{(2)}(x) = x^{4} − 3x^{2}a + a^{2}, D_{5}^{(2)}(x) = x^{5} − 4x^{3}a + 3xa^{2}.

Dickson polynomials can also be defined recursively and the definition extended to all ℤ as follow:

D_{0}(x, a) = 2, D_{1}(x, a) = x, D_{n+1}(x, a) = xD_{n}(x, a) − aD_{n−1}(x, a)
D_{0}^{(2)}(x, a) = 1, D_{1}^{(2)}(x, a) = x, D_{n+1}^{(2)}(x, a) = xD_{n}^{(2)}(x, a) − aD_{n−1}^{(2)}(x, a)

Dickson polynomials are closely related to some well-known polynomials like the first and the second Chebychev polynomials T_{n}(x) and U_{n}(x) defined as follows:

T_{n}(x) = \begin{cases}1, & if n = 0; \\ x, & if n = 1; \\ 2xT_{n−1}(x) + T_{n−2}(x), & if n ≥ 2; \end{cases} and T_{−n}(x) = (−1)^{n}T_{n}(x),

U_{n}(x) = \begin{cases}1, & if n = 0; \\ 2x, & if n = 1; \\ 2xU_{n−1}(x) + U_{n−2}(x), & if n ≥ 2; \end{cases} and U_{−n}(x) = (−1)^{n}U_{n}(x).

Dickson polynomials are also related to Lucas polynomials L_{n}(x) and the Fibonacci polynomials F_{n}(x), see for instance Piero [10]. These are defined as follows:

Definition 6. For n ∈ ℕ,

L_{n}(x) = ∑_{j=0}^{⌊k/2⌋} \frac{k}{k − j} \binom{k − j}{j}x^{j}

F_{n+1}(x) = ∑_{j=1}^{⌊k/2⌋} \binom{k − j}{j}x^{j}

Like the Dickson polynomials, these two classes of polynomials can also be defined recursively in ℤ as follow:

L_{n}(x) = \begin{cases}2, & if n = 0; \\ x, & if n = 1; \\ xL_{n−1}(x) + L_{n−2}(x), & if n ≥ 2; \end{cases} and L_{−n}(x) = (−1)^{n}L_{n}(x),

F_{n}(x) = \begin{cases}0, & if n = 0; \\ 1, & if n = 1; \\ xF_{n−1}(x) + F_{n−2}(x), & if n ≥ 2; \end{cases} and F_{−n}(x) = (−1)^{n−1}F_{n}(x).

The first few Lucas and Fibonacci polynomials are

L_{0}(x) = 2, L_{1}(x) = x, L_{2}(x) = x^{2} + 2, L_{3}(x) = x^{3} + 3x,

L_{4}(x) = x^{4} + 4x^{2} + 2, L_{5}(x) = x^{5} + 5x^{3} + 5x.
F_{0}(x) = 0, F_{1}(x) = 1, F_{2}(x) = x, F_{3}(x) = x^{2} + 1,
F_{4}(x) = x^{3} + 2x, F_{5}(x) = x^{4} + 3x^{2} + 1.

Note that (L_{n}(1))_{n=0}^{∞} is the Lucas sequence (L_{n})_{n=0}^{∞} and that (F_{n}(1))_{n=0}^{∞} is the Fibonacci sequence (F_{n})_{n=0}^{∞}. It is important for the current purposes to note that Dickson polynomials D_{n}(x, a) and Lucas polynomials L_{n}(x) are even functions for even n and odd functions for odd n, whereas Fibonacci polynomials F_{n}(x) are even functions for odd n and odd functions for even n. For the Lucas and Fibonacci polynomials, it is well-known that for m, n ∈ ℤ,

L_{m+n}(x) = L_{m}(x)L_{n}(x) − (−1)^{n}L_{m−n}(x),  (5)
F_{m+n}(x) = F_{m}(x)F_{n}(x) + F_{m−1}(x)F_{n−1}(x).  (6)

In particular, if m, n ∈ 2ℕ − 1, then the relation (6) implies

F_{m+n+1}(x) − F_{m−n+1}(x) = F_{m+1}(x) (F_{n+1}(x) + F_{n−1}(x)) .  (7)

Similar properties can be easily proved by induction for the Dickson polynomials. More precisely, for n ∈ ℤ, we have

D_{m+n}(x, a) = D_{m}(x, a)D_{n}(x, a) − aD_{m−n}(x, a),  (8)
D_{m+n}^{(2)}(x, a) = D_{m}^{(2)}(x, a)D_{n}^{(2)}(x, a) − aD_{m−1}^{(2)^{2}}(x, a)D_{n−1}^{(2)^{2}}(x, a),  (9)

which imply

D_{2n}(x, a) = D_{n}^{2}(x, a) − 2.  (10)
D_{2n}^{(2)}(x, a) = D_{n}^{(2)}(x, a) − aD_{n−1}^{(2)^{2}}(x, a).  (11)

Further for any n ∈ ℤ, we have:

D_{n}(2xa, a^{2}) = 2a^{n}T_{n}(x) and D_{n}^{(2)}(2xa; a^{2}) = a^{n}U_{n}(x),  (12)

in particular

D_{n}(x; 1) = 2T_{n}(x/2) and D_{n}^{(2)}(x; 1) = U_{n}(x/2).  (13)

Many other interesting properties about Dickson polynomials can be found in Lidl, Mullen, and Turnwald [6].

3 An identity involving Bernoulli polynomials

Theorem 7. If n ∈ 2ℕ, then

∑_{(a,b,u,v)∈O′(n)} (∑_{l=0}^{2b−1}(a − b + l)^{n−1} + \frac{(a + b)^{n−1} − (a − b)^{n−1}}{2} )

= (∑_{l=0}^{n−1} \binom{n}{l}B_{l}n^{n−l−1} + \frac{1}{2}n^{n}) \frac{ϕ(n)}{n}.

Proof. Let n ∈ 2ℕ and let f : ℤ → ℂ be defined by:

f(k) = B_{n}(k) − nB_{1}k^{n−1} = B_{n}(k) + \frac{n}{2}k^{n−1} = ∑_{l=0}^{\frac{n}{2}} \binom{n}{2l}B_{2l}k^{n−2l}.

Then clearly f is an even function and thus by Theorem 2 we find

∑_{(a,b,u,v)∈O′(n)} ((B_{n}(a + b) − B_{n}(a − b)) + \frac{n}{2}(a + b)^{n−1} − \frac{n}{2}(a − b)^{n−1})  (14)

= (B_{n}(n) + \frac{n}{2}n^{n−1} − B_{n}(0))ϕ(n) = (B_{n}(n) + \frac{n^{n}}{2} − B_{n})ϕ(n).

By the relations (2) we have

B_{n}(n) − B_{n} = ∑_{l=0}^{n} \binom{n}{l}B_{l}n^{n−l} − B_{n} = n ∑_{l=0}^{n−1} \binom{n}{l}B_{l}n^{n−l−1}

and by the relation (3) we have

B_{n}(a + b) − B_{n}(a − b) = B_{n}(a − b + 2b) − B_{n}(a − b) = n ∑_{l=0}^{2b−1}(a − b + l)^{n−1}.

Now put in (14) and divide by n to obtain the desired identity. Remark 8. If n ∈ 2ℕ, then by Theorem 2 applied to the function f(k) = k^{n} we have

∑_{(a,b,u,v)∈O′(n)}((a + b)^{n} − (a − b)^{n}) = n^{n}ϕ(n) (n ∈ 2ℕ).

On the other hand, if n ∈ 2ℕ, we do not know an evaluation for the sum

∑_{(a,b,u,v)∈O′(n)}((a + b)^{n−1} − (a − b)^{n−1})

which appears in the left hand side of the formula in Theorem 7.

4 An identity for Euler polynomials

Theorem 9. If n ∈ 2ℕ, then

∑_{(a,b,u,v)∈O′(n)} (∑_{l=0}^{2b−1}(a − b + \frac{1}{2} + l)^{n} − E_{n}(a − b + \frac{1}{2} + l))

= ϕ(n) ∑_{l=0}^{n−1}((\frac{1}{2} + l)^{n} − E_{n}(\frac{1}{2} + l)).

Proof. Let n ∈ 2ℕ and let f : ℤ → ℂ be defined as follows:

f(k) = E_{n}(k + \frac{1}{2}) = ∑_{l=0}^{\frac{n}{2}} \binom{n}{2l}\frac{E_{2l}}{4^{l}} k^{n−2l}.

Then clearly f is an even function and so by Theorem 2 we get

∑_{(a,b,u,v)∈O′(n)} (E_{n}(a + b + \frac{1}{2}) − E_{n}(a − b + \frac{1}{2}))  (15)

= (E_{n}(n + \frac{1}{2}) − E_{n}(\frac{1}{2})) ϕ(n).

By the property (4) we have

E_{n}(a + b + 1/2) − E_{n}(a − b + 1/2) = E_{n}(a − b + 1/2 + 2b) − E_{n}(a − b + 1/2)

= 2 (∑_{l=0}^{2b−1}(a − b + 1/2 + l)^{n} − E_{n}(a − b + 1/2 + l))

and

E_{n}(n + 1/2) − E_{n}(1/2) = 2 ∑_{l=0}^{n−1}((1/2 + l)^{n} − E_{n}(1/2 + l)),

which put in (15) yield the desired formula.

5 Identities for Lucas polynomials

Theorem 10. If n ∈ 2ℕ and x ∈ ℝ, then

∑_{(a,b,u,v)∈O′(n)} L_{a}(x)L_{b}(x) = ϕ(n)(L_{n}(x) − 2).

Proof. Let x ∈ ℝ and let the function f(k) be defined as follows:

f(k) = \begin{cases}0, & if k ∈ 2ℤ − 1; \\ L_{k}(x), & if k ∈ 2ℤ. \end{cases}

Clearly f is an even function. Then application of Theorem 2 to the function f yields

∑_{(a,b,u,v)∈O′(n)} (L_{a+b}(x) − L_{a−b}(x)) = (L_{n}(x) − L_{0}(x)) ϕ(n)  (16)
= (L_{n}(x) − 2)ϕ(n),

which by virtue of the relation (5) and the fact that L_{0}(x) = 2 gives

∑_{(a,b,u,v)∈O′(n)} (L_{a}(x)L_{b}(x) − (−1)^{b}L_{a−b}(x) − L_{a−b}(x)) = (L_{n}(x) − 2) ϕ(n),

and the desired formula follows since (−1)^{b} = −1.

Taking x = 1 in Theorem 10 we have the following result on Lucas numbers.

Corollary 11. If n ∈ 2ℕ, then

∑_{(a,b,u,v)∈O′(n)} L_{a}L_{b} = ϕ(n)(L_{n} − 2).

Theorem 12. If n ∈ 2ℕ, then

∑_{(a,b,u,v)∈O′(n)}(L_{a+b}(a + b) − L_{a−b}(a − b)) = ∑_{(a,b,u,v)∈O′(n)}(L_{n}(a + b) − L_{n}(a − b))

= ∑_{(a,b,u,v)∈O′(n)}(L_{a+b}(n) − L_{a−b}(n)).

Proof. Note first that by formula (16) the rightmost sum equals

(L_{n}(n) − 2)ϕ(n).

Next let f(k) = L_{k}(−k) and g(k) = L_{n}(k). Then it is easily verified that both f and g are even functions. Thus by Theorem 2 applied to f and g we find respectively

∑_{(a,b,u,v)∈O′(n)}(L_{a+b}(−a − b) − L_{a−b}(b − a)) = (L_{n}(n) − L_{0}(0))ϕ(n)

and

∑_{(a,b,u,v)∈O′(n)}(L_{n}(a + b) − L_{n}(a − b)) = (L_{n}(n) − L_{n}(0))ϕ(n).

Now combine the previous two relations with the facts that L_{n}(0) = L_{0}(0) = 2, L_{a+b}(−a − b) = L_{a+b}(a + b), and L_{a−b}(b − a) = L_{a−b}(a − b) to deduce the equality of the three sums.

6 Identities for Fibonacci polynomials

Theorem 13. Let n ∈ 2ℕ and x ∈ ℝ. Then

∑_{(a,b,u,v)∈O′(n)}(F_{a+1}(x) (F_{b+1}(x) + F_{b−1}(x))) = ϕ(n)(F_{n+1}(x) − 1).

Proof. Note first that if f is even, then

∑_{(a,b,u,v)∈O′(n)} (f(a + b) − f(a − b))

= (f(2) − f(0)) ϕ(n) + 2 ∑_{\begin{matrix}(a,b,u,v)∈O′(n) \\ a>b \end{matrix}} (f(a + b) − f(a − b)) . (17)

Let f be the function defined by:

f(k) = \begin{cases}0, & if k ∈ 2ℤ − 1; \\ F_{k+1}(x), & if 0 ≤ k ∈ 2ℤ; \\ F_{k−1}(x), & otherwise. \end{cases}

It is easily verified that f is an even function. Then by Theorem 2 we obtain

∑_{(a,b,u,v)∈O′(n)} (F_{a+b+1}(x) − F_{a−b+1}(x))  (18)
= ϕ(n)(F_{n+1}(x) − F_{1}(x)) = ϕ(n)(F_{n+1}(x) − 1).

By virtue of the relation (17) the previous formula is equivalent to

(F_{3}(x) − F_{1}(x)) ϕ(n) + 2 ∑_{\begin{matrix}(a,b,u,v)∈O′(n) \\ a>b \end{matrix}} (F_{a+b+1}(x) − F_{a−b+1}(x))

= (F_{n+1}(x) − F_{1}(x)) ϕ(n).

That is,

∑_{\begin{matrix}(a,b,u,v)∈O′(n) \\ a>b \end{matrix}} (F_{a+b+1}(x) − F_{a−b+1}(x)) = \frac{F_{n+1}(x) − F_{3}(x)}{2}ϕ(n),

which by identity (6) and the fact that F_{3}(x) = x^{2} + 1 gives

∑_{\begin{matrix}(a,b,u,v)∈O′(n) \\ a>b \end{matrix}}(F_{a+1}(x)(F_{b+1}(x) + F_{b−1}(x))) = \frac{F_{n+1}(x) − x^{2} − 1}{2}ϕ(n).

Now use the previous formula together with the facts that F_{0}(x) = 0, F_{2}(x) = x, and

∑_{(a,b,u,v)∈O′(n)} (F_{a+1}(x)(F_{b+1}(x) + F_{b−1}(x))) = F_{2}(x) (F_{2}(x) + F_{0}(x)) ϕ(n)

+ 2 ∑_{\begin{matrix}(a,b,u,v)∈O′(n) \\ a>b \end{matrix}} (F_{a+1}(x)(F_{b+1}(x) + F_{b−1}(x)))

to derive the result.

Taking x = 1 in Theorem 13 we have the following identity involving Fibonacci numbers. Corollary 14. If n ∈ 2ℕ, then

∑_{(a,b,u,v)∈O′(n)} (F_{a+1}(F_{b+1} + F_{b−1})) = ϕ(n)(F_{n+1} − 1).

Theorem 15. Let n ∈ 2ℕ. Then we have

(a) ∑_{(a,b,u,v)∈O′(n)}(F_{a+b}(a + b) − F_{a−b}(a − b)) = F_{n}(n)ϕ(n).

(b) ∑_{(a,b,u,v)∈O′(n)}(F_{n+1}(a + b) − F_{n+1}(a − b)) = ∑_{(a,b,u,v)∈O′(n)}(F_{a+b+1}(n) − F_{a−b+1}(n)).

Proof. To prove part (a) apply Theorem 2 to the even functions f(k) = F_{k}(k). As to part (b) note first that by the relation (18) the sum on the right equals

(F_{n+1}(n) − 1)ϕ(n).

Next by Theorem 2 applied to the even function g(k) = F_{n+1}(k) we have

∑_{(a,b,u,v)∈O′(n)}(F_{n+1}(a + b) − F_{n+1}(a − b)) = (F_{n+1}(n) − F_{n+1}(0))ϕ(n) = (F_{n+1}(n) − 1)ϕ(n).

This completes the proof.

7 Identities involving Dickson polynomials

To simplify the notation in this section, we write D_{n}(x) for D_{n}(x, 1). Theorem 16. Let n ∈ ℕ and x ∈ ℝ. Then

∑_{(a,b,u,v)∈O′(n)}(D_{a}(x)D_{b}(x))(D_{a}(x)D_{b}(x) − 2D_{a−b}(x)) = ϕ(n)(D_{2n}(x) − 2).

Proof. Clearly,

D_{2n}(x) = D_{n}^{2}(x) − 2

is an even function. Therefore if we define f on ℤ by

f(k) = \begin{cases}0, & if k ≤ 0; \\ D_{2k}(x), & if k > 0; \end{cases}

then f is also even. By Theorem 2 we have

∑_{(a,b,u,v)∈O′(n)} f(a + b) − f(a − b) = (f(n) − f(0))ϕ(n),

or equivalently,

∑_{(a,b,u,v)∈O′(n)} (D_{2(a+b)}(x) − D_{2(a−b)}(x)) = (D_{2n}(x) − D_{0}(x))ϕ(n),

which by identity (10) and the fact that D_{0}(x) = 2 gives

∑_{(a,b,u,v)∈O′(n)}(D_{a+b}^{2}(x) − D_{a−b}^{2}(x)) = (D_{2n}(x) − 2)ϕ(n).
∑_{(a,b,u,v)∈O′(n)}(D_{a+b}(x) + D_{a−b}(x))(D_{a+1}(x) − D_{a−b}(x)) = (D_{2n}(x) − 2)ϕ(n).

Now application of the formula (8) yields

(D_{a+b}(x) + D_{a−b}(x)) = D_{a}(x)D_{b}(x),

and

(D_{a+b}(x) − D_{a−b}(x)) = D_{a}(x)D_{b}(x) − 2D_{a−b}(x).

That is,

∑_{(a,b,u,v)∈O′(n)}(D_{a+b}(x) + D_{a−b}(x))(D_{a+1}(x) − D_{a−b}(x)) = (D_{2n}(x) − 2)ϕ(n) or
∑_{(a,b,u,v)∈O′(n)}(D_{a}(x)D_{b}(x))(D_{a}(x)D_{b}(x) − 2D_{a−b}(x)) = ϕ(n)(D_{2n}(x) − 2),

which completes the proof. Theorem 17. Let n ∈ 2ℕ and x ∈ ℝ. Then

∑_{(a,b,u,v)∈O′(n)}(D_{a}(x)D_{b}(x) − 2D_{a−b}(x))(D_{a}(x)D_{b}(x) − 1)) = ϕ(n)(D_{2n}(x) − D_{n}(x)).

Proof. Using the same argument as in the previous theorem with

g(k) = \begin{cases}0, & if k ∈ 2ℤ − 1; \\ D_{k}(x), & if k ∈ 2ℤ; \end{cases}

we obtain:

∑_{(a,b,u,v)∈O′(n)} (D_{(a+b)}(x) − D_{(a−b)}(x)) = (D_{n}(x) − D_{0}(x))ϕ(n),

or equivalently

∑_{(a,b,u,v)∈O′(n)}(D_{a}(x)D_{b}(x) − 2D_{a−b}(x)) = ϕ(n)(D_{n}(x) − 2).

Combining with the result of Theorem 16 we obtain:

∑_{(a,b,u,v)∈O′(n)}(D_{a}(x)D_{b}(x))(D_{a}(x)D_{b}(x) − 2D_{a−b}(x))

+ ∑_{(a,b,u,v)∈O′(n)}(D_{a}(x)D_{b}(x) − 2D_{a−b}(x))

= ϕ(n)(D_{2n}(x) − 2) + ϕ(n)(D_{n}(x) − 2).

or equivalently

∑_{(a,b,u,v)∈O′(n)}(D_{a}(x)D_{b}(x) − 2D_{a−b}(x))(D_{a}(x)D_{b}(x) − 1)) = ϕ(n)(D_{2n}(x) − D_{0}(x)).

As a corollary, we have

Corollary 18. Let n ∈ 2ℕ and x ∈ ℝ. Then

∑_{(a,b,u,v)∈O′(n)}(T_{a}(x)T_{b}(x) − 2T_{a−b}(x))(T_{a}(x)T_{b}(x) − 1)) = \frac{1}{2}ϕ(n)(T_{2n}(x) − T_{n}(x)).

Proof. The proof is immediate from the previous theorem and relation (13).

Remark 19. Similar identities can be proven for the Dickson polynomials of the second kind.

8 Concluding remarks

It should be noticed that in deducing our formulas involving the classical polynomials, we restricted ourselves to sums over the set O′(n) as the formulas we obtain when considering sums over the sets B(n), O(n), and B′(n) are not easy to simplify. However, under some restrictions on the positive integer n one can get to nice identities. For our example in this matter we need the following particular case of a sum of Liouville [8] (Problem 13 of Chapter 10 in Williams [12]). If N is twice an odd positive integer, then

∑_{(a,b,x,y)∈O(N)}(f(a + b) − f(a − b)) = −f(0)σ(N/2) + ∑_{d|N/2} df(2d).  (19)

Let N be twice an odd positive integer. Using relation (19) one can show that the analogue of Theorem 7 is

∑_{(a,b,u,v)∈O(N)} (∑_{l=0}^{2b−1}(a − b + l)^{N−1} + \frac{(a + b)^{N−1} − (a − b)^{N−1}}{2} )

= 2^{N−2}σ_{N}(N/2) − \frac{B_{N}}{N} σ(N/2) + \frac{1}{N} ∑_{d|N/2} dB_{N}(2d).

9 Acknowledgment

The authors are grateful to the referee for valuable comments and interesting suggestions.

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2010 Mathematics Subject Classification: Primary 11B39; Secondary 11B68, 11B83, 11A25, 11B75. Keywords: Liouville sum, Bernoulli polynomial, Chebychev polynomial, Dickson polynomial, Euler polynomial, Fibonacci polynomial.

Received June 3 2012; revised version received July 25 2012. Published in Journal of Integer Sequences, August 5 2012.

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