FOURIER COEFFICIENTS OF A CLASS OF ETA QUOTIENTS

Transcription

1 #A INTEGERS () FOURIER COEFFICIENTS OF A CLASS OF ETA QUOTIENTS Barış Kendirli Department of Mathematics, Istanbul Aydın University, Turkey bariskendirli@aydin.edu.tr Received: //, Revised: //, Accepted: //, Published: // Abstract Recently, Williams, and then Yao, Xia and Jin, discovered explicit formulas for the coe cients of the Fourier series expansions of a class of eta quotients. Williams expressed all coe cients of one hundred and twenty-six eta quotients in terms of (n), ( n ), ( n ) and ( n ), and Yao, Xia and Jin, following the method of Williams proof, expressed only the even coe cients of one hundred and four eta quotients in terms of (n), ( n ), ( n ) and ( n ). Here, by using the method of Williams proof, we will express the odd Fourier coe cients of seventy-four eta quotients f(q) n in terms of (n ) and, i.e., the Fourier coe cients of the di erence f(q) f( q) of seventy-four eta quotients; and we will express the even Fourier coe cients of sixty eta quotients, i.e., the Fourier coe cients of the sum f(q) + f( q) of sixty eta quotients, in terms of (n), n, n, n, n and n.. Introduction The divisor function i (n) is defined for a positive integer i by P d positive integer,d n i (n) = di, if n is a positive integer, if n is not a positive integer The Dedekind eta function is defined by. (z) = q / Y n= ( q n ), where q = e iz, z H = {x + iy : y > },

2 INTEGERS: () and an eta quotient of level n is defined by f(z) = Y m n (mz) am, n, m N, a m Z. It is interesting and important to find explicit formulas for the Fourier coe cients of eta quotients since they are the building blocks of modular forms of level n and weight k. The book of Koehler (see [], p. ) describes such expansions by means of Hecke theta series, and it develops algorithms for the determination of suitable eta quotients. One can find more information in [,,,, ]. Additionally, the present author has determined the Fourier coe cients of the theta series associated with some quadratic forms (see [], [], [], [], [] and []). Recently, Williams [] discovered explicit formulas for the coe cients of the Fourier series expansions of a class of one hundred and twenty-six eta quotients in terms of (n), ( n ), ( n ) and ( n ). An example is where (z) (z) (z) (z) (z) (z) = + X c(n)q n, n= c(n) = (n) (n/) + (n/) + (n/) (n/). Then, Yao, Xia and Jin [] expressed the even Fourier coe cients of one hundred and four eta quotients in terms of (n), ( n ), ( n ) and ( n ). One example is (z) (z) (z) (z) (z) (z) = + X c(n)q n, n= where c(n) = (n) (n/) (n/) + (n/). Motivated by these two results, we find that we can express the odd Fourier coe cients of seventy-four eta quotients in terms of (n + ) and ( n+ ); see Table/A. An example is where (z) (z) (z) (z) = + X c(n)q n, c(n ) =. n=

3 INTEGERS: () We can also express the even Fourier coe cients of sixty eta quotients in terms of (n), n, n, n, n n and ; see Table/A. An example is where (z) (z) (z) (z) = + X c(n)q n, n= c(n) = (n) (n) + +. Now we define some rational numbers and functions for the theorem: let b, b,, b be non-negative integers satisfying b + b + + b apple. Define the integers a, a, a, a, a and a by a = b + b b b b +, a = b + b + b + b + b, a = b + b + b + b + b, a = b b b b + b +, a = b b b b b +, a = b + b + b + b + b. () Now define the integers k, k, k, k, k, k, k, k, k, k, k, k and k by b+b xb ( x) b ( + x) b ( + x) b ( + x) b = k + k x + k x + k x + k x + k x + k x +k x + k x + k x + k x + k x + k x. () Let f = f = f = f = f = X f (n) q n = (z) (z) (z) (z), n= X f (n) q n = (z) (z) (z) (z), n= X f (n) q n = (z) (z) (z) (z), X f (n) q n = (z) (z) (z) (z), X f (n) q n = (z), n= n= n=

4 INTEGERS: () f = f = X n= X n= f (n) q n = (z) (z) (z) (z), f (n) q n = (z) (z) (z) (z). Now define integers k, k, k, k, k, k, k, k, k, k, k, k and k by b+b xb ( x) b ( + x) b ( + x) b ( + x) b = k + k x + k x + k x + k x + k x + k x + k x + k x +k x + k x + k x + k x. Define the rational numbers c, c, c, c, c, c, r, r, r, r, r, r and r by c = k c = k + k + k k + k k + k k + k + c = k k k k + k, k k k + k k + k + k k + + k + k c = k k + k, k + k k k + k k k + k k k + k + k + k + k k, k + k k k k, k + k

5 INTEGERS: () c = k k + k k + k k k + k k + k k + k + k, c = k + k + k r = k k + k k k + k k + k k k, k + k k + k k + k k + k k k + k k + k, r = k + k k + k k + k r = k r = k + k k + k k + k, k + k k + k + k + k + k k + k k + k + k k + k k + k k k k, k + k k + k k,

6 INTEGERS: () r = k k + k k + k k k + k k + k k + k, r = k + k k k + k k + k r = k + k k + k Now we can state our main theorem: k + k k + k k + k k + k k, k k k + k. Theorem. The functions f, f,..., f are in S ( ()), while f is in M ( ()), and X a (z) a (z) a (z) a (z) a (z) a (z) = (b ) + c(n)q n, where for n N, c(n) = c (n) c c c c c +r f (n) + r f (n) + r f (n) + r f (n) + r f (n) + r f (n) + r f (n). In particular, c(n) = c (n) c (n) c (c + c ) n (c c ) + r f (n) + r f (n) + r f (n), n c(n ) = c (n ) c +r f (n ) + r f (n ) + r f (n ) + r f (n ), for n N. Proof. It follows from () that a + a + a + a + a + a = b, n=

7 INTEGERS: () and a a + a + a + a + a + a = a a a a a = b b. Now we will use (p, k), the parametrization of Alaca, Alaca and Williams (see []): p (q) = ' (q) ' q ' (q ), k (q) = ' q ' (q), where the theta function ' (q) is defined by ' (q) = X q n. Setting x = p in () and multiplying both sides by k, we obtain k b +b pb ( p) b ( + p) b ( + p) b ( + p) b = (k + k p + k p + k p + k p + k p + k p +k p + k p + k p + k p + k p + k p )k. Alaca, Alaca and Williams [] have established the following representations in terms of p and k: (q) = / p / ( p) / ( + p) / ( + p) / ( + p) / k /, () q = / p / ( p) / ( + p) / ( + p) / ( + p) / k /, () q = / p / ( p) / ( + p) / ( + p) / ( + p) / k /, () q = / p / ( p) / ( + p) / ( + p) / ( + p) / k /, () q = / p / ( p) / ( + p) / ( + p) / ( + p) / k /, () q = / p / ( p) / ( + p) / ( + p) / ( + p) / k /. () We also have the following: E (q) = X n= (n) q n = ( p p p p p p p p p p p + p )k,

8 INTEGERS: () E (q ) = ( + p p p p p p p p p p +p + p )k, E (q ) = ( + p + p p p p p p p p + p + p + p )k, E (q ) = ( + p + p + p p p p + p + p p p + p + p )k, E (q ) = ( + p + p + p p p p p p + p + p + p + p )k, E (q ) = ( + p + p + p p p p + p + p + p + p + p + p )k. It is easy to check the following expressions using ()-( ): X f = f (n) q n = (z) (z) (z) (z) = f = f = n= p + p + p p p p X f (n) q n = (z) (z) (z) (z) n= = ( p + p + p p p p + p + p + p + p )k, X n= f (n) q n = (z) (z) (z) (z) = ( p + p + p p p + p + p + p )k, p k,

9 INTEGERS: () f = X n= f (n) q n = (z) (z) (z) (z) = ( p + p + p + p p p p p )k, X f = f (n) q n = (z) f = n= p = ( p + p + p p p + p + p p p p )k, X n= f (n) q n = (z) (z) (z) (z) = ( p + p + p p p p + p + p + p )k, f = X f (n) q n = (z) (z) (z) (z) n= = ( p + p + p + p + p + p + p + p )k. We see that f,... f, S ( ()), f M ( ()) and by []. Now ord / f =, ord / f =, a (z) a (z) a (z) a (z) a (z) a (z) Y = q b ( q n ) a q n a q n a q n a q n a n a q n= = a a a a a a a p + a + a + a + a + a a ( p) + a + a + a + a + a ( + p) a + a + a + a + a + a a ( + p) + a + a + a + a + a a ( + p) + a + a + a + a + a k a +a +a +a +a +a = k b+b pb ( p) b ( + p) b ( + p) b ( + p) b

10 INTEGERS: () = k (k + k p + k p + k p + k p + k p + k p +k p + k p + k p + k p + k p + k p )! = c X (n) q n + c X + c + c +r q Y n= +r q Y n= +r q Y +r q +r.q +r q = (b ) n= Y n= Y n= Y n= +c n= X n= X n= (n) q n! (n) q n! + c + c n= X n= X n= q n q n q n q n q n q n q n q n q n q n ( q n ) ( q n ) q n n q ( q n ) ( q n ) q n + r q Y n= q n n q ( q n ) ( q n ) X (c (n) + c n= + c q n n q ( q n ) ( q n ) (n) q n! + c + c (n) q n! (n) q n! )q n + r f (n)q n r f (n)q n, where So (b ) = if b = if b =. c(n) = (c (n) + c + c n +c + c + c ) + r f (n) r f (n).

11 INTEGERS: () Therefore, for n =,,..., c(n) = c (n) c (n) c (c + c ) n (c c ) + r f (n) + r f (n) + r f (n), n c(n ) = c (n ) c +r f (n ) + r f (n ) + r f (n ) + r f (n ), since it is easy to see that n =, and, for n =,,..., f (n ) = f (n ) = f (n ) =, f (n) = f (n) = f (n) = f (n) =. These formulas are valid for nontrivial eta quotients; see quotients. Among them, we have found eta quotients (see Table/A), such that c(n) = c (n) c (n) c (c + c ) n (c c ) + r f (n) + r f (n) + r f (n), n c(n ) = c (n ) c = ; and eta quotients ( see Table/A), such that c(n) = c (n) c (n) c (c + c ) n (c c ), n c(n ) = c (n ) c + r f (n ) + r f (n ) +r f (n ) + r f (n ). Remark : The coe cients k, k, k, k, k, k, k, k, k, k, k, k, k and r, r, r, r, r, r, r are given in Table/B and Table/B.

12 INTEGERS: () Remark : If f is an eta quotient, then f( q) is also an eta quotient, the coe - cients of (f(q) f( q)) are exactly the odd coe cients of f, and the coe cients of (f(q) + f( q)) are exactly the even coe cients of f. In particular, this means that we have obtained all coe cients of the di erence of seventy-four eta quotients and the sum of sixty eta quotients. Remark : The space S ( ()) is seven-dimensional(see [, Chapter and ]), and it is generated by,,, (z),, (z),,,, (z),,,, (z), where, is the unique newform in S ( ()),, is the unique newform in S ( ()), and, is the unique newform in S ( ()). By simple calculation, we see that f =, (z) +, (z) +, (z), f =, (z), (z) +, (z), f =, (z), (z) +, (z), f =, (z) +, (z) +, (z) +, (z) +, (z) +, (z), (z), f =, (z), f =, (z) +, (z) +, (z) +, (z), (z) +, (z), (z). Note that f is in M ( ())\S ( ()), so it can be written in the form f =, (z) +, (z) +, (z) +, (z) +, (z) +, (z), (z) E (z) + E (z) + E (z) E (z) E (z) + E (z). Acknowledgement. The author thanks the referee for his/her useful suggestions.

13 INTEGERS: () References [] P S. Alaca A. Alaca and K. S. Williams. Evaluation of the convolution sums l+m=n (l) (m) and P l+m=n (l) (m). Adv. Theor. Appl. Math., :,. [] S. Alaca A. Alaca and K. S. Williams. On the two-dimensional theta functions of Borweins. Acta Arith., :,. [] J. Shurman F. Diamond. A First Course in Modular Forms, volume. Springer Graduate Texts in Math.,. [] B. Gordon. Some identities in combinatorial analysis. Quart. J. Math. Oxford Ser., :,. [] B. Gordon and S. Robins. Lacunarity of Dedekind eta-products. Glasgow Math. J., :,. [] V. G. Kac. Infinite-dimensional algebras, Dedekind s eta-function, classical Moebius function and the very strange formula. Adv. Math., :,. [] B. Kendirli. Cusp forms in S(Gamma()) and the number of representations of positive integers by some direct sum of binary quadratic forms with discriminant -. Int. J. Math. Math. Sci., pages,. [] B. Kendirli. Cusp forms in S(Gamma()) and the number of representations of positive integers by some direct sum of binary quadratic forms with discriminant -. Bull. Korean Math. Soc., :,. [] B. Kendirli. The bases of M(Gamma()), M(Gamma()) and the number of representations of integers. Math. Probl. Eng., :,. [] B. Kendirli. Evaluation of some convolution sums and the representation numbers. Ars Combin., :,. [] B. Kendirli. Evaluation of some convolution sums and representation numbers of quadratic forms of discriminant -. British J. Math. Comp. Sci., :,. [] B. Kendirli. Evaluation of some convolution sums by quasimodular forms. Adv. Math., :,. [] G. Koehler. Eta Products and Theta Series Identities. Springer-Verlag, Berlin,. [] I. G. Macdonald. A ne root systems and Dedekind s eta-function. Invent. Math., :,. [] Ernest X. W. Xia Olivia X. M. Yao and J. Jin. Explicit formulas for the Fourier coe cients of a class of eta quotients. Int. J. Number Theory, :,. [] K. S. Williams. Fourier series of a class of eta quotients. Int. J. Number Theory., :,. [] I. J. Zucker. Further relations amongst infinite series and products: II. the evaluation of three dimensional lattice sums. J. Phys., :,. [] I. J. Zucker. A systematic way of converting infinite series into infinite products. J. Phys., :L L,.

14 INTEGERS: () Appendix Table /A No a a a a a a c c c c / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /

15 INTEGERS: () / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /

16 INTEGERS: () Table /B No k k k k k k k k k k k k k r r r / /

17 INTEGERS: () Table /A No b b b b b a a a a a a c c c c c c

18 INTEGERS: ()

19 INTEGERS: () Table /B No k k k k k k k k k k k k k r r r r

20 INTEGERS: ()