Fourıer coeffıcıents of a class of ETA quotıents of weıght 20 wıth level 12

  • Authors

    • Barış Kendirli Aydın University Istanbul/Turkey
    2015-10-02
    https://doi.org/10.14419/ijams.v3i2.5247
  • Dedekind eta function, eta quotients, Fourier series.

  • Abstract

    Williams and later Yao, Xia and Jin discovered explicit formulas for the coefficients of the Fourier series expansions of a class of eta quotients. Williams expressed all coefficients of 126 eta quotients in terms of σ(n),σ((n/2)),σ((n/3)) and σ((n/6)) and Yao, Xia and Jin, following the method of proof of Williams, expressed only even coefficients of 104 eta quotients in terms of σ₃(n),σ₃((n/2)),σ₃((n/3)) and σ₃((n/6)).Here, we will express the even Fourier coefficients of 570 eta quotients in terms of σâ‚₉(n),σâ‚₉((n/2)),σâ‚₉((n/3)),σâ‚₉((n/4)),σâ‚₉((n/6)) and σâ‚₉((n/(12))).

  • References

    1. [1] Alaca, S. Alaca and K. S. Williams, On the two-dimensional theta functions of Borweins, Acta Arith. 124 (2006), 177-195.

      [2] Alaca, S. Alaca and K. S. Williams, Evaluation of the convolution sums ∑_{l+12m=n}σ(l)σ(m) and ∑_{3l+4m=n}σ(l)σ(m), Adv. Theor. Appl. Math. 1(2006), 27-48.

      [3] Gordon, Some identities in combinatorial analysis, Quart. J. Math. Oxford Ser.12 (1961), 285-290.

      [4] Gordon and S. Robins, Lacunarity of Dedekind η-products, Glasgow Math. J. 37 (1995), 1-14.

      [5] F. Diamond, J. Shurman, A First Course in Modular Forms, Springer Graduate Texts in Mathematics 228, 2005.

      [6] V. G. Kac, Infinite-dimensional algebras, Dedekind's η-function, classical Möbius function and the very strange formula, Adv. Math. 30 (1978), 85-136.

      [7] Kendirli, Evaluation of Some Convolution Sums by Quasimodular Forms, European Journal of Pure and Applied Mathematics ISSN 13075543 Vol.8., No. 1, Jan. 2015, 81-110.

      [8] Kendirli, Evaluation of Some Convolution Sums and Representation Numbers of Quadratic Forms of Discriminant 135, British Journal of Mathematics and Computer Science, Vol 6/6, Jan. 2015, 494-531.

      [9] B. Kendirli, Evaluation of Some Convolution Sums and the Representation numbers, Ars Combinatoria Volume CXVI, July 2014, 65-91.

      [10] B. Kendirli, Cusp Forms in S₄(Γ₀(79)) and the number of representations of positive integers by some direct sum of binary quadratic forms with discriminant -79, Bulletin of the Korean Mathematical Society, Vol 49/3, 2012, 529-572.

      [11] B. Kendirli, Cusp Forms in S₄(Γ₀(47)) and the number of representations of positive integers by some direct sum of binary quadratic forms with discriminant -47, International Journal of Mathematics and Mathematical Sciences Vol. 2012, Article ID 303492, 10 pages.

      [12] B. Kendirli, The Bases of M₄(Γ₀(71)),M₆(Γ₀(71)) and the Number of Representations of Integers, Mathematical Problems in Engineering Vol 2013, Article ID 695265, 34 pages

      [13] G. Köhler, Eta Products and Theta Series Identities (Springer-Verlag, Berlin, 2011).

      [14] G. Macdonald, Affine root systems and Dedekind's η-function, Invent. Math. 15 (1972), 91-143.

      [15] Olivia X. M. Yao, Ernest X. W. Xia and J. Jin, Explicit Formulas for the Fourier coefficients of a class of eta quotients, International Journal of Number Theory Vol. 9, No. 2 (2013), 487-503.

      [16] J. Zucker, A systematic way of converting infinite series into infinite products, J. Phys. A 20 (1987) L13-L17.

      [17] J. Zucker, Further relations amongst infinite series and products:II. The evaluation of three-dimensional lattice sums, J. Phys. A23 (1990), 117-132.

      [18] K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), 993-1004.

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    Kendirli, B. (2015). Fourıer coeffıcıents of a class of ETA quotıents of weıght 20 wıth level 12. International Journal of Advanced Mathematical Sciences, 3(2), 121-146. https://doi.org/10.14419/ijams.v3i2.5247