Fórmula polinómica para la suma de potencias de los primeros n enteros positivos impares

• Autores: José Luis Cereceda Berdiel
• Localización: Pensamiento Matemático, ISSN-e 2174-0410, Vol. 9, Nº. 2, 2019
• Idioma: español
• Títulos paralelos:
  • Polynomial formula for the sum of powers of the first n positive odd integers
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    En este artículo consideramos la suma de potencias de los primeros n enteros positivos impares, Tk(n) = 1k + 3k + 5k + · · · + (2n − 1)k , donde n y k son enteros no negativos, y derivamos una fórmula polinómica para Tk (n) que es válida para todo k ≥ 0, en contraste con otras representaciones polinómicas que solo se aplican para k impar o para k par.

  • English

    This paper presents a simple approach to obtaining a unified polynomial formula accounting for the sums of powers of the first n odd integers, Tk (n) = 1 k + 3 k + 5 k + · · · + (2n − 1) k, where n and k are nonnegative integers. Our polynomial formula for Tk (n) applies to any k ≥ 0, in contrast with other polynomial representations specialized to the case where k is odd or even

• Referencias bibliográficas
  • APOSTOL, T. M., A primer on Bernoulli numbers and polynomials, Mathematics Magazine, Vol. 81, Nº 3, pp. 178–190, 2008.
  • BENJAMIN, A. T., QUINN, J. J., Proofs that Really Count. The Art of Combinatorial Proof, The Mathematical Association of America, 2003.
  • CERECEDA, J. L., Explicit polynomial for sums of powers of odd integers, International Mathematical Forum, Vol. 9, Nº 30, pp. 1441–1446, 2014.
  • DEEBA, E. Y., RODRIGUEZ, D. M., Bernoulli numbers and trigonometric functions, International Journal of Mathematical Education in Science...
  • DEEBA, E. Y., RODRIGUEZ, D. M., Stirling’s series and Bernoulli numbers, The American Mathematical Monthly, Vol. 98, Nº 5, pp. 423–426, 1991.
  • GOULD, H. W., Explicit formulas for Bernoulli numbers, The American Mathematical Monthly, Vol. 79, Nº 1, pp. 44–51, 1972.
  • GUO, S., SHEN, Y., On sums of powers of odd integers, Journal of Mathematical Research with Applications, Vol. 33, Nº 6, pp. 666–672, 2013.
  • NAMIAS, V., A simple derivation of Stirling’s asymptotic series, The American Mathematical Monthly, Vol. 93, Nº 1, pp. 25–29, 1986.
  • NUNEMACHER, J., YOUNG, R. M., On the sum of consecutive Kth powers, Mathematics Magazine, Vol. 60, Nº 4, pp. 237–238, 1987.
  • RUIZ LÓPEZ, F., Sobre las sumas de Bernoulli, Pensamiento Matemático, Vol. VIII, Nº 2, pp. 109–134, 2018.
  • WILLIAMS, K. S., Bernoulli’s identity without calculus, Mathematics Magazine, Vol. 70, Nº 1, pp. 47–50, 1997.
  • WITMER, E. E., The sums of powers of integers, The American Mathematical Monthly, Vol. 42, Nº 9, pp. 540–548, 1935.
  • WU, D. W., Bernoulli numbers and sums of powers, International Journal of Mathematical Education in Science and Technology, Vol. 32, Nº 3,...