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

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.

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