Note on two extensions of the classical formula for sums of powers on arithmetic progressions
- José A Adell^{1}Email author and
- Alberto Lekuona^{1}
https://doi.org/10.1186/s13662-017-1250-y
© The Author(s) 2017
Received: 3 March 2017
Accepted: 20 June 2017
Published: 27 June 2017
Abstract
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.
Keywords
sum of powers formula forward difference binomial mixture binomial transform Bernoulli polynomialsMSC
05A19 60C051 Introduction
2 Main results
Theorem 2.1
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.
Corollary 2.2
Corollary 2.3
Observe that both corollaries extend formula (3) by choosing \(f=\psi_{m}\) and \(p=q=1\) in Corollary 2.2, and \(f=\psi_{m}\) and \(j=1\) in Corollary 2.3.
3 The proofs
Proof of Theorem 2.1
Proof of Corollary 2.2
Proof of Corollary 2.3
Declarations
Acknowledgements
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.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Kannappan, PL, Zhang, W: Finding sum of powers on arithmetic progressions with application of Cauchy’s equation. Results Math. 42(3-4), 277-288 (2002). doi:https://doi.org/10.1007/BF03322855 MathSciNetView ArticleMATHGoogle Scholar
- Guo, VJW, Zeng, J: A q-analogue of Faulhaber’s formula for sums of powers. Electron. J. Comb. 11(2), R19 (2005) MathSciNetMATHGoogle Scholar
- Rosen, KH: Handbook of Discrete and Combinatorial Mathematics. CRC Press, Boca Raton (2000) MATHGoogle Scholar
- Spivey, MZ: Combinatorial sums and finite differences. Discrete Math. 307(24), 3130-3146 (2007). doi:https://doi.org/10.1016/j.disc.2007.03.052 MathSciNetView ArticleMATHGoogle Scholar
- Luo, Q-M, Qi, F, Debnath, L: Generalizations of Euler numbers and polynomials. Int. J. Math. Math. Sci. 61, 3893-3901 (2003). doi:https://doi.org/10.1155/S016117120321108X MathSciNetView ArticleMATHGoogle Scholar
- Adell, JA, Anoz, JM: Signed binomial approximation of binomial mixtures via differential calculus for linear operators. J. Stat. Plan. Inference 138(12), 3687-3695 (2008). doi:https://doi.org/10.1016/j.jspi.2007.11.018 MathSciNetView ArticleMATHGoogle Scholar
- Flajolet, P, Vepstas, L: On differences of zeta values. J. Comput. Appl. Math. 220(1-2), 58-73 (2008). doi:https://doi.org/10.1016/j.cam.2007.07.040 MathSciNetView ArticleMATHGoogle Scholar
- Mu, Y-P: Symmetric recurrence relations and binomial transforms. J. Number Theory 133(9), 3127-3137 (2013). doi:https://doi.org/10.1016/j.jnt.2013.03.003 MathSciNetView ArticleMATHGoogle Scholar
- Galaz Fontes, F, Solís, FJ: Iterating the Cesàro operators. Proc. Am. Math. Soc. 136(6), 2147-2153 (2008). doi:https://doi.org/10.1090/S0002-9939-08-09197-1 View ArticleMATHGoogle Scholar
- Adell, JA, Lekuona, A: Rates of convergence for the iterates of Cesàro operators. Proc. Am. Math. Soc. 138(3), 1011-1021 (2010). doi:https://doi.org/10.1090/S0002-9939-09-10127-2 View ArticleMATHGoogle Scholar
- Hardy, GH: Divergent Series. Clarendon, Oxford (1949) MATHGoogle Scholar