Fourier Coefficients of a Class of Eta Quotients of Weight 16 with Level 12
Author(s)
Barış Kendirli
ABSTRACT
Recently, Williams [1] and then Yao, Xia and Jin [2] 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
and 
and Yao, Xia and Jin, following the method of proof of Williams, expressed only even coefficients of 104 eta quotients in terms of 
and 
. Here, by using the method of proof of Williams, we will express the even Fourier coefficients of 360 eta quotients i.e., the Fourier coefficients of the sum, f(q) + f(?q), of 360 eta quotients in terms of 
and 
.
Recently, Williams [1] and then Yao, Xia and Jin [2] 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
and and Yao, Xia and Jin, following the method of proof of Williams, expressed only even coefficients of 104 eta quotients in terms of and . Here, by using the method of proof of Williams, we will express the even Fourier coefficients of 360 eta quotients i.e., the Fourier coefficients of the sum, f(q) + f(?q), of 360 eta quotients in terms of and .
Cite this paper
Kendirli, B. (2015) Fourier Coefficients of a Class of Eta Quotients of Weight 16 with Level 12. Applied Mathematics, 6, 1426-1493. doi: 10.4236/am.2015.68133.
Kendirli, B. (2015) Fourier Coefficients of a Class of Eta Quotients of Weight 16 with Level 12. Applied Mathematics, 6, 1426-1493. doi: 10.4236/am.2015.68133.
References
[1] Williams, K.S. (2012) Fourier Series of a Class of Eta Quotients. International Journal of Number Theory, 8, 993-1004.
http://dx.doi.org/10.1142/S1793042112500595 [2] Yao, O.X.M., Xia, E.X.W. and Jin, J. (2013) Explicit Formulas for the Fourier Coefficients of a Class of Eta Quotients. International Journal of Number Theory, 9, 487-503.
http://dx.doi.org/10.1142/S179304211250145X [3] Köhler, G. (2011) Eta Products and Theta Series Identities. Springer-Verlag, Berlin.
http://dx.doi.org/10.1007/978-3-642-16152-0 [4] Gordon, B. (1961) Some Identities in Combinatorial Analysis. Quarterly Journal of Mathematics, 12, 285-290. [5] Kac, V.G. (1978) Infinite-Dimensional Algebras, Dedekind’s η-Function, Classical Möbius Function and the Very Strange Formula. Advances in Mathematics, 30, 85-136.
http://dx.doi.org/10.1016/0001-8708(78)90033-6 [6] Macdonald, I.G. (1972) Affine Root Systems and Dedekind’s η-Function. Inventiones Mathematicae, 15, 91-143.
http://dx.doi.org/10.1007/BF01418931 [7] Zucker, I.J. (1987) A Systematic Way of Converting Infinite Series into Infinite Products. Journal of Physics A, 20, L13-L17.
http://dx.doi.org/10.1088/0305-4470/20/1/003 [8] Zucker, I.J. (1990) Further Relations amongst Infinite Series and Products: II. The Evaluation of Three-Dimensional Lattice Sums. Journal of Physics A, 23, 117-132.
http://dx.doi.org/10.1088/0305-4470/23/2/009 [9] Kendirli, B. (2015) Evaluation of Some Convolution Sums by Quasimodular Forms. European Journal of Pure and Applied Mathematics, 8, 81-110. [10] Kendirli, B. (2015) Evaluation of Some Convolution Sums and Representation Numbers of Quadratic Forms of Discriminant -135. British Journal of Mathematics and Computer Science, 6, 494-531.
http://dx.doi.org/10.9734/BJMCS/2015/13973 [11] Kendirli, B. (2014) Evaluation of Some Convolution Sums and the Representation Numbers. Ars Combinatoria, CXVI, 65-91. [12] Kendirli, B. (2012) Cusp Forms in 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, 49, 529-572.
http://dx.doi.org/10.4134/BKMS.2012.49.3.529 [13] Kendirli, B. (2012) Cusp Forms in 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, 2012, Article ID: 303492. [14] Kendirli, B. (2012) The Bases of , and the Number of Representations of Integers. Mathematical Problems in Engineering, 2013, Article ID: 695265. [15] Alaca, A., Alaca, S. and Williams, K.S. (2006) On the Two-Dimensional Theta Functions of Borweins. Acta Arithmetica, 124, 177-195.
http://dx.doi.org/10.4064/aa124-2-4 [16] Alaca, A., Alaca, S. and Williams, K.S. (2006) Evaluation of the convolution sums and . Advances in Theoretical and Applied Mathematics, 1, 27-48. [17] Gordon, B. and Robins, S. (1995) Lacunarity of Dedekind η-Products. Glasgow Mathematical Journal, 37, 1-14.
http://dx.doi.org/10.1017/S0017089500030329 [18] Diamond, F. and Shurman, J. (2005) A First Course in Modular Forms. Springer Graduate Texts in Mathematics 228. Springer, New York.
[1] Williams, K.S. (2012) Fourier Series of a Class of Eta Quotients. International Journal of Number Theory, 8, 993-1004.
http://dx.doi.org/10.1142/S1793042112500595 [2] Yao, O.X.M., Xia, E.X.W. and Jin, J. (2013) Explicit Formulas for the Fourier Coefficients of a Class of Eta Quotients. International Journal of Number Theory, 9, 487-503.
http://dx.doi.org/10.1142/S179304211250145X [3] Köhler, G. (2011) Eta Products and Theta Series Identities. Springer-Verlag, Berlin.
http://dx.doi.org/10.1007/978-3-642-16152-0 [4] Gordon, B. (1961) Some Identities in Combinatorial Analysis. Quarterly Journal of Mathematics, 12, 285-290. [5] Kac, V.G. (1978) Infinite-Dimensional Algebras, Dedekind’s η-Function, Classical Möbius Function and the Very Strange Formula. Advances in Mathematics, 30, 85-136.
http://dx.doi.org/10.1016/0001-8708(78)90033-6 [6] Macdonald, I.G. (1972) Affine Root Systems and Dedekind’s η-Function. Inventiones Mathematicae, 15, 91-143.
http://dx.doi.org/10.1007/BF01418931 [7] Zucker, I.J. (1987) A Systematic Way of Converting Infinite Series into Infinite Products. Journal of Physics A, 20, L13-L17.
http://dx.doi.org/10.1088/0305-4470/20/1/003 [8] Zucker, I.J. (1990) Further Relations amongst Infinite Series and Products: II. The Evaluation of Three-Dimensional Lattice Sums. Journal of Physics A, 23, 117-132.
http://dx.doi.org/10.1088/0305-4470/23/2/009 [9] Kendirli, B. (2015) Evaluation of Some Convolution Sums by Quasimodular Forms. European Journal of Pure and Applied Mathematics, 8, 81-110. [10] Kendirli, B. (2015) Evaluation of Some Convolution Sums and Representation Numbers of Quadratic Forms of Discriminant -135. British Journal of Mathematics and Computer Science, 6, 494-531.
http://dx.doi.org/10.9734/BJMCS/2015/13973 [11] Kendirli, B. (2014) Evaluation of Some Convolution Sums and the Representation Numbers. Ars Combinatoria, CXVI, 65-91. [12] Kendirli, B. (2012) Cusp Forms in 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, 49, 529-572.
http://dx.doi.org/10.4134/BKMS.2012.49.3.529 [13] Kendirli, B. (2012) Cusp Forms in 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, 2012, Article ID: 303492. [14] Kendirli, B. (2012) The Bases of , and the Number of Representations of Integers. Mathematical Problems in Engineering, 2013, Article ID: 695265. [15] Alaca, A., Alaca, S. and Williams, K.S. (2006) On the Two-Dimensional Theta Functions of Borweins. Acta Arithmetica, 124, 177-195.
http://dx.doi.org/10.4064/aa124-2-4 [16] Alaca, A., Alaca, S. and Williams, K.S. (2006) Evaluation of the convolution sums and . Advances in Theoretical and Applied Mathematics, 1, 27-48. [17] Gordon, B. and Robins, S. (1995) Lacunarity of Dedekind η-Products. Glasgow Mathematical Journal, 37, 1-14.
http://dx.doi.org/10.1017/S0017089500030329 [18] Diamond, F. and Shurman, J. (2005) A First Course in Modular Forms. Springer Graduate Texts in Mathematics 228. Springer, New York.