Pure and Applied Mathematics Journal
Volume 4, Issue 4, August 2015, Pages: 178-188

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

Baris Kendirli

Dept. of Math,. Aydın University, Istanbul, Turkey

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To cite this article:

Barış Kendirli. Fourier Coefficients of a Class of Eta Quotients of Weight 18 with Level 12.Pure and Applied Mathematics Journal.Vol.4, No. 4, 2015, pp. 178-188. doi: 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  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, we will express the even Fourier coefficients of 324 eta quotients in terms of  and

Keywords: Dedekind Eta Function, Eta Quotients, Fourier Series


1. Introduction

The divisor function  is defined for a positive integer  by

(1)

The Dedekind eta function is defined by

(2)

Where

(3)

and an eta quotient of level  is defined by

(4)

It is interesting and important to determine explicit formulas of the Fourier coefficients of eta quotients, because they are the building blocks of modular forms of level n and weight k. The book of Köhler [13] (Chapter 3, pg.39) describes such expansions by means of Hecke Theta series and develops algorithms for the determination of suitable eta quotients. One can find more information in [3], [6], [14]. I have determined the Fourier coefficients of the theta series associated to some quadratic forms, see [7], [8], [9][10], [11] and [12].

It is known that Williams, see [16] discovered explicit formulas for the coefficients of Fourier series expansions of a class of 126 eta quotients in terms of  and  One example is as follows:

gives the expansion found by Williams.

Then Yao, Xia and Jin [15] expressed the even Fourier coefficients of 104 eta quotients in terms of  and One example is as follows:

where the even coefficients are obtained. After that we find that we can express the even Fourier coefficients of 324 eta quotients in terms of  and see Table 3. One example is as follows:

We also see that the odd Fourier coefficients of 650 eta quotients are zero and even coefficients can be expressed by simple formula.

2. Main Body

Now let

Now we can state our main Theorem:

Theorem 1. Let  be non-negative integers satisfying

(5)

Define the integers  by

(6)

(7)

(8)

(9)

(10)

(11)

They are functions of  by (3). Now define integers

by

(12)

(13)

(14)

(15)

(16)

Define the rational numbers

and  as in www.fatih.edu.tr/~bkendirli/weight18/Table 1. Here    and

where for n

In particular,

for n

Proof. It follows from (6-11) that

(17)

(18)

Now we will use p-k parametrization of Alaca, Alaca and Williams, see [1]:

(19)

where the theta function  is defined by

etting  in (12), and multiplying both sides by we obtain

Alaca, Alaca and Williams [2] have established the following representations in terms of p and k:

(20)

(21)

(22)

(23)

(24)

(25)

Therefore, since

we have



















































































We can also similarly determine  and  in terms of  and  as in www.fatih.edu.tr/~bkendirli/weight18/Table 3. Obviously,  are functions of , see (3),( 19). We see that    by [4]. Now










































where

So


Therefore, for n=1,2,…



since it is easy to see that

hence,

and, for n=1,2,…

3. Conclusion

1. We have found 324 eta quotients, see www.fatih.edu.tr/~bkendirli/weight18/Table 4, such that, for n=1,2, 


and 650 eta quotients, such that for n=1,2, 



2.  is 31 dimensional,  is 37 dimensional,see [5] (Chapter 3, pg.87 and Chapter 5, pg.197), and generated by









where  is the unique newform in ;  is the unique newform in ;  and  are the unique newforms in ,  are the unique newforms in ,  and  are the unique newforms in  and  are the unique newforms in . By taking  as a root of , and  as the root of  we see  as linear combinations of this basis in www.fatih.edu.tr/~bkendirli/weight18/Table 3.


References

  1. Ayşe Alaca, Şaban Alaca and Kenneth S. Williams, On the twodimensional 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 and Adv. Theor. Appl. Math. 1(2006), 2748.
  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 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 USAE-mail:fdiamond@brandeis.edu, jerry@reed.edu
  6. Victor G. Kac, Infinitedimensional algebras, Dedekind’s 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 35, British Journal of Mathematics and Computer Science, Vol6/6, Jan. 2015, pp. 494-531.
  9. Evaluation of Some Convolution Sums andthe Representation numbers, Ars Combinatoria Volume CXVI, July, pp 65-91.
  10. Cusp Forms in and the number of representations of positive integers by some direct sum of binary quadratic forms with discriminant79, Bulletin of the Korean Mathematical Society Vol 49/3 2012
  11. Cusp Forms in and the number of representations of positive integers by some direct sum of binary quadratic forms with discriminant47,Hindawi , International Journal of Mathematics and Mathematical Sciences Vol 2012, 303492 10 pages
  12. The Bases of MM 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 (SpringerVerlag, Berlin, 2011).University of Würzburg Am Hubland 97074 Würzburg GermanyE-mail:koehler@mathematik.uni-wuerzburg.de
  14. Ian Grant Macdonald, Affine root systems and Dedekind’s 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. ChinaE-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 5B6E-mail: williams@math.carleton.ca.

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