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CERTAIN COMBINATORIC CONVOLUTION SUMS AND THEIR RELATIONS TO BERNOULLI AND EULER POLYNOMIALS
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 Title & Authors
CERTAIN COMBINATORIC CONVOLUTION SUMS AND THEIR RELATIONS TO BERNOULLI AND EULER POLYNOMIALS
Kim, Daeyeoul; Bayad, Abdelmejid; Ikikardes, Nazli Yildiz;
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 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 applications, 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.
 Keywords
Bernoulli polynomials;Euler polynomials;convolution sums;divisor functions;
 Language
English
 Cited by
1.
A PRODUCT FORMULA FOR COMBINATORIC CONVOLUTION SUMS OF ODD DIVISOR FUNCTIONS, Honam Mathematical Journal, 2016, 38, 2, 243  crossref(new windwow)
2.
TRIPLE AND FIFTH PRODUCT OF DIVISOR FUNCTIONS AND TREE MODEL, Journal of applied mathematics & informatics, 2016, 34, 1_2, 145  crossref(new windwow)
 References
1.
A. Bazso, A. Pinter, and H. M. Srivastava, A refinement of Faulhaber's theorem concerning sums of powers of natural numbers, Appl. Math. Lett. 25 (2012), no. 3, 486-489. crossref(new window)

2.
B. C. Berndt, Ramanujan's Notebooks. Part II, Springer-Verlag, New York, 1989.

3.
M. Besge, Extrait d'une lettre de M. Besge a M. Liouville, J. Math. Pures Appl. 7 (1862), 256.

4.
B. Cho, D. Kim, and H. Park, Evaluation of a certain combinatorial convolution sum in higher level cases, J. Math. Anal. Appl. 406 (2013), no. 1, 203-210. crossref(new window)

5.
W. Chu and R. R. Zhou, Convolutions of Bernoulli and Euler polynomials, Sarajevo J. Math. 6(19) (2010), no. 2, 147-163.

6.
A. Erdelyi, Higher Transcendental Functions. Vol 1, McGraw Hill, New York, 1953.

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 (1885), 156-163.

8.
H. Hahn, Convolution sums of some functions on divisors, Rocky Mountain J. Math. 37 (2007), no. 5, 1593-1622. crossref(new window)

9.
J. G. Huard, Z. M. Ou, B. K. Spearman, and K. S. Williams, Elementary evaluation of certain convolution sums involving divisor functions, Number theory for the millennium, II (Urbana, IL, 2000), 229-274, A K Peters, Natick, MA, 2002.

10.
D. Kim and A. Bayad, Convolution identities for twisted Eisenstein series and twisted divisor functions, Fixed Point Theory Appl. 2013 (2013), 81, 23 pp.

11.
D. Kim and N. Y. Ikikardes, Certain combinatoric Bernoulli polynomials and convolution sums of divisor functions, Adv. Difference Equ. 2013 (2013), 310, 11 pp.

12.
A. Kim, D. Kim, and L. Yan, Convolution sums arising from divisor functions, J. Korean Math. Soc. 50 (2013), no. 2, 331-360. crossref(new window)

13.
D. Kim and Y. K. Park, Bernoulli identities and combinatoric convolution sums with odd divisor functions, Abstr. Appl. Anal. 2014 (2014), Article ID 890973, 8 pages.

14.
D. Kim and Y. K. Park, Certain Combinatoric convolution sums involving divisor functions product formulae, Taiwanese J. Math. 18 (2014), no. 3, 973-988.

15.
D. B. Lahiri, On Ramanujan's function ${\tau}$(n) and the divisor function ${\sigma}$(n). I, Bull. Calcutta Math. Soc. 38 (1946), 193-206.

16.
G. Melfi, On some modular identities, Number Theory (Eger, 1996), 371-382, de Gruyter, Berlin, 1998.

17.
L. Navas, F. J. Ruiz, and J. L. Varona, Old and new identities for Bernoulli polynomials via Fourier series, Int. J. Math. Math. Sci. 2012 (2012), Article ID 129126, 14 pp.

18.
S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22 (1916), no. 9, 159-184.

19.
Y. Simsek, Elliptic analogue of the Hardy sums related to elliptic Bernoulli functions, Gen. Math. 15 (2007), no. 3, 3-23.

20.
H. M. Srivastava and A. Pinter, Remarks on some relationships between the Bernoulli and Euler polynomials, Appl. Math. Lett. 17 (2004), no. 4, 375-380. crossref(new window)

21.
Z.-H. Sun, Congruences concerning Bernoulli numbers and Bernoulli polynomials, Discrete Appl. Math. 105 (2000), no. 1-3, 193-223. crossref(new window)

22.
Z.-H. Sun, Legendre polynomials and supercongruences, Acta Arith. 159 (2013), no. 2, 169-200. crossref(new window)

23.
Z.-H. Sun, Congruences concerning Legendre polynomials. III, Int. J. Number Theory 9(2013), no. 4, 965-999. crossref(new window)

24.
K. S. Williams, The convolution sum ${\Sigma}_{m(n-8m), Pacific J. Math. 228(2006), no. 2, 387-396. crossref(new window)

25.
K. S. Williams, Number Theory in the Spirit of Liouville, London Mathematical Society, Student Texts 76, Cambridge, 2011.

26.
K. S. Williams, The parents of Jacobi's four squares theorem are unique, Amer. Math. Monthly 120 (2013), no. 4, 329-345. crossref(new window)