On some new Eisenstein series identities involving the Broweins' cubic theta function $a(q)$ and the convolution sum $\ds\sum_{i+3j=m}\sigma(i)\sigma(j)$

SHARATH G

doi:10.5269/bspm.67975

SHARATH G PES College of Engineering, Mandya

Resumo

In this paper, employing the $(p, k)$-parametrization of Srinivasa Ramanujan's Eisenstein series $L(q), M(q)$ and the Borweins' cubic theta functions due to Alaca et. al \cite{Alaca1, Alaca12, Alaca13}, we derive some new Eisenstein series identities involving the Borweins' cubic theta function $a(q)$ with the help of Computer. Further, as an application of these, we deduce the
convolution sum $\ds\sum_{i+3j=m}\sigma(i)\sigma(j)$.

Downloads

Não há dados estatísticos.

Referências

Alaca, A., Alaca, S., Williams, K.S.,On the Two-Dimensional Theta Functions of the Borweins, Acta Arith., 124(2), 177–195, (2006).

Alaca, A., Alaca, S., Williams, K.S.,Evaluation of the convolution sums Pl+18m=n σ(l)σ(m) and P2l+9m=n σ(l)σ(m), International Mathematical Forum. J. Theory Appl., 14(2), 45–68, (2007).

Alaca, S¸., Williams, K.S., Evaluation of the convolution sums Pl+6m=n σ(l)σ(m) and P2l+3m=n σ(l)σ(m), J. Number Theory, 124(2), 491–510, (2007).

Alaca, A., Alaca, S., Williams, K.S., Evaluation of the convolution sums Pl+12m=n σ(l)σ(m) and P3l+4m=n σ(l)σ(m), Adv. Theor. Appl. Math., 1(1), 27–48, (2006).

Alaca, A., Alaca, S., Williams, K.S., Evaluation of the convolution sums Pl+24m=n σ(l)σ(m) and P3l+8m=n σ(l)σ(m), Math. J. Okayama Univ., 49, 93–111, (2007).

Alaca, A., Alaca, S., Williams, K.S., Evaluation of the convolution sums Pm< n 16 σ(m)σ(n − 16m), Canadian Mathematical Bulletin. Bull. Can. Math., 51(1), 3–14, (2008).

Alaca, S¸., Kesicioglu, Y., Evaluation of the convolution sums Pl+27m=n σ(l)σ(m) and Pl+32m=n σ(l)σ(m), Int. J. of Number Theory, 12(1), 1–13, (2016).

Alaca, A., Alaca, S., Ntienjem, E., The Convolution Sum Pal+bm=n σ(l)σ(m) for (a; b) = (1; 28); (4; 7), (1; 14); (2; 7); (1; 7), KYUNGPOOK Math. J., 59, 377-389, (2019).

Baruah, N.D., Berndt, B.C., Ramanujan’s Eisenstein series and new hypergeometric-like series for 1 π2 , Journal of Approximation Theory, 160, 135-153, (2008).

Berndt, B.C., Bhargava, S., Garvan, F.G., Ramanujan’s theories of elliptic functions to alternative bases, Trans. Am. Math. Soc., 347(11), 4163–4244, (1995).

Berndt, B.C., Chan, H.H., Sohn, J., Son, S.H., Eisenstein Series in Ramanujan’s lost notebook, The Ramanujan Journal, 4, 81–114, (2000).

Bhuvan, E.N., Vasuki K.R., On a Ramanujan’s Eisenstein series identity of level fifteen, Proc Math Sci., 129, (2019).

Borwein, J.M., Browein, P.B., A cubic counterpart of Jacobi’s identity and the AGM, Trans. Am. Math. Soc., 323(2), 691–701, (1991).

Borwein, J.M., Browein, P.B., Garvan, F.G., Some cubic modular identities of Ramanujan, Trans. Am. Math. Soc., 343(1), 35–47, (1994).

Chan, H.H., On Ramanujan’s cubic transformation formula for 2F1( 1 3 , 2 3 ; 1; z), Math. Proc. Camb. Philos. Soc., 124(2), 193–204, (1998).

Chan, H.H., Cooper, S., Toh, P.C., Ramanujan’s Eisenstein series and powers of Dedekind’s eta-function, Journal of the London Mathematical Society, 75, 225–242, (2007).

Cooper, S., Lam, H.Y., Sixteen Eisenstein series, Ramanujan J., 18, 33–59, (2009).

Cooper, S., Ye, D., Evaluation of the convolution sums Pl+20m=n σ(l)σ(m), P4l+5m=n σ(l)σ(m) and P2l+5m=n σ(l)σ(m), Int. J. of Number Theory, 10(6), 1385–1394, (2014).

Huard, J.G., Ou, Z.M., Spearman, B.K., Williams, K.S., Elementary evaluation of certain convolution sums involving divisor functions, Number Theory for the Millennium, II, Bennett, M.A., Berndt, B.C., Boston, N., Diamond, H.G., Hildebrand, A.J. and Phillipp, W. Eds., Peters, A.K. Natick, Mass USA, 229-274 (2002).

Kendirli, B., Evaluation of some convolution sums by quasimodular forms, European Journal of Pure and Applied Mathematics, 8(1), 81–110, (2015).

Lemire, M., Williams, K.S., Evaluation of two convolution sums involving the sum of divisors function, Bull. Aust. Math. Soc., 73(1), 107–115, (2006).

Liu, Z.-G., The Borweins’ cubic theta function identity and some cubic modular identities of Ramanujan, Ramanujan J., 4(1), 43–50, (2000).

Liu, Z.-G., Some Eisenstein series identities, J. Number Theory, 85(2), 231–252, (2000).

Liu, Z.-G., Some Eisenstein series identities associated with the Borwein functions, In: Garvan, F. G., Ismail, M. E. H. (eds) Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Developments in Mathematics, 4, (2001), Springer, Boston, MA.

Ntienjem, E., Evaluation of the convolution sum involving the sum of divisors function for 14, 22 and 26, Open Mathematics, 15(1), 446-458, (2007).

Park, Y.K., Evaluation of the convolution sums Pal+bm=n lσ(l)σ(m) with ab ≤ 9, Open Mathematics, 15(1), 1389-1399, (2017).

Pushpa, K., Vasuki, K.R., Evaluation of convolution sums Pl+15m=n σ(l)σ(m) and P3l+5m=n σ(l)σ(m), Indian J. Pure Appl. Math., 53, 1110–1121, (2022).

Pushpa, K., Vasuki, K.R., Evaluation of Convolution Sums Pl+km=n σ(l)σ(m) and Pal+bm=n σ(l)σ(m) for k = a·b = 21, 33, and 35, Glasgow Mathematical Journal, 64(2), 434-453, (2022).

Ramakrishnan, B., Sahu, B., Evaluation of the Convolution Sums Pl+15m=n σ(l)σ(m) and P3l+5m=n σ(l)σ(m) and an Application, Int. J. of Number Theory, 9(3), 799-809, (2013).

Ramanujan, S., Notebooks (2 Volumes), Tata Institute of Fundamental Research, Mumbai, (1957)

Ramanujan, S., Collected Papers, Chelsea, New York, 1962.

Ramanujan, S., The lost notebook and other unpublished papers, Narosa, New Delhi, 1988.

Royer, E., Evaluating convolution sums of the divisor function by quasimodular forms, Int. J. Number Theory, 3(2), 231–261, (2007).

Shruthi and Srivatsa Kumar, B.R., Some new Eisenstein series containing the Borweins’ cubic theta functions and convolution sum Pi+4j=n σ(i)σ(j), Afr. Mat., 31, 971–982, (2020).

Srivatsa Kumar, B.R., Shruthi, Anu Radha, D., Relation between Borweins’ Cubic Theta Functions and Ramanujan’s Eisenstein Series, J. Appl. Math., 2021, (2021).

Vasuki, K.R., Veeresha, R.G., Ramanujan’s Eisenstein series of level 7 and 14, J. of Number Theory, 159, 59-75, (2016).

Williams, K.S., The convolution Pm

Williams, K.S., The convolution Pm< n 8 σ(m)σ(n − 8m), Pac. J. Math., 228(2), 387–396, (2006).

Xia, E.X.W., Yao, O.X.M., Eisenstein series identities involving the Borweins’ cubic theta functions, J. Appl. Math., 2012, (2012).

Ye, D., Evaluation of the convolution sums Pl+36m=n σ(l)σ(m) and P4l+9m=n σ(l)σ(m)., Int. J. of Number Theory, 11(1), 171–183, (2015).