Abstract

We present some general formulas related to sum of powers, also with alternating sign, involving Lucas functions sequences. In particular, our formulas give a synthesis of various identities involving sum of powers of well-known polynomial sequences such as Fibonacci, Lucas, Pell, Jacobsthal, and Chebyshev polynomials. Finally, we point out some interesting divisibility properties between polynomials arising from our results.

1. Introduction

The spirit of this paper is to develop the interesting ideas of Horadam presented in [1, 2], finding some general formulas for special sum of powers (also with alternating signs) related to a wide class of functions and in particular to some important classes of well-known polynomials. We will use a slight different notation with respect to Horadam.

Definition 1. One defines the Lucas functions sequence as the sequence of functions satisfying the recurrence relation where is an integer and are polynomials such that One supposes that all the functions in (1) and (2) are well defined for suitable values of the variable .

Clearly from the previous definition, we have and, in particular, when is a positive integer, the sequence (1) is a polynomial sequence. It is worth noting that very important families of polynomials satisfy the recurrence (1), such as Chebyshev, Fibonacci, Lucas, and Jacobsthal polynomials as Horadam pointed out in [1]. For example, considering , we retrieve for the sequence and for the sequence involving Chebyshev polynomials of second and first kind, respectively.

In the next section, we present our results on power sums of the kind with , , integers, , and , positive integers. Finally we discuss interesting consequences of our formulas, related to some divisibility properties for polynomials obtained generalizing the so-called Melham convolutions. These convolutions were introduced in [3] for Fibonacci and Lucas numbers and studied in many recent papers, for example, [4–9], also with their extensions to Fibonacci, Lucas, and Chebyshev polynomials. For the sake of simplicity from now on we omit the dependence on .

2. Power Sums

First of all, we give some straightforward calculation rules related to the functions , which we will use along our proofs.

Proposition 2. For the functions defined by the recurrence (1), the following rules hold:where , , , , , are integers.

Proof. These rules are direct consequences of relations (3) and easy calculations. Some of them are also listed in the paper of Horadam [2].

We start with two useful lemmas which will enable us to obtain our general formulas.

Lemma 3. The following equality holds: where , , are integers, is positive integer, and with .

Proof. Using the binomial theorem, we have from relations (3) or equivalently and if we sum (8) and (9) picking up we have from which we easily find (7).

Now, from identity (7), it is straightforward to obtain expressions for the powers in the cases odd or even. We only need to apply some little calculations and rule (4) to change sign in the subscripts. After a suitable rearrangement of the summations involved, if we getotherwise, if , we find

Lemma 4. Let one consider , , integers , with , and positive integers , . Then one has where and

Proof. From rule (6), we haveand we observe that since the sum telescopes. Thus we easily obtain (13) using rule (5).

Thanks to rule (6), we point out that the right member of (13) could also be expressed in the following equivalent form:which becomeswhen , since in this case , orwhen , in other words, by definition (1), when and .

Now we have all what we need to find out our general formulas.

Theorem 5. For all integers , , , , , , with , , , and , the following identities hold: where , , , and where

Proof. The identities (19) and (21) are direct consequences of (11) and (12), combined with (13) of Lemma 4. In the case , we have ; on the other hand, if , we have . Clearly in the latter, we also have to take in account that , so, applying rule (5), we obtainFinally, when we consider the sum related to the term in (12), we observe that .

In order to highlight in the next section some interesting divisibility properties between polynomials, we end with some results which made us able to rewrite equalities (19) and (21). For our purposes, we recall two important formulas due to Girard and Waring.

Lemma 6 (Girard–Waring formulas). For all nonnegative integers and real numbers , , the following identities hold:

Clearly these two formulas have a long history and should be widely known, so we only refer the reader to the original books of Girard [10] and Waring [11]. We also mention the paper of Gould [12], in which the reader will find some interesting remarks about the history and the use of these formulas and their generalizations. We observe that formula (26) also holds in the case ; indeed and corresponds to the right member of (26) via the identity Applying these formulas, we can find alternative expressions for in the cases odd or even, under the convention that ratios of the kind when and odd will take the value , according to the result of the limit .

Proposition 7. Let one consider integers , , , , , and nonnegative integer . Then the following equality holds:where

Proof. Thanks to the Girard–Waring formula (25) of Lemma 6, we can rewrite obtaining Now, since the last member of (33) is equal to where, by definition (31) of and by the Girard–Waring formula (26) of Lemma 6, we have with simple calculations