References
- A. A. Aygunes, Y. Simsek, The action of Hecke operators to families of Weierstrass-type functions and Weber-type functions and their applications, Applied Mathematics and Computation 218 (2011), 678-682. https://doi.org/10.1016/j.amc.2011.03.090
- B. C. Berndt, Ramanujan's Notebooks, Part II. Springer-Verlag, New York, 1989.
- B. C. Berndt and R. J. Evans, Chapter 15 of Ramanujan's second notebook, Part II. Modular forms, Acta Arith 47 (1986), 123-142. https://doi.org/10.4064/aa-47-2-123-142
-
C.-H. Chang, H. M. Srivastava, A note on Bernoulli identities associated with the Weierstrass
${\wp}$ -function, Integral Transform. Spec. Funct. 18 (2007), 245-253. https://doi.org/10.1080/10652460701210276 - C.-H. Chang, H. M. Srivastava, T.-C. Wu, Some families of Weierstrass-type functions and their applications, Integral Transform. Spec. Funct. 19 (2008), 621-632. https://doi.org/10.1080/10652460802230546
- N. Cheng and K. S. Williams, Evaluation of some convolution sums involving the sum of divisors functions, Yokohama Mathematical J. 52 (2005), 39-57.
- B. Cho, D. Kim and J. K. Koo, Modular forms arising from divisor functions, J. Math. Anal. Appl. 356 (2009), 537-547. https://doi.org/10.1016/j.jmaa.2009.03.003
- B. Cho, D. Kim and J. K. Koo, Divisor functions arising from q-series, Publ. Math. Debrecen 76 (2010), 495-508.
- L. E. Dickson, History of the Theory of Numbers, Vol. I, Chelsea Publ. Co., New York, 1952.
- L. E. Dickson, History of the Theory of Numbers, Vol. II, Chelsea Publ. Co., New York, 1952.
- N. J. Fine, Basic hypergeometric series and applications, American Mathematical Society, Providence, RI, 1988.
- J. W. L. Glaisher, Expressions for the first five powers of the series in which the coefficients are sums of the divisors of the exponents, Mess. Math. 15 (1885), 33-36.
- 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, 2002, 229-274.
- D. Kim, A. Kim, and Y. Li, Convolution sums arising from the divisor functions, J. Korean Math. Soc. 50 (2013), 331-360. https://doi.org/10.4134/JKMS.2013.50.2.331
- D. Kim and J. K. Koo, Algebraic integer as values of elliptic functions, Acta Arith. 100 (2001), 105-116. https://doi.org/10.4064/aa100-2-1
-
D. B. Lahiri, On Ramanujan's function
${\tau}$ (n) and the divisor function${\sigma}_1$ (n)-I, Bull. Calcutta Math. Soc. 38 (1946), 193-206. -
D. B. Lahiri, On Ramanujan's function
${\tau}$ (n) and the divisor function${\sigma}_k$ (n)-II, Bull. Calcutta Math. Soc. 39 (1947), 33-52. - J. Levitt, On a Problem of Ramanujan, M. Phil thesis, University of Nottingham, 1978.
- G. Melfi, On some modular identities, de Gruyter, Berlin, 1998, 371-382.
- K. Ono, S. Robins and D. T. Wahl, On the representation of integer as sum of triangular numbers, Aequationes Math. 50 (1995), 73-94. https://doi.org/10.1007/BF01831114
- V. Ramamani, On some identities conjectured by Srinivasa Ramanujan found in his lithographed notes connected with partition theory and elliptic modular functions, Ph. D Thesis, University of Mysore, 1970.
- S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22 (1916), 159-184.
- S. Ramanujan, Collected papers, AMS Chelsea Publishing, Providence, RI, 2000.
- J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, Springer-Verlag, 1994.
-
K. S. Williams, The convolution sum
${\Sigma}_{m<\frac{n}{8}}{\sigma}_1(m){\sigma}_1$ (n-8m), Pacific J. Math. 228 (2006), 387-396. https://doi.org/10.2140/pjm.2006.228.387 - K. S. Williams, On Liouville's twelve squares theorem, Far East J. Math. Sci. 29 (2008), 239-242.
- K. S. Williams, Number Theory in the Spirit of Liouville, London Mathematical Society, Student Texts 76, Cambridge, 2011.