- Open Access
Note on two extensions of the classical formula for sums of powers on arithmetic progressions
© The Author(s) 2017
- Received: 3 March 2017
- Accepted: 20 June 2017
- Published: 27 June 2017
We give two extensions of the classical formula for sums of powers on arithmetic progressions. This is achieved by using an identity involving binomial mixtures, which can be viewed as a generalization of the binomial transform.
- sum of powers formula
- forward difference
- binomial mixture
- binomial transform
- Bernoulli polynomials
Every choice of the function f and the random variable T in Theorem 2.1 gives us a different binomial identity. Whenever the probability density of T includes the uniform density on \((0,1)\) as a particular case, we are able to obtain a different extension of formula (3). In this respect, we give the following two corollaries of Theorem 2.1.
Proof of Theorem 2.1
Proof of Corollary 2.2
Proof of Corollary 2.3
The authors would like to thank the reviewers for their careful reading of the manuscript and for their suggestions, which greatly improved the final outcome. This work was supported by research grants MTM2015-67006-P, DGA (E-64), and by FEDER funds.
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