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The Ramanujan Journal

, Volume 33, Issue 1, pp 23–53 | Cite as

The power series expansion of certain infinite products \(q^{r}\prod_{n=1}^{\infty}(1-q^{n})^{a_{1}}(1-q^{2n})^{a_{2}}\cdots(1-q^{mn})^{a_{m}}\)

  • Lerna Pehlivan
  • Kenneth S. Williams
Article
  • 316 Downloads

Abstract

Let ℕ, \(\mathbb{N}_{0}\), ℤ, ℚ, and ℂ denote the sets of positive integers, nonnegative integers, integers, rational numbers, and complex numbers, respectively. If \(f(q)\) is a complex-valued function with
$$f(q)= \sum_{n=0}^{\infty} f_{n} q^{n} \quad\bigl(q \in\mathbb{C}, \vert q \vert <1 \bigr) $$
we define
$$\bigl[f(q)\bigr]_{n}:=f_{n} \quad(n \in\mathbb{N}_{0}). $$
For \(k \in\mathbb{N}\) we define
$$E_{k}:= \prod_{n=1}^{\infty} \bigl(1-q^{kn}\bigr) \quad\bigl(q \in\mathbb{C}, \vert q \vert <1 \bigr). $$
We show how modular equations of a special form can be used in conjunction with the representation numbers of certain quadratic forms to determine
$$\bigl[q^{r}E_{1}^{a_{1}}\cdots E_{m}^{a_{m}} \bigr]_{n} \quad(r \in\mathbb{N}_{0},m \in \mathbb{N},a_{1}, \ldots,a_{m} \in\mathbb{Z}) $$
for certain products \(q^{r}E_{1}^{a_{1}}\cdots E_{m}^{a_{m}}\). For example, we show that
$$\biggl[q^{2}\frac{E_{1}^{4}E_{16}^{4}}{E_{2}^{2}E_{8}^{2}} \biggr]_{n} = \begin{cases} 0 & \mbox{if}\ n \equiv1\ (\mbox{mod}\ 4),\\ - \sigma(N) & \mbox{if}\ n \equiv3\ (\mbox{mod}\ 4),\\ \sigma(N) & \mbox{if}\ n \equiv2\ (\mbox{mod}\ 4),\\ 4\sigma(N) & \mbox{if}\ n \equiv4\ (\mbox{mod}\ 8),\\ 0 & \mbox{if}\ n \equiv0\ (\mbox{mod}\ 8), \end{cases} $$
where \(N\) denotes the odd part of the positive integer \(n\) and
$$\sigma(n): =\sum_{ \begin{array}{c} \scriptstyle d \in\mathbb{N}\\[-3pt] \scriptstyle d \mid n \end{array}} d. $$

Keywords

Infinite products Quadratic forms Representations Theta functions Modular equations 

Mathematics Subject Classification

11E25 11F27 11B65 05A19 

References

  1. 1.
    Alaca, A.: Representations by quaternary quadratic forms whose coefficients are 1, 3 and 9. Acta Arith. 136, 151–166 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Alaca, A., Alaca, Ş., Lemire, M.F., Williams, K.S.: Nineteen quaternary quadratic forms. Acta Arith. 130, 277–310 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Alaca, A., Alaca, Ş., Lemire, M.F., Williams, K.S.: Jacobi’s identity and representations of integers by certain quaternary quadratic forms. Int. J. Mod. Math. 2, 143–176 (2007) zbMATHMathSciNetGoogle Scholar
  4. 4.
    Alaca, A., Alaca, Ş., Lemire, M.F., Williams, K.S.: The number of representations of a positive integer by certain quaternary quadratic forms. Int. J. Number Theory 5, 13–40 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Alaca, A., Alaca, Ş., Williams, K.S.: On the two-dimensional theta functions of the Borweins. Acta Arith. 124, 177–195 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Alaca, A., Alaca, Ş., Williams, K.S.: On the quaternary forms \(x^{2}+y^{2}+z^{2}+5t^{2}\), \(x^{2}+y^{2}+5z^{2}+5t^{2}\) and \(x^{2}+5y^{2}+5z^{2}+5t^{2}\). JP J. Algebra Number Theory Appl. 9, 37–53 (2007) zbMATHMathSciNetGoogle Scholar
  7. 7.
    Alaca, A., Alaca, Ş., Williams, K.S.: Liouville’s sextanary quadratic forms \(x^{2}+y^{2}+z^{2}+t^{2}+2u^{2}+2v^{2}\), \(x^{2}+y^{2}+2z^{2}+2t^{2}+2u^{2}+2v^{2}\) and \(x^{2}+2y^{2}+2z^{2}+2t^{2}+2u^{2}+4v^{2}\), Far East. J. Math. Sci. 30, 547–556 (2008) zbMATHMathSciNetGoogle Scholar
  8. 8.
    Alaca, Ş., Williams, K.S.: The number of representations of a positive integer by certain octonary quadratic forms. Funct. Approx. Comment. Math. 43, 45–54 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Berndt, B.C.: Ramanujan’s Notebooks Part III. Springer, New York (1991) CrossRefzbMATHGoogle Scholar
  10. 10.
    Berndt, B.C.: Number Theory in the Spirit of Ramanujan. Am. Math. Soc., Providence (2006) zbMATHGoogle Scholar
  11. 11.
    Bhargava, S.: Some applications of Ramanujan’s remarkable summation formula. In: Ramanujan Rediscovered. Proceedings of a Conference on Elliptic Functions, Partitions, and \(q\)-Series in Memory of K. Venkatachaliengar, Bangalore, 1–5 June 2009. Ramanujan Mathematical Society Lecture Notes Series, vol. 14, pp. 63–72. Ramanujan Mathematical Society, Thiruchirappalli (2010) Google Scholar
  12. 12.
    Blecksmith, R., Brillhart, J., Gerst, I.: Parity results for certain partition functions and identities similar to theta function identities. Math. Comput. 48, 29–38 (1987) CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Chen, S.-L., Huang, S.-S.: Identities for certain products of theta functions. Ramanujan J. 8, 5–12 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Cooper, S., Hirschhorn, M.: On some sum-to-product identities. Bull. Aust. Math. Soc. 63, 353–365 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 4th edn. Oxford University Press, London (1960) zbMATHGoogle Scholar
  16. 16.
    Huard, J.G., Ou, Z.M., Spearman, B.K., Williams, K.S.: Elementary evaluation of certain convolution sums involving divisor functions. In: Bennett, M.A., et al. (eds.) Number Theory for the Millennium, II, pp. 229–274. A K Peters, Natick (2002) Google Scholar
  17. 17.
    Liouville, J.: Sur la forme \(x^{2}+y^{2}+3(z^{2}+t^{2})\). J. Math. Pures Appl. 5, 147–152 (1860) Google Scholar
  18. 18.
    Liouville, J.: Sur la forme \(x^{2}+y^{2}+2(z^{2}+t^{2})\). J. Math. Pures Appl. 5, 269–272 (1860) Google Scholar
  19. 19.
    Liouville, J.: Sur la forme \(x^{2}+y^{2}+4(z^{2}+t^{2})\). J. Math. Pures Appl. 5, 305–308 (1860) Google Scholar
  20. 20.
    Liouville, J.: Sur la forme \(x^{2}+2y^{2}+4z^{2}+8t^{2}\). J. Math. Pures Appl. 6, 409–416 (1861) Google Scholar
  21. 21.
    Liouville, J.: Sur la forme \(x^{2}+2y^{2}+4z^{2}+4t^{2}\). J. Math. Pures Appl. 7, 62–64 (1862) Google Scholar
  22. 22.
    Liouville, J.: Sur la forme \(x^{2}+2y^{2}+8z^{2}+8t^{2}\). J. Math. Pures Appl. 7, 65–68 (1862) Google Scholar
  23. 23.
    Liouville, J.: Sur la forme \(x^{2}+y^{2}+2z^{2}+4t^{2}\). J. Math. Pures Appl. 7, 99–100 (1862) Google Scholar
  24. 24.
    Liouville, J.: Sur la forme \(x^{2}+y^{2}+4z^{2}+8t^{2}\). J. Math. Pures Appl. 7, 103–104 (1862) Google Scholar
  25. 25.
    Liouville, J.: Sur la forme \(x^{2}+y^{2}+8z^{2}+8t^{2}\). J. Math. Pures Appl. 7, 109–112 (1862) Google Scholar
  26. 26.
    Liouville, J.: Sur la forme \(x^{2}+xy+y^{2}+z^{2}+zt+t^{2}\). J. Math. Pures Appl. 8, 141–144 (1863) Google Scholar
  27. 27.
    Liouville, J.: Sur la forme \(x^{2}+xy+y^{2}+2z^{2}+2zt+2t^{2}\). J. Math. Pures Appl. 8, 308–310 (1863) Google Scholar
  28. 28.
    Liouville, J.: Sur la forme \(2x^{2}+2xy+2y^{2}+3z^{2}+3zt+3t^{2}\). J. Math. Pures Appl. 9, 183–184 (1864) Google Scholar
  29. 29.
    Liouville, J.: Sur la forme \(x^{2}+xy+y^{2}+3z^{2}+3zt+3t^{2}\). J. Math. Pures Appl. 9, 223–224 (1864) Google Scholar
  30. 30.
    Williams, K.S.: On the representations of a positive integer by the forms \(x^{2}+y^{2}+z^{2}+2t^{2}\) and \(x^{2}+2y^{2}+2z^{2}+2t^{2}\). Int. J. Mod. Math. 3, 225–230 (2008) zbMATHMathSciNetGoogle Scholar
  31. 31.
    Williams, K.S.: Number Theory in the Spirit of Liouville. Cambridge University Press, Cambridge (2011) zbMATHGoogle Scholar
  32. 32.
    Williams, K.S.: Fourier series of a class of eta quotients. Int. J. Number Theory 8, 993–1004 (2012) CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsYork UniversityNorth YorkCanada
  2. 2.Centre for Research in Algebra and Number Theory, School of Mathematics and StatisticsCarleton UniversityOttawaCanada

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