Discrete Dynamics in Nature and Society
Volume 2017 (2017), Article ID 9092515, 7 pages
https://doi.org/10.1155/2017/9092515

Research Article

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

Miomir Andjić¹ and Romeo Meštrović²

¹Faculty for Information Technology, University “Mediterranean”, Vaka Djurovića BB, Podgorica, Montenegro
²Maritime Faculty Kotor, University of Montenegro, Dobrota 36, 85330 Kotor, Montenegro

Correspondence should be addressed to Romeo Meštrović; romeo@ac.me

Received 19 December 2016; Accepted 8 February 2017; Published 27 February 2017

Academic Editor: Piergiulio Tempesta

Copyright © 2017 Miomir Andjić and Romeo Meštrović. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 known

Notice 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 that

Furthermore, 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 form

where () 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 that

Comparing (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 that

As 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 obtain

Notice 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.

References

J. Faulhaber, Academia Algebrae, Darinnen die miraculosische Inventiones zu den höchsten Cossen weiters continuirt und profitiert werden, Johann Ulrich Schönig, Augsburg, Germany, 1631.

J. Bernoulli, Ars Conjectandi, Thurneysen Brothers, Basel, Switzerland, 1713.

R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, Reading, Mass, USA, 1989.

S. Araci, M. Acikgoz, and E. g. Şen, “Some new formulae for Genocchi numbers and polynomials involving Bernoulli and Euler polynomials,” International Journal of Mathematics and Mathematical Sciences, vol. 2014, Article ID 760613, 7 pages, 2014.

D. S. Kim, T. Kim, S.-H. Rim, and S. H. Lee, “Hermite polynomials and their applications associated with Bernoulli and Euler numbers,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 974632, 13 pages, 2012.

N. I. Mahmudov and M. Momenzadeh, “On a class of -Bernoulli, -Euler, and -Genocchi polynomials,” Abstract and Applied Analysis, vol. 2014, Article ID 696454, 10 pages, 2014.

D. S. Kim, D. V. Dolgy, T. Kim, and S.-H. Rim, “Some formulae for the product of two Bernoulli and Euler polynomials,” Abstract and Applied Analysis, vol. 2012, Article ID 784307, 15 pages, 2012.

T. Kim, S.-H. Rim, Y. Simsek, and D. Kim, “On the analogs of Bernoulli and Euler numbers, related identities and zeta and L-functions,” Journal of the Korean Mathematical Society, vol. 45, no. 2, pp. 435–453, 2008.

N. I. Mahmudov, “q-analogues of the Bernoulli and Genocchi polynomials and the Srivastava-Pintér addition theorems,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 169348, 8 pages, 2012.

C. S. Ryoo, H. I. Kwon, J. Yoon, and Y. S. Jang, “Fourier series of the periodic Bernoulli and Euler functions,” Abstract and Applied Analysis, vol. 2014, Article ID 856491, 4 pages, 2014.

J. Sándor and B. Crstici, Handbook of Number Theory II, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2004.

K. Dilcher, L. Skula, and I. Sh. Slavutsky, “Bernoulli numbers bibliography (1713–1990),” Queen's Papers in Pure and Applied Mathematics, vol. 87, p. 175, 1991, Appendix, Queen's University, Kingston, Ontario, http://www.mathstat.dal.ca/~dilcher/bernoulli.html.

M. Kaneko, “A recurrence formula for the Bernoulli numbers,” Japan Academy. Proceedings. Series A. Mathematical Sciences, vol. 71, no. 8, pp. 192–193, 1995.

G. Dattoli, C. Cesarano, and S. Lorenzutta, “Bernoulli numbers and polynomials from a more general point of view,” Rendiconti di Matematica, Serie VII, vol. 22, pp. 193–202, 2002.

G. Dattoli, S. Lorenzutta, and C. Cesarano, “Finite sums and generalized forms of Bernoulli polynomials,” Rendiconti di Matematica e delle sue Applicazioni. Serie VII, vol. 19, no. 3, pp. 385–391, 1999.

Y. Simsek, “Complete sum of products of (h, q)-extension of Euler polynomials and numbers,” Journal of Difference Equations and Applications, vol. 16, no. 11, pp. 1331–1348, 2010.

D. E. Knuth, “Johann Faulhaber and sums of powers,” Mathematics of Computation, vol. 61, pp. 277–294, 1993.

R. Vakil, A Mathematical Mosaic: Patterns and Problem Solving, Brendan Kelly Publishing, Burlington, Canada, 1996.

B. Pascal, “Sommation des puissances numériques,” in Oeuvres Completés, vol. III, pp. 341–367, Desclée-Brouwer, Paris, France, 1964, English translation by A. Knoebel, R. Laubenbacher, J. Lodder, and D. Pengelley, “Sums of numerical powers”, in Mathematical Masterpieces: Further Chronicles by the Explorers, pp. 32–37, Springer, New York, NY, USA, 2007.

T. Kim, S.-H. Rim, and Y. Simsek, “A note on the alternating sums of powers of consecutive -integers,” Advanced Studies in Contemporary Mathematics, vol. 13, no. 2, pp. 159–164, 2006.

R. Meštrović, “On a congruence modulo involving two consecutive sums of powers,” Journal of Integer Sequences, vol. 17, no. 8, article 14.8.4, 20 pages, 2014.

P. S. Bruckman and O. P. Lossers, “Problem 6446,” The American Mathematical Monthly, vol. 92, no. 6, pp. 434–435, 1985.