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

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1 Pure and Applied Mathematics Journal 2015; 4(4): 17-1 Published online August ( doi: /j.pamj ISSN: (Print); ISSN: (Online) Fourier Coefficients of a Class of Eta Quotients of Weight 1 with Level 12 Baris Kendirli Dept. of Math. Aydın University Istanbul Turkey address: baris.kendirli@gmail.com To cite this article: Barış Kendirli. Fourier Coefficients of a Class of Eta Quotients of Weight 1 with Level 12. Pure and Applied Mathematics Journal. Vol. 4 No pp doi: /j.pamj 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 ():=! if is a positive integer and (1) ():=0 if n is not a positive integer. The Dedekind eta function is defined by Where 1(2):=3 / (2) 3:= : ;< 2 > =?@B:B > 0D (3) and an eta quotient of level is defined by E(2):= 1 F (G2) H IG NK F L. (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.3) 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] [] [][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: 1 1 (42)1 (62) 1 (2)1 (32)1 (122) 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: 1 Q 1 (32) 1 (2)1 Q (42)1 (62)1(122) 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: 1 R 1 S (42)1 (62)1 (122). We also see that the odd Fourier coefficients of 650 eta quotients are zero and even coefficients can be expressed by

2 Pure and Applied Mathematics Journal 2015; 4(4): simple formula. 2. Main Body Now let E := TE S S S S E Q = TE Q S E = TE S S E R = TE R S E U = TE U S () = 1 (42)1 U (62)1 (122) () = 1 (42)1 S (62)1 S (122) 1 R () = 1 (42)1 S (62)1 R (122) 1 R () = 1R (42)1 (62)1 R (122) 1 () = 1 (42)1 (62)1 (122) 1 Q () = 1R (42)1 (62)1 (122) () = 1S (42)1 (62)1 (122) () = 1 1 R (42)1 R (62) 1 (122) () = 1 1 (42)1 (122) 1(62) E S = TE S () =1 R 1 S (42)1 (62)1 (122) S S S S S E Q = TE Q S () = 1R (42)1 R (62)1 R (122) 1 R () = 1 1 (42)1 (122) 1 Q (62) () = 1 1 (42)1 U (62) (122) () = 1S 1 S (42)1 (122) 1 R (62) () = 1 1 U (62)1 (122) (42) S S () = 1S 1 (62)1 (122) (42) () = 1 1 (62)1 (122) (42) E R = TE R () = 1 1 Q (42)1 (62)1(122) S E U = TE U S E S = TE S S S S S S () = 1R (42)1 (62)1 R (122) () = 1 (42)1 (62)1 (122) () = 1 (42)1 (62)1 (122) () = 1 (42)1 (62)1 S (122) () = 1 (42)1 (62)1 R (122) () = 1R 1 R (62)1 (122) (42) E Q = TE Q () =1 R (42)1 R (122) S S S E R = TE S E U = TE S E S = TE S S S () = 11 (42)1 Q (62)1 (122) () 1 1 U (42)1 (62) 1(122) () 1 1 R (42)1 R (122) (62) () 1 1 R (42)1 (62) 1 (122) () = 1 1 (42)1 (122) (62) () 1R 1 (42)1 R (62) 1. (122) Now we can state our main Theorem: Theorem 1. Let V V V Q be non-negative integers satisfying

3 10 Barış Kendirli: Fourier Coefficients of a Class of Eta Quotients of Weight 1 with Level 12 V V V Q 36. (5) Define the integers K K K K K K by K :=V 2V 2V 4V V Q 36 (6) K :=3V V 3V 10V V Q 0 (7) K :=3V 2V 6V 4V 3V Q 10 () K :=2V V V 4V 2V Q 36 () K :=V 7V V 10V 7V Q 270 (10) K := 6V 3V 3V 4V 2V Q 10. (11) by They are functions of 3 by (3). Now define integers \ S \ \ \ \ \ Q \ \ \ R \ U \ S \ \ \ \ \ Q \ \ \ R \ U \ S \ \ \ \ \ Q \ \ \ R \ U \ S \ \ \ \ \ Q and \ ]^_]`@a^(1@) a b(1@) a c(12@) a d(2@) a` = \ @ \ Q \ R (12) Define the rational numbers \ U \ S \ Q (13) \ R \ U \ S (14) \ Q (15) \ R \ U \ S \ Q (16) e e e e e e f f f S and f as in 1. Here?E E D\?E D i R jγ S (12)l E m R jγ S (12)l\ i R jγ S (12)l and where for n N 1 H^(2)1H b1 H c(32)1 H d(42)1 H n(62)1 H^b(122) =o(v )Te()3 e() =e ()e 2 e 3 e 4 e 6 e 12 f E () f E (). In particular e(2) =e (2)e ()e 2 ( e e ) Q 3 (e e ) Q 6 f E (2) f E (2) for n N. Proof. It follows from (6-11) that e(21) = e Q (21)e Q r 21 sf 3 E (21) f E (21) K_12K_23K_34K_46K_612K_12 =24V_1 (17) K K K K K K = 36 (1) K 6 K 3 K 6 2K 3 K 3 2K 3 = V V Q. Now we will use p-k parametrization of Alaca Alaca and Williams see [1]: where the theta function y(3) is defined by u(3):= vb (w)xv b jw c l \(3):= vc jw c l (1) v b (w c ) v(w)

4 Pure and Applied Mathematics Journal 2015; 4(4): y(3) =T3 b. = u in (12) and multiplying both sides by \ R we obtain x \ R 2 a^za`u a^(1u) a b(1u) a c(12u) a d(2u) a` = (\ S \ u\ u \ u \ u \ Q u Q \ u \ u \ R u R \ U u U \ S u S \ u \ u \ u \ u \ Q u Q \ u \ u \ R u R \ U u U \ S u S \ u \ u \ u \ u \ Q u Q \ u \ u \ R u R \ U u U \ S u S \ u \ u \ u \ u \ Q u Q \ u )\ R. Alaca Alaca and Williams [2] have established the following representations in terms of p and k: Therefore since 1(3) =2 x/ u / (1u) / (1u) / (12u) /R (2u) /R \ / (20) 1(3 )=2 x/ u / (1u) / (1u) / (12u) / (2u) / \ / (21) 1(3 )=2 x/ u /R (1u) / (1u) / (12u) / (2u) / \ / 1(3 )=2 x/ u / (1u) /R (1u) / (12u) /R (2u) / \ / (23) 1(3 )=2 x/ u / (1u) / (1u) / (12u) / (2u) / \ / (24) 1(3 )=2 x/ u / (1u) / (1u) /R (12u) / (2u) / \ / (25) { (3):= 1504T Q ()3 = (1246u5532u 3614u 13536u 27604u Q 34024u 27604u 13536u R 3614u U 5532u S 246u u )\ { (3):= 1240T ()3 =(1124u64u 27u 310u 27u Q 64u 124u u R )\. we have { R (3) = 5500 { (3) 3367 { (3){ (3) { R (3) =(u uq S U R u u Q u u

5 12 Barış Kendirli: Fourier Coefficients of a Class of Eta Quotients of Weight 1 with Level S R S U R Q u1)\r { R (3 )=(u 1u Q u u U R Q U Q S U R u u u S u S u R u u Q u 1u1)\ R { R (3 )=(u 1u Q 144u u u u us U R u u Q u Q u U u

6 Pure and Applied Mathematics Journal 2015; 4(4): S U R Q S U R u u Q u u 144u 1u1)\ R { R ( )=( u uq u u u S U u R u Q u u S U R Q S U R u u Q 1127 u 663u 144u 1u1)\ R { R (3 )=(u 1u Q 144u 663u u 252u us U R u u Q

7 14 Barış Kendirli: Fourier Coefficients of a Class of Eta Quotients of Weight 1 with Level u S u U u R u Q u u u S U R u 252u Q u 663u 144u 1u1)\ R We can also similarly determine E E S and E in terms of u and \ as in 3. Obviously E E are functions of see (3)( 1). We see that 1 { R (3 ) =( u uq 1634 u u u u us uu ur u 3616 u uq u u 4410 u S uu u R u u Q u u 1521 u S 1011 U R u 252u Q u 663u 144u 1u1)\ R. 1 H^(2)1 H b1 H c(32)1 H d(42)1 H n(62)1 H^b(122)?E E D\?E D i R jγ S (12)l E m R jγ S (12)l\ i R jγ S (12)l by [4]. Now = 3 a^}(13 ) H^ H b H c H d H n H^b = 2 x~^ n x~ b c x~ c n xb~ d c x~ n xb~^b c c u ~^ bd z~ b ^b z~ c z~ d n z~ n z~^b d b (1u) ~^ b z~ b d z~ c n z~ d z~ n z~^b ^b bd (1u) ~^ n z~ b ^b z~ c b z~ d bd z~ n z~^b d (12u) ~^ z~ b d z~ c bd z~ d z~ n z~^b ^b bd(2u) ~^ z~ b d z~ c bd z~ d b z~ n z~^b ^b n \ ~^_~b_~c_~d_~n_~^b b = \R 2 a^za`u a^(1u) a b(1u) a c(12u) a d(2u) a`

8 Pure and Applied Mathematics Journal 2015; 4(4): =\ R (\ S \ u\ u \ u \ u \ Q u Q \ u \ u \ R u R \ U u U \ S u S \ u \ u \ u \ u \ Q u Q \ u \ u \ R u R \ U u U \ S u S \ u \ u \ u \ u \ Q u Q \ u \ u \ R u R \ U u U \ S u S \ u \ u \ u \ u \ Q u Q \ u ) = e e T T e T f 3 } (13 ) U f 3 Q } (13 ) S S R f 3 } (13 ) S R R f 3 } (13 ) R R f Q.3 } ((13 ) Q f 3 } (13 ) R f 3 U } (13 ) S f R 3 Q } (13 ) R R ()3 e T ()3 e T ()3 e T f U 3 } (13 ) f S 3 } R S f 3 Q } (13 ) R R R R f 3 U } (13 ) Q f 3 } (13 ) U f 3 Q } (13 ) S S R f Q 3 } (13 ) U ()3 ()3 ()3

9 16 Barış Kendirli: Fourier Coefficients of a Class of Eta Quotients of Weight 1 with Level 12 f 3 } (13 ) S f 3 S } (13 ) f R 3 } Q f U 3 } (13 ) R R f S 3 } (13 ) f 3 } (13 ) f 3 } (13 ) S f 3 S } (13 ) R f 3 S } (13 ) R R f Q 3 } R R f 3 } Q f 3 } (13 ) U where So f R 3 S } (13 ) R R f U 3 } (13 ) R f S 3 S } (13 ) f 3 } (13 ) R R =o(v )T( e ()e 2 e 3 e 4 e 6 e 12 )f E ()...f E () o(v )= 0 if V 0 1 if V =0.

10 Pure and Applied Mathematics Journal 2015; 4(4): e() =(e ()e 2 e 3 e 4 e 6 e 12 )f E ()...f E (). Therefore for n=12 since it is easy to see that hence and for n=12 3. Conclusion e(2) =e (2)e ()e 2 ( e e ) 3 1. We have found 324 eta quotients see (e e ) 6 f E (2) f E (2) e(21) =e Q (21)e Q r 21 s 3 f E (21) f E (21) r 2 3 s =(2 1) r 2 3 s= E (2) = =E (2) = 0 E Q (21) = =E (21) = such that for n=12 e(2) =e (2)e ()e 2 ( e e ) 3 (e e ) 6 e(21) =e (21)e r 21 sf 3 E (21) f E (21). and 650 eta quotients such that for n=12 e(2) =e (2)e ()e 2 e 3 e(21) =0. e Q 6 f E (2) f E (2) 2. i R (Γ S (12)) is 31 dimensional m R (Γ S (12)) is 37 and generated by dimensionalsee [5] (Chapter 3 pg.7 and Chapter 5 pg.17) Δ R Δ R Δ R (32)Δ R (42)Δ R (62)Δ R (122) Δ R Δ R Δ R (32)Δ R (62) Δ R Δ R Δ R (42) Δ R (2)Δ R Δ R (42) Δ R (2)Δ R Δ R (42)(conjugate of Δ R 54@4212) Δ R Δ R (32) Δ R Δ R (32)(conjugate of Δ R 50@ ) Δ RΔ RΔ RΔ RΔ RΔ R Δ R Δ R

11 1 Barış Kendirli: Fourier Coefficients of a Class of Eta Quotients of Weight 1 with Level 12 where Δ R is the unique newform in i R (Γ S (1)); Δ R is the unique newform in i R (Γ S (2)) ; Δ R Δ R and Δ R are the unique newforms in i R (Γ S (3)) Δ R Δ R are the unique newforms in i R (Γ S (4)) Δ RΔ R and Δ R are the unique newforms in i R (Γ S (6)) and Δ R Δ R are the unique newforms in i R (Γ S (12)). By taking as a root 54@4212 and Œ as the root 50@ we see E E as linear combinations of this basis in 3. References [1] Ayşe Alaca Şaban Alaca and Kenneth S. Williams On the twodimensional theta functions of Borweins Acta Arith. 124 (2006) Carleton University Ottawa Ontario Canada K1S 5B6. aalaca@math.carleton.ca salaca@math.carleton.ca williams@math.carleton.ca [2] Evaluation of the convolution sums zf (Ž)(G) and zf (Ž)(G) Adv. Theor. Appl. Math. 1(2006) 274. [3] Basil Gordon Some identities in combinatorial analysis Quart. J. Math. Oxford Ser.12 (161) University of California Los Angeles [4] Basil Gordon 1 and Sınai Robins 2 Lacunarity of Dedekind 1products Glasgow Math. J. 37 (15) University of California Los Angeles 2 University of Northern Colorado Greeley [5] Fred Diamond 1 Jerry Shurman 2 A First Course in Modular FormsSpringer Graduate Texts in Mathematics 22 1 Brandeis University Waltham MA USA 2 Reed College Portland OR 7202 USA fdiamond@brandeis.edu jerry@reed.edu [6] Victor G. Kac Infinitedimensional algebras Dedekind s 1function classical Möbius function and the very strange formula Adv. Math. 30 (17) MIT Cambridge Massachusetts 0213 [] "Evaluation of Some Convolution Sums and Representation Numbers of Quadratic Forms of Discriminant 135" British Journal of Mathematics and Computer Science Vol6/6 Jan pp [] Evaluation of Some Convolution Sums and the Representation numbers Ars Combinatoria Volume CXVI July pp [10] Cusp Forms in i jγ S (7)l and the number of representations of positive integers by some direct sum of binary quadratic forms with discriminant 7 Bulletin of the Korean Mathematical Society Vol 4/ [11] Cusp Forms in i jγ S (47)l and the number of representations of positive integers by some direct sum of binary quadratic forms with discriminant 47Hindawi International Journal of Mathematics and Mathematical Sciences Vol pages [12] The Bases of M jγ S (71)lM jγ S (71)l and the Number of Representations of Integers Hindawi Mathematical Problems in Engineering Vol pages [13] Günter Köhler Eta Products and Theta Series Identities (SpringerVerlag Berlin 2011). University of Würzburg Am Hubland 7074 Würzburg Germany koehler@mathematik.uni-wuerzburg.de [14] Ian Grant Macdonald Affine root systems and Dedekind s 1function Invent. Math. 15 (172) [15] Olivia X. M. Yao 1 Ernest X. W. Xia 2 and J. Jin 3 Explicit Formulas for the Fourier coefficients of a class of eta quotients International Journal of Number Theory Vol. No. 2 (2013) Jiangsu University Zhenjiang Jiangsu P. R. China 3 Nanjing Normal University Taizhou College Jiangsu P. R. China yaoxiangmei@163.com ernestxwxia@163.com jinjing141@126.com [16] Kenneth S. Williams Fourier series of a class of eta quotients Int. J. Number Theory (2012) Carleton University Ottawa Ontario Canada K1S 5B6 williams@math.carleton.ca. [7] Barış Kendirli "Evaluation of Some Convolution Sums by Quasimodular Forms" European Journal of Pure and Applied Mathematics ISSN Vol.. No. 1 Jan pp Aydın University Istanbul/Turkey baris.kendirli@gmail.com