An Identity in Commutative Rings with Unity with Applications to Various Sums of Powers

Abstract

Let be a commutative ring of characteristic ( may be equal to ) with unity and zero 0. Given a positive integer and the so-called -symmetric set such that for each , define the th power sum as , for We prove that for each positive integer there holds As an application, we obtain two new Pascal-like identities for the sums of powers of the first positive integers.

1. Introduction and Statements Results

1.1. Recurrence Formulas for -Symmetric Sets of Commutative Rings with Unity

Here, as always in the sequel, (briefly ) will denote a commutative ring of characteristic ( may be equal to ) with unity and zero 0. Throughout this paper , , and () will, respectively, denote the set of positive integers, the ring of integers, and the ring of residues modulo .

Definition 1. Given a positive integer such that , we say that a subset of is -symmetric if it is satisfied: belongs to if and only if also belongs to .
Similarly, is called -symmetric tuple if it satisfies for all .

It is easy to see that if is not invertible in , then a finite subset of is -symmetric if and only if where is a finite subset of such that for all and with . Clearly, in this case is a -symmetric -tuple of .

If is invertible element in with the inverse , then a finite -symmetric set is of the above form or of the form where is a finite subset of such that for all and for all (in particular, is a finite -symmetric set). Clearly, in this case, is a -symmetric -tuple of .

Our main result is as follows.

Theorem 2. Let be a commutative ring of characteristic with unity and zero . For a nonnegative integer and a -symmetric set with , define the th power sum as Then for each positive integer it holds

Using the obvious facts that , is -symmetric set of ring and is -symmetric set of (with ) for each ; Theorem 2 immediately yields the following result.

Corollary 3. Let be a commutative ring of characteristic with unity and zero . For positive integers and and a set let (we also define ). Then for each positive integer it holds

1.2. The Application of Theorem 2 for the Sums of Powers on a Finite Arithmetic Progression in

Using the obvious fact that for and with , is a -symmetric set of the ring of integers, as a consequence of Theorem 2 we immediately obtain the following recurrence formula for the sums of powers on a finite arithmetic progression in .

Corollary 4. Let , , , and be integers, and let (we also define ). Then for each positive integer it holds that

If and are arbitrary real numbers, then, expanding via binomial formula every term () of the power sum (with a fixed ), it follows by Bernoulli’s formula given in Remark 8 that can be expressed as a polynomial in variables , and . Hence, this is also true for the sum on the left hand side of (8) which vanishes for all integers and . This yields the following extension of Corollary 4.

Corollary 5. For complex numbers and , and positive integers and set Then formula (8) of Corollary 4 is satisfied for each positive integer .

Remark 6. A calculation of sums can be traced back to Faulhaber [1] in 1631 and Bernoulli [2] in 1713. The recurrence formula (8) is in fact a summation formula which expresses as a sum involving power sums with the corresponding “coefficients” .

In particular, since the set is -symmetric (the case when with instead of in Corollary 4) and the set is -symmetric (the case , in Corollary 4), the formula (8) immediately yields the following two Pascal-like identities for the sums of powers of the first positive integers (cf. (6) with and ).

Corollary 7. For nonnegative integers and define Then for each positive integer it holds that

Remark 8. Finding formulas for sums of powers has interested mathematicians for more than 300 years since the early 17th-century mathematical publications of Faulhaber (1580–1635) [1]. Bernoulli (1654–1705) had given a comprehensive account of these sums in his famous work Ars Conjectandi [2], published posthumously in 1713. The second section of Ars Conjectandi (also see [3, pp. 269-270]) contains the fundamental Bernoulli’s formula which expresses the sum as a th-degree polynomial function on whose coefficients involve Bernoulli numbers. Namely, the celebrated Bernoulli’s formula (sometimes called Faulhaber’s formula) gives the sum explicitly given as a th-degree polynomial of (see, e.g., [3, Section  6.5, pp. 269-270], where it is given an induction proof on ). where are Bernoulli numbers defined by the generating function (see, e.g., [4–6]) It is easy to find the values , , , , and for odd . Furthermore, for all . It is well known that , where is the classical Bernoulli polynomial (see, e.g., [4, 7–10]). These and many other properties can be found, for instance, in [11, Section  5.3. pp. 525–538] or [12]. In particular (see, e.g., [13], where a similar formula was established), the usual recurrence is well knownNotice that Bernoulli numbers and polynomials from a more general point of view were studied by many authors (e.g., in [14, 15] the method of generating function was applied to introduce new forms of Bernoulli numbers and polynomials; also see [16]). Finding formulas for has interested mathematicians for more than 300 years since the time of Bernoulli (see, e.g., [17, 18]). Recall that, by the well-known Pascal’s identity proven by Pascal (1623–1662) in 1654 [19], Recently, -analogues of the sums of powers of consecutive integers have been investigated extensively (see, e.g., [20]).
For given positive integer , the recurrence (11) presents a formula for expressing as a sum involving power sums with the corresponding “coefficients” . For example, taking into (11), we, respectively, obtain where .
Substituting (17) into (18), we get Substituting the above formula and (17) into (19), we obtain Substituting the above two formulas and (17) into (20), we find that More generally, for any fixed , iterating the formula (11) we can express the sum as a linear combination of the polynomials in with . This is given by the following result.

Corollary 9. Let , , , be real numbers recursively defined as with , , and . Then for each it holds thatFurthermore, for each (25) is a unique representation of the power sum as a linear combination of the functions .

Remark 10. Unfortunately, the analogous formula to (11) with instead of does not exist. Namely, suppose that for some even there exist real numbers such that for every Considering the sum as a polynomial in , then it is well known that is divisible by for each , and is divisible by if and only if is odd (see, e.g., [17]; this also immediately follows by Bernoulli’s formula). However, the first of these facts and the equality (26) show that is divisible by for each , a contradiction.
Although formula (11) cannot be directly used for recursive determination of the expressions for , it can be useful for establishing various congruence involving these sums [21].

1.3. The Application of Theorem 2 to the Sums

The Euler totient function is defined to be equal to the number of positive integers less than which are relatively prime to . Each of these integers is called a totative (or “totitive”) of (see [11, Section  3.4, p. 242], where this notion is attributed to J. J. Sylvester). Let denote the set of all totatives of ; that is, . Given any fixed nonnegative integer , in 1850 A. Thacker (see [11, p. 242]) introduced the function defined as where the summation ranges over all totatives of (in addition, we define for all ). Notice that and there holds if and only if or is a prime number.

Using the obvious fact that is a -symmetric set in the ring , Theorem 2 immediately yields the following recurrence relation involving the functions .

Corollary 11. Let and be positive integers. Then

Remark 12. The following recurrence relation for the functions was established in 1857 by J. Liouville (cf. [11, p. 243]): which for reduces to Gauss’ formula . Furthermore, in 1985 Bruckman and Lossers [22] established an explicit Bernoulli’s-like formula for the Dirichlet series of defined as (there is called generalized Euler function).

For given integers , , , and define the function as Notice that with the function defined above. From the obvious fact that if and only if , it follows immediately that the set is a -symmetric set in the ring . Therefore, by Theorem 2 we immediately obtain the following recurrence formula for the functions .

Corollary 13. Let be a positive integer. Then

2. Proofs of Theorem 2 and Corollary 9

Proof of Theorem 2. Suppose first that is not invertible element in . Then as noticed above, the set has the formwhere () is a finite subset of such that for all with .
Since the binomial formula holds in any commutative ring with unity, we have for all positive integers and with . After summation in (33) over we obtain On the other hand, from (32) we see that and we can assume that for each and for each . Using this and observing that for each , we find thatComparing (34) and (35) immediately gives the desired identity (4).
Similarly, if is invertible element in , then as noticed in the previous section, the set may be of the form (32) or of the form Then since the central element of satisfies the equality , in the same manner as in the first case, we arrive to (32). This completes the proof.

Proof of Corollary 9 is based on Corollary 7 and the following lemma.

Lemma 14. Let , be a sequence of polynomials of the real variable defined as and where are Bernoulli numbers. Then for each nonnegative integer the polynomials are linearly independent over the field of real numbers .

Proof. The proof easily follows by induction on . The base of induction is satisfied because . Suppose that the polynomials are linearly independent for some . Now consider the polynomials and assume that where are real numbers. Note that the equality (39) can be written as From (37) we see that the linear term of the polynomial is . This together with (40) yields that the linear term of the polynomial on the left hand side of (40) is . Accordingly, in view of the fact that for each , the identical equality (40) yields , which substituting in (40) gives It follows from (41) that because the polynomials are linearly independent by induction hypothesis. Therefore, (39) yields . This completes the induction proof.

Proof of Corollary 9. By using the identity (11) of Corollary 7 with instead of , we find thatAs noticed above, iterating the formula (11), for any we can express the power sum as a linear combination of the polynomials of with . This means that for each there exist real numbers such that Replacing by into formula (43) yields Substituting (44) into (42) we obtainNotice that, for each by Bernoulli’s formula (see the first formula of Remark 8), From (37) and (46) we see that for each , and thus the polynomial identity (45) with respect to the positive integer variable can be extended to the real variable instead of ; that is, (45) is identically satisfied (on ) if we replace by (with and ). Accordingly, in view of Lemma 14, by comparing in (46) the coefficients with , for , we immediately obtain Finally, since , from the recurrence relations (24) and (47) we conclude that for all and .
Notice also that, by Lemma 14, (25) is a unique representation of the power sum as a linear combination of the functions . This completes proof of Corollary 9.

Competing Interests

The authors declare that they have no conflict of interests.