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

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. Let R = (R, +, ⋅) be a commutative ring of characteristic m > 0 (m may be equal to +∞) with unity e and zero 0. Given a positive integer n < m and the so-called n-symmetric set A = {a1, a2, . . . , a2l−1, a2l} such that al+i = ne − ai for each i = 1, . . . , l, define the rth power sum Sr(A) as Sr(A) = ∑2l i=1 ar i , for r = 0, 1, 2, . . . . We prove that for each positive integer k there holds ∑2k−1 i=0 (−1)i ( 2k−1 i ) 22k−1−iniS2k−1−i(A) = 0. As an application, we obtain two new Pascal-like identities for the sums of powers of the first n − 1 positive integers.


Recurrence Formulas for 𝑛-Symmetric Sets of Commutative
Rings with Unity.Here, as always in the sequel,  = (, +, ⋅) (briefly ) will denote a commutative ring of characteristic  > 0 ( may be equal to +∞) with unity  and zero 0. Throughout this paper N, Z, and Z  ( = 2, 3, . ..) 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 .
If 2 is invertible element in  with the inverse (2) −1 , then a finite -symmetric set  ⊆  is of the above form or of the form  = { 1 , . . .,   , () ⋅ (2) where Our main result is as follows.

The Application of Theorem 2 for the Sums of Powers on a
Finite Arithmetic Progression in Z.Using the obvious fact that for 1 <  <  and ,  ∈ Z with  ̸ = 0, {,  + , . . .,  + ( − 1) } is a (2+(−1) )-symmetric set of the ring Z 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 Z.
Corollary 5.For complex numbers  and  ̸ = 0, and positive integers  ≥ 1 and  ≥ 2 set Then formula (8) of Corollary 4 is satisfied for each positive integer .
Remark 10.Unfortunately, the analogous formula to (11) [17]; this also immediately follows by Bernoulli's formula).However, the first of these facts and the equality (26) show that  2 () is divisible by  2 for each  ≥ 1, 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].

The Application of Theorem 2 to the Sums 𝜑 𝑘 (𝑛).
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, () = { ∈ N : 1 ≤  < , gcd(, ) = 1}.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   (1) = 0 for all ).Notice that  0 () = () and there holds   () =   () if and only if  = 1 or  is a prime number.
Using the obvious fact that () is a -symmetric set in the ring Z, Theorem 2 immediately yields the following recurrence relation involving the functions   ().

Proofs of Theorem 2 and Corollary 9
Proof of Theorem 2. Suppose first that 2 is not invertible element in .Then as noticed above, the set  has the form where { 1 , . . .,   } ( = 2) is a finite subset of  such that  −   ̸ =   for all ,  with 1 ≤  <  ≤ .
Since the binomial formula holds in any commutative ring with unity, we have for all positive integers  and  with 1 ≤  ≤ .After summation in (33) over  = 1, . . ., 2 we obtain On the other hand, from (32) we see that  = 2 and we can assume that   =   for each  = 1, . . .,  and   = −  for each  =  + 1, . . ., 2.Using this and observing that −(2   − ) = 2( −   ) −  for each  = 1, . . ., , we find that Comparing (34) and (35) immediately gives the desired identity (4).Similarly, if 2 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 () ⋅ (2) −1 of  satisfies the equality 2() ⋅ (2) −1 −  = 0, in the same manner as in the first case, we arrive to (32).This completes the proof.
From (37) we see that the linear term of the polynomial S2+2 () is  2+2 .This together with (40) yields that the linear term of the polynomial on the left hand side of (40) is  +1  2+2 .Accordingly, in view of the fact that  2+2 ̸ = 0 for each  ≥ 0, the identical equality (40) yields  +1 = 0, which substituting in (40) gives
with 2 instead of 2 − 1 does not exist.Namely, suppose that for some even  ≥ 2 there exist real numbers  1 ,  2 , . . .,   such that for every    () = () as a polynomial in , then it is well known that   () is divisible by  for each  ≥ 1, and   () is divisible by  2 if and only if  is odd (see, e.g.,