Fourier Coefficients of a Class of Eta Quotients of Weight 18 with Level 12

Baris Kendirli

doi:doi:10.11648/j.pamj.20150404.17

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Fourier Coefficients of a Class of Eta Quotients of Weight 18 with Level 12

Baris Kendirli

Received: 20 June 2015 Accepted: 4 August 2015 Published: 12 August 2015

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Abstract

Williams [16] and later Yao, Xia and Jin[15] 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 σ_3 (n),σ_3 (n/2),σ_3 (n/3) and σ_3 (n/6). Here, we will express the even Fourier coefficients of 324 eta quotients in terms of σ_17 (n),σ_17 (n/2),σ_17 (n/3),σ_17 (n/4),σ_17 (n/6) and σ_17 (n/12).

DOI	10.11648/j.pamj.20150404.17
Published in	Pure and Applied Mathematics Journal (Volume 4, Issue 4, August 2015)
Page(s)	178-188
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Dedekind Eta Function, Eta Quotients, Fourier Series

References

[1]	Ayşe Alaca, Şaban Alaca and Kenneth S. Williams, On the two-dimensional theta functions of Borweins, Acta Arith. 124 (2006) 177-195. Carleton University, Ottawa, Ontario, Canada K1S 5B6. E-mail: aalaca@math.carleton.ca, salaca@math.carleton.ca, williams@math.carleton.ca
[2]	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]	Basil Gordon, Some identities in combinatorial analysis, Quart. J. Math. Oxford Ser.12 (1961), 285-290. University of California, Los Angeles
[4]	Basil Gordon1 and Sınai Robins2 , Lacunarity of Dedekind η-products, Glasgow Math. J. 37 (1995), 1-14. 1University of California, Los Angeles, 2University of Northern Colorado, Greeley
[5]	Fred Diamond1, Jerry Shurman2, A First Course in Modular Forms,Springer Graduate Texts in Mathematics 228 1Brandeis University Waltham, MA 02454 USA, 2Reed College Portland, OR 97202 USA E-mail: fdiamond@brandeis.edu, jerry@reed.edu
[6]	Victor G. Kac, Infinite-dimensional algebras, Dedekind’s η-function, classical Möbius function and the very strange formula, Adv. Math. 30 (1978) 85-136. MIT, Cambridge, Massachusetts 02139
[7]	Barış Kendirli, "Evaluation of Some Convolution Sums by Quasimodular Forms", European Journal of Pure and Applied Mathematics ISSN 13075543 Vol.8., No. 1, Jan. 2015, pp. 81-110 Aydın University Istanbul/Turkey E-mail:baris.kendirli@gmail.com
[8]	"Evaluation of Some Convolution Sums and Representation Numbers of Quadratic Forms of Discriminant 135", British Journal of Mathematics and Computer Science, Vol6/6, Jan. 2015, pp. 494-531.
[9]	Evaluation of Some Convolution Sums and the Representation numbers, Ars Combinatoria Volume CXVI, July, pp 65-91.
[10]	Cusp Forms in S_4 (Γ_0 (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
[11]	Cusp Forms in S_4 (Γ_0 (47)) and the number of representations of positive integers by some direct sum of binary quadratic forms with discriminant -47,Hindawi , International Journal of Mathematics and Mathematical Sciences Vol 2012, 303492 10 pages
[12]	The Bases of M〖_4〗(Γ_0 (71)),M〖_6〗(Γ_0 (71)) and the Number of Representations of Integers, Hindawi, Mathematical Problems in Engineering Vol 2013, 695265, 34 pages
[13]	Günter Köhler, Eta Products and Theta Series Identities (Springer-Verlag, Berlin, 2011). University of Würzburg Am Hubland 97074 Würzburg Germany E-mail:koehler@mathematik.uni-wuerzburg.de
[14]	Ian Grant Macdonald, Affine root systems and Dedekind’s η-function, Invent. Math. 15 (1972), 91-143.
[15]	Olivia X. M. Yao1, Ernest X. W. Xia2 and J. Jin3, Explicit Formulas for the Fourier coefficients of a class of eta quotients, International Journal of Number Theory Vol. 9, No. 2 (2013) 487-503. 1,2Jiangsu University Zhenjiang, Jiangsu, 212013, P. R. China 3Nanjing Normal University Taizhou College, 225300, Jiangsu, P. R. China E-mail:yaoxiangmei@163.com, ernestxwxia@163.com, jinjing19841@126.com
[16]	Kenneth S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), 993-1004. Carleton University, Ottawa, Ontario, Canada K1S 5B6 E-mail: williams@math.carleton.ca.

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Baris Kendirli. (2015). Fourier Coefficients of a Class of Eta Quotients of Weight 18 with Level 12. Pure and Applied Mathematics Journal, 4(4), 178-188. https://doi.org/10.11648/j.pamj.20150404.17

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ACS Style

Baris Kendirli. Fourier Coefficients of a Class of Eta Quotients of Weight 18 with Level 12. Pure Appl. Math. J. 2015, 4(4), 178-188. doi: 10.11648/j.pamj.20150404.17

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Baris Kendirli. Fourier Coefficients of a Class of Eta Quotients of Weight 18 with Level 12. Pure Appl Math J. 2015;4(4):178-188. doi: 10.11648/j.pamj.20150404.17

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@article{10.11648/j.pamj.20150404.17,
  author = {Baris Kendirli},
  title = {Fourier Coefficients of a Class of Eta Quotients of Weight 18 with Level 12},
  journal = {Pure and Applied Mathematics Journal},
  volume = {4},
  number = {4},
  pages = {178-188},
  doi = {10.11648/j.pamj.20150404.17},
  url = {https://doi.org/10.11648/j.pamj.20150404.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20150404.17},
  abstract = {Williams [16] and later Yao, Xia and Jin[15] 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 σ_3 (n),σ_3 (n/2),σ_3 (n/3) and σ_3 (n/6). Here, we will express the even Fourier coefficients of 324 eta quotients in terms of σ_17 (n),σ_17 (n/2),σ_17 (n/3),σ_17 (n/4),σ_17 (n/6) and σ_17 (n/12).},
 year = {2015}
}

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TY  - JOUR
T1  - Fourier Coefficients of a Class of Eta Quotients of Weight 18 with Level 12
AU  - Baris Kendirli
Y1  - 2015/08/12
PY  - 2015
N1  - https://doi.org/10.11648/j.pamj.20150404.17
DO  - 10.11648/j.pamj.20150404.17
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 178
EP  - 188
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20150404.17
AB  - Williams [16] and later Yao, Xia and Jin[15] 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 σ_3 (n),σ_3 (n/2),σ_3 (n/3) and σ_3 (n/6). Here, we will express the even Fourier coefficients of 324 eta quotients in terms of σ_17 (n),σ_17 (n/2),σ_17 (n/3),σ_17 (n/4),σ_17 (n/6) and σ_17 (n/12).
VL  - 4
IS  - 4
ER  -

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Author Information

Baris Kendirli

Dept. of Math,. Ayd?n University, Istanbul, Turkey

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APA Style

Baris Kendirli. (2015). Fourier Coefficients of a Class of Eta Quotients of Weight 18 with Level 12. Pure and Applied Mathematics Journal, 4(4), 178-188. https://doi.org/10.11648/j.pamj.20150404.17

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ACS Style

Baris Kendirli. Fourier Coefficients of a Class of Eta Quotients of Weight 18 with Level 12. Pure Appl. Math. J. 2015, 4(4), 178-188. doi: 10.11648/j.pamj.20150404.17

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AMA Style

Baris Kendirli. Fourier Coefficients of a Class of Eta Quotients of Weight 18 with Level 12. Pure Appl Math J. 2015;4(4):178-188. doi: 10.11648/j.pamj.20150404.17

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@article{10.11648/j.pamj.20150404.17,
  author = {Baris Kendirli},
  title = {Fourier Coefficients of a Class of Eta Quotients of Weight 18 with Level 12},
  journal = {Pure and Applied Mathematics Journal},
  volume = {4},
  number = {4},
  pages = {178-188},
  doi = {10.11648/j.pamj.20150404.17},
  url = {https://doi.org/10.11648/j.pamj.20150404.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20150404.17},
  abstract = {Williams [16] and later Yao, Xia and Jin[15] 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 σ_3 (n),σ_3 (n/2),σ_3 (n/3) and σ_3 (n/6). Here, we will express the even Fourier coefficients of 324 eta quotients in terms of σ_17 (n),σ_17 (n/2),σ_17 (n/3),σ_17 (n/4),σ_17 (n/6) and σ_17 (n/12).},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Fourier Coefficients of a Class of Eta Quotients of Weight 18 with Level 12
AU  - Baris Kendirli
Y1  - 2015/08/12
PY  - 2015
N1  - https://doi.org/10.11648/j.pamj.20150404.17
DO  - 10.11648/j.pamj.20150404.17
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 178
EP  - 188
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20150404.17
AB  - Williams [16] and later Yao, Xia and Jin[15] 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 σ_3 (n),σ_3 (n/2),σ_3 (n/3) and σ_3 (n/6). Here, we will express the even Fourier coefficients of 324 eta quotients in terms of σ_17 (n),σ_17 (n/2),σ_17 (n/3),σ_17 (n/4),σ_17 (n/6) and σ_17 (n/12).
VL  - 4
IS  - 4
ER  -

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