Barış Kendirli
Istanbul Kultur University, Istanbul, Turkey.
DOI: 10.4236/am.2015.68133   PDF    HTML   XML   3,363 Downloads   4,260 Views   Citations

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 .

Keywords

Dedekind Eta Function, Eta Quotients, Fourier Series

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The divisor function is defined for a positive integer i by

(1)

The Dedekind eta function is defined by

(2)

where

(3)

And an eta quotient of level n is defined by

(4)

It is interesting and important to determine explicit formulas of the Fourier coefficients of eta quotients since they are the building blocks of modular forms of level n and weight k. The book of Köhler [3] (Chapter 3, p. 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 [4] -[8] . I have determined the Fourier coefficients of the theta series associated to some quadratic forms, see [9] - [14] .

Recently, Williams, see [1] 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 [2] expressed the even Fourier coefficients of 104 eta quotients in terms of

and. One example is as follows:

where the even coefficients are obtained. Motivated by these two results, we find that we can express the even Fourier coefficients of 360 eta quotients in terms of and,

see Table 2. One example is as follows:

We see that the odd Fourier coefficients of 875 eta quotients are zero and even coefficients can be expressed by simple formula. 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 q by (3). Now define integers

by

(12)

(13)

(14)

(15)

(16)

Define the rational numbers

and as in Table 1. Here

and

where for

In particular,

for

Proof. It follows from (6)-(11) that

(17)

(18)

Now we will use p-k parametrization of Alaca, Alaca and Williams, see [15] :

(19)

where the theta function is defined by

Setting x = p in (12), and multiplying both sides by k¹⁶ we obtain

Alaca, Alaca and Williams [16] have established the following representations in terms of p and k:

(20)

(21)

(22)

(23)

(24)

(25)

Therefore, since

we immediately obtain:

It is easy to check the following expressions by (20)-(25)

Obviously, are functions of q, see (3), (19). We see that

by [17] . Now

where

So

Therefore, for

since it is easy to see that

hence,

and, for

Remark 2 We have found 360 eta quotients, see Table 2, such that, for

and 875 eta quotients, such that for

Remark 3 If f is an eta quotient, then is also an eta quotient, so the coefficients of

are exactly the even coefficients of f. In particular, it means that we have obtained all coefficients of some sum of 360 eta quotients.

Remark 4 is 27 dimensional, is 33 dimensional, see [18] (Chapter 3, p. 87 and Chapter 5, p. 197), and generated by

where is the unique newform in; is the unique newform in;

are the unique newforms in, is the unique newform in, are

the unique newforms in and are the unique newforms in. By

taking t as a root of, we see as linear combinations in Table 3.

Conflicts of Interest

The authors declare no conflicts of interest.

References

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[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
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