본문 바로가기
KERD
논문KCI 등재

CERTAIN COMBINATORIC CONVOLUTION SUMS AND THEIR RELATIONS TO BERNOULLI AND EULER POLYNOMIALS

Abdelmejid Bayad|김대열|Nazli Yildiz Ikikardes
대한수학회 (The Korean Mathematical Society)|2015.05|대한수학회지 (Journal of the KMS)|제52권 제3호|pp. 537-565

2

조회

0

다운로드

2

인용

0

참고문헌

키워드

Euler polynomialsBernoulli polynomialsdivisor functionsconvolution sums
주제분류수학

초록

In this paper, we give relationship between Bernoulli-Euler polynomials and convolution sums of divisor functions. First, we establish two explicit formulas for certain combinatoric convolution sums of divisor functions derived from Bernoulli and Euler polynomials. Second, as ap- plications, we show five identities concerning the third and fourth-order convolution sums of divisor functions expressed by their divisor functions and linear combination of Bernoulli or Euler polynomials.

Abstract

In this paper, we give relationship between Bernoulli-Euler polynomials and convolution sums of divisor functions. First, we establish two explicit formulas for certain combinatoric convolution sums of divisor functions derived from Bernoulli and Euler polynomials. Second, as ap- plications, we show five identities concerning the third and fourth-order convolution sums of divisor functions expressed by their divisor functions and linear combination of Bernoulli or Euler polynomials.

참고문헌 (26)

  1. A. Bazs´o, "A refinement of Faulhaber’s theorem con-cerning sums of powers of natural numbers", Appl. Math. Lett, 25(3), pp.486~489 (2012)
  2. B. C. Berndt, "Ramanujan’s Notebooks. Part II" (1989), Springer-Verlag
  3. M. Besge, "Extrait d’une lettre de M. Besge `a M. Liouville", J. Math. Pures Appl, 7, pp.256~ (1862)
  4. B. Cho, "Evaluation of a certain combinatorial convolution sum in higher level cases", J. Math. Anal. Appl, 406(1), pp.203~210 (2013)
  5. W. Chu, "Convolutions of Bernoulli and Euler polynomials", Sarajevo J. Math, 6(19)(2), pp.147~163 (2010)
  6. A. Erd´elyi, "Higher Transcendental Functions. Vol 1" (1953), McGraw Hill
  7. J. W. L. Glaisher, "On the square of the series in which the coefficients are the sums of the divisors of the exponents", Messenger Math, 14, pp.156~163 (1885)
  8. H. Hahn, "Convolution sums of some functions on divisors", Rocky Mountain J. Math, 37(5), pp.1593~1622 (2007)
  9. J. G. Huard, "Elementary evaluation of certain convolution sums involving divisor functions, Number theory for the millennium, II", pp.229~274 (2002), A K Peters
  10. D. Kim, "Convolution identities for twisted Eisenstein series and twisted divisor functions", Fixed Point Theory Appl, 2013(81), pp.23~ (2013)
  11. D. Kim, "Certain combinatoric Bernoulli polynomials and convolu-tion sums of divisor functions", Adv. Difference Equ, (310), pp.11~ (2013)
  12. 김애란, "Convolution sums arising from divisor functions", 대한수학회지, 50(2), pp.331~360 (2013), 대한수학회
  13. D. Kim, "Bernoulli identities and combinatoric convolution sums with odd divisor functions", Abstr. Appl. Anal, 2014, pp.8~ (2014)
  14. D. Kim, "Certain Combinatoric convolution sums involving divisor functions product formulae", Taiwanese J. Math, 18(3), pp.973~988 (2014)
  15. D. B. Lahiri, "On Ramanujan’s function τ(n) and the divisor function σ(n). I", Bull. Calcutta Math. Soc, 38, pp.193~206 (1946)
  16. G. Melfi, "On some modular identities, Number Theory", pp.371~382 (1998), de Gruyter
  17. L. Navas, "Old and new identities for Bernoulli polynomials via Fourier series", Int. J. Math. Math. Sci, 2012, pp.14~ (2012)
  18. S. Ramanujan, "On certain arithmetical functions", Trans. Cambridge Philos. Soc, 22(9), pp.159~184 (1916)
  19. Y. Simsek, "Elliptic analogue of the Hardy sums related to elliptic Bernoulli functions", Gen. Math, 15(3), pp.3~23 (2007)
  20. H. M. Srivastava, "Remarks on some relationships between the Bernoulli and Euler polynomials", Appl. Math. Lett, 17(4), pp.375~380 (2004)
  21. Z.-H. Sun, "Congruences concerning Bernoulli numbers and Bernoulli polynomials", Dis-crete Appl. Math, 105(1-3), pp.193~223 (2000)
  22. Z.-H. Sun, "Legendre polynomials and supercongruences", Acta Arith, 159(2), pp.169~200 (2013)
  23. Z.-H. Sun, "Congruences concerning Legendre polynomials. III", Int. J. Number Theory, 9(4), pp.965~999 (2013)
  24. K. S. Williams, "The convolution sum Pm<n", 8(m)(n − 8m), 2006, pp.228(2), 387~396
  25. K. S. Williams, "Number Theory in the Spirit of Liouville", Stu-dent Texts, 76 (2011), London Mathematical Society
  26. K. S. Williams, "The parents of Jacobi’s four squares theorem are unique", Amer. Math. Monthly, 120(4), pp.329~345 (2013)

관련 논문AI추천

SOME IDENTITIES FOR THE BERNOULLI, THE EULER AND THE GENOCCHI NUMBERS AND POLYNOMIALS김태균장전수학회 · Advanced Studies in Contemporary Mathematics2010 · 제20권 제1호A note on Eulerian polynomials associated with Bernoulli and Euler numbers and polynomials김대산, 김영희, D. V. Dolgy, T. Kim장전수학회 · Advanced Studies in Contemporary Mathematics2012 · 제22권 제3호IDENTITIES FOR THE BERNOULLI, THE EULER AND THE GENOCCHI NUMBERS AND POLYNOMIALS김태균, Bayad장전수학회 · Advanced Studies in Contemporary Mathematics2010 · 제20권 제2호SOME IDENTITIES INVOLVING FROBENIUS-EULER POLYNOMIALS AND NUMBERS김태균, 서종진장전수학회 · Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집)2016 · 제19권 제1호Some identities of the type 2 degenerate Bernoulli and Euler numbers권종겸, 김대산, 장관우, 정세교장전수학회 · Advanced Studies in Contemporary Mathematics2019 · 제29권 제4호Convolution sums and their relations to Eisenstein series김대열, 김애란, Ayyadurai Sankaranarayanan대한수학회 · 대한수학회보2013 · 제50권 제4호

이 간행물의 다른 논문 (11건)

PRECISE LARGE DEVIATIONS FOR AGGREGATE LOSS PROCESS IN A MULTI-RISK MODELJianming Bai, Fengqin Tangpp. 447-467HYBRID ON-OFF CONTROLS FOR AN HIV MODEL BASED ON A LINEAR CONTROL PROBLEM장태수, 이지현, 김정은, 권희대pp. 469-487DUO RING PROPERTY RESTRICTED TO GROUPS OF UNITS박상원, 이양, 한준철pp. 489-501PARAMETER CHANGE TEST FOR NONLINEAR TIME SERIES MODELS WITH GARCH TYPE ERRORS이상열, 이지연pp. 503-522TRANSLATION AND HOMOTHETICAL SURFACES IN EUCLIDEAN SPACE WITH CONSTANT CURVATUREMarilena Moruz, Rafael L\'opezpp. 523-535
원문 보기