A family of Eta Quotients and an Extension of the Ramanujan-Mordell Theorem
aa r X i v : . [ m a t h . N T ] A p r A FAMILY OF ETA QUOTIENTS AND AN EXTENSION OFTHE RAMANUJAN-MORDELL THEOREM
AYS¸E ALACA, S¸ABAN ALACA, ZAFER SELCUK AYGIN
Abstract.
Let k ≥ j an integer satisfying 1 ≤ j ≤ k − { C j,k ( z ) } ≤ j ≤ k − of eta quotients, and prove that this fam-ily constitute a basis for the space S k (Γ (12)) of cusp forms of weight 2 k andlevel 12. We then use this basis together with certain properties of modularforms at their cusps to prove an extension of the Ramanujan-Mordell formula.Key words and phrases: Ramanujan-Mordell formula, Dedekind eta function,eta quotients, eta products, theta functions, Eisenstein series, Eisenstein forms,modular forms, cusp forms, Fourier coefficients, Fourier series.2010 Mathematics Subject Classification: 11F11, 11F20, 11F27, 11E20, 11E25,11F30, 11Y35 Introduction
Let N , N , Z , Q and C denote the sets of positive integers, non-negative integers,integers, rational numbers and complex numbers, respectively. Let N ∈ N . LetΓ ( N ) be the modular subgroup defined byΓ ( N ) = (cid:26)(cid:18) a bc d (cid:19) | a, b, c, d ∈ Z , ad − bc = 1 , c ≡ N ) (cid:27) . Let k ∈ Z . We write M k (Γ ( N )) to denote the space of modular forms of weight k for Γ ( N ), and E k (Γ ( N )) and S k (Γ ( N )) to denote the subspaces of Eisensteinforms and cusp forms of M k (Γ ( N )), respectively. It is known that M k (Γ ( N )) = E k (Γ ( N )) ⊕ S k (Γ ( N )) . (1.1)The Dedekind eta function η ( z ) is the holomorphic function defined on the upperhalf plane H = { z ∈ C | Im( z ) > } by the product formula η ( z ) = e πiz/ ∞ Y n =1 (1 − e πinz ) . An eta quotient is defined to be a finite product of the form f ( z ) = Y δ η r δ ( δz ) , (1.2) where δ runs through a finite set of positive integers and the exponents r δ arenon-zero integers. By taking N to be the least common multiple of the δ ’s we canwrite the eta quotient (1.2) as f ( z ) = Y ≤ δ | N η r δ ( δz ) , (1.3)where some of the exponents r δ may be 0. When all the exponents r δ are nonneg-ative, f ( z ) is said to be an eta product.As in [11] throughout the paper we use the notation q = e ( z ) := e πiz with z ∈ H , and so | q | < q / = e ( z/ ϕ ( z ) isdefined by ϕ ( z ) = ∞ X n =0 q n . It is known that ϕ ( z ) can be expressed as an eta quotient as ϕ ( z ) = η (2 z ) η ( z ) η (4 z ) . (1.4)For a j ∈ N , 1 ≤ j ≤ k , we define N ( a , . . . , a k ; n ) := card { ( x , . . . , x k ) ∈ Z k | n = a x + · · · + a k x k } . Then we have ϕ ( a z ) · · · ϕ ( a k z ) = ∞ X n =0 N ( a , . . . , a k ; n ) q n . (1.5)The value of N ( a , . . . , a k ; n ) is independent of the order of the a j ’s.Let k ≥ a j ∈ { , } , 1 ≤ j ≤ k , with an even numberof a j ’s equal to 3. Then we write N ( a , . . . , a k ; n ) = N (1 k − i , i ; n ) , where i is an integer with 0 ≤ i ≤ k . Ramanujan [17] stated a formula for N (1 k , ; n ), which was proved by Mordell in [14], see also [7, 3].In this paper we define a family { C j,k ( z ) } ≤ j ≤ k − of eta quotients, and provethat this family constitute a basis for the space S k (Γ (12)) of cusp forms of weight2 k and level 12. We then use this basis together with certain properties of modularforms at their cusps to prove an extension of the Ramanujan-Mordell formula, thatis, we give a formula for N (1 k − i , i ; n ).For n, k ∈ N we define the sum of divisors function σ k ( n ) by σ k ( n ) = X ≤ m | n m k . FAMILY OF ETA QUOTIENTS AND THE RAMANUJAN-MORDELL THEOREM 3 If n N we set σ k ( n ) = 0. We define the Eisenstein series E k ( z ) by E k ( z ) := − B k k + ∞ X n =1 σ k − ( n ) q n , (1.6)where B k are Bernoulli numbers defined by the generating function xe x − ∞ X n =0 B n x n n ! . (1.7)The cusps of Γ ( N ) can be represented by rational numbers a/c , where a ∈ Z , c ∈ N , c | N and gcd( a, c ) = 1, see [15, p. 320] and [8, p. 103]. We can choose therepresentatives of cusps of Γ (12) as1 , / , / , / , / , ∞ . Throughout the paper we use ∞ and 1 /
12 interchangeably as they are equivalentcusps for Γ (12).Let f ( z ) be an eta quotient given by (1.3). A formula for the order v a/c ( f ) of f ( z ) at the cusp a/c (see [15, p. 320] and [12, Proposition 3.2.8]) is given by v a/c ( f ) = N
24 gcd( c , N ) X ≤ δ | N gcd( δ, c ) · r δ δ . (1.8)We use the following theorem to determine if a given eta quotient is in M k (Γ ( N )).See [13, Proposition 1, p. 284], [11, Corollary 2.3, p. 37], [9, p. 174], [12] and [10]. Theorem 1.1. (Ligozat)
Let f ( z ) be an eta quotient given by (1.3) which satis-fies the following conditions: (L1) X ≤ δ | N δ · r δ ≡ , (L2) X ≤ δ | N Nδ · r δ ≡ , (L3) For each d | N , X ≤ δ | N gcd( d, δ ) · r δ δ ≥ , (L4) s Y ≤ δ | N δ r δ ∈ Q , (L5) k = 12 X ≤ δ | N r δ an even integer.Then f ( z ) ∈ M k (Γ ( N )) . Furthermore if all inequalities in (L3) are strict then f ( z ) ∈ S k (Γ ( N )) . AYS¸E ALACA, S¸ABAN ALACA, ZAFER SELCUK AYGIN
For N = 12 in (L4) of Theorem 1.1, we have s Y ≤ δ | δ r δ = 2 r + r √ r + r r + r + r . (1.9)Thus the expresion in (1.9) is a rational number if and only if r + r ≡ r + r + r ≡ . Statements of main results
For j, k ∈ Z we define an eta quotient C j,k ( z ) by C j,k ( z ) := (cid:16) η (2 z ) η (3 z ) η (4 z ) η (6 z ) η ( z ) η (12 z ) (cid:17)(cid:16) η (2 z ) η (3 z ) η (12 z ) η ( z ) η (4 z ) η (6 z ) (cid:17) j (cid:16) η ( z ) η (6 z ) η (2 z ) η (3 z ) (cid:17) k = q j + ∞ X n = j +1 c j,k ( n ) q n . (2.1)In the following theorem we give a basis for M k (Γ (12)) when k ≥ Theorem 2.1.
Let k ≥ be an integer. (a) The family { C j, k ( z ) } ≤ j ≤ k − constitute a basis for S k (Γ (12)) . (b) The set of Eisenstein series { E k ( z ) , E k (2 z ) , E k (3 z ) , E k (4 z ) , E k (6 z ) , E k (12 z ) } constitute a basis for E k (Γ (12)) . (c) The set { E k ( δz ) | δ = 1 , , , , , } ∪ { C j, k ( z ) | ≤ j ≤ k − } constitute a basis for M k (Γ (12)) . For convenience we set α k = − k (2 k − k − B k , (2.2)where B k are Bernoulli numbers given in (1.7). Also we write [ j ] f ( z ) := a j for f ( z ) = ∞ X n =0 a n q n . We now give an extension of the Ramanujan-Mordell Theorem. Theorem 2.2.
Let k ≥ be an integer and i an integer satisfying ≤ i ≤ k .Let α k be as in (2.2) . Then ϕ k − i ( z ) ϕ i (3 z ) = X r | b ( r,i,k ) E k ( rz ) + X ≤ j ≤ k − a ( j,i,k ) C j, k ( z ) , where b (1 ,i,k ) = ( − k (3 k − i + ( − i +1 ) · α k , (2.3) FAMILY OF ETA QUOTIENTS AND THE RAMANUJAN-MORDELL THEOREM 5 b (2 ,i,k ) = ( − i +1 (1 + ( − i + k )(3 k − i + ( − i +1 ) · α k , (2.4) b (3 ,i,k ) = ( − i + k k − i (3 i + ( − i +1 ) · α k , (2.5) b (4 ,i,k ) = ( − i k (3 k − i + ( − i +1 ) · α k , (2.6) b (6 ,i,k ) = − (1 + ( − i + k )3 k − i (3 i + ( − i +1 ) · α k , (2.7) b (12 ,i,k ) = 2 k k − i (3 i + ( − i +1 ) · α k , (2.8) and for ≤ j ≤ k − a ( j,i,k ) = N (1 k − i , i ; j ) − X r | b ( r,i,k ) σ k − ( j/r ) − X ≤ l ≤ j − a ( l,i,k ) c l, k ( j ) . (2.9)The following theorem follows immediately from Theorem 2.2. Theorem 2.3.
Let k ≥ be an integer and i an integer with ≤ i ≤ k . Then N (1 k − i , i ; n ) = X r | b ( r,i,k ) σ k − ( n/r ) + X ≤ j ≤ k − a ( j,i,k ) c j, k ( n ) , where a ( j,i,k ) , c j, k ( n ) and b ( r,i,k ) ( r = 1 , , , , , are given in (2.9), (2.1) , and (2.3)–(2.8) respectively. Proof of Theorem 2.1 (a)
By Theorem 1.1 we see that C j, k ( z ) ∈ S k (Γ (12)) for 1 ≤ j ≤ k −
5. Itfollows from (2.1) that { C j, k ( z ) } ≤ j ≤ k − is a linearly independent set. We deducefrom the formulae in [18, Section 6.3, p. 98] thatdim( S k (Γ (12))) = 4 k − . Thus the set { C j, k ( z ) } ≤ j ≤ k − is a basis for S k (Γ (12)). (b) It follows from [18, Theorem 5.9] that { E k ( z ) , E k (2 z ) , E k (3 z ) , E k (4 z ) , E k (6 z ) , E k (12 z ) } constitute a basis for E k (Γ (12)). (c) Appealing to (1.1), the assertion follows from (a) and (b).4.
Fourier series expansions of η ( rz ) and E k ( rz ) at certain cusps Let k ≥ η r ( z ) = η ( rz ) for r ∈ N . Wealso set A c = (cid:20) − c − (cid:21) ∈ SL ( Z ) . (4.1)The Fourier series expansions of η r ( z ) for r = 1 , , , , ,
12 at the cusp 1 /c aregiven by the Fourier series expansions of η r ( A − c z ) at the cusp ∞ . AYS¸E ALACA, S¸ABAN ALACA, ZAFER SELCUK AYGIN
In [11, Theorem 1.7 and Proposition 2.1] we take L = (cid:20) x yu v (cid:21) = L r as L = (cid:20) − − (cid:21) , L = (cid:20) − − (cid:21) , L = (cid:20) − − (cid:21) ,L = (cid:20) − − (cid:21) , L = (cid:20) − − (cid:21) , L = (cid:20) −
12 1 − (cid:21) , and A = (cid:20) a bc d (cid:21) = A , where A is given by (4.1). We obtain the Fourier seriesexpansions of η r ( z ) for r = 1 , , , , ,
12 at the cusp 1 as η ( A − z ) = e πi/ ( − z − / X n ≥ (cid:16) n (cid:17) e (cid:16) n
24 ( z + 1) (cid:17) , (4.2) η ( A − z ) = e πi/ / ( − z − / X n ≥ (cid:16) n (cid:17) e (cid:16) n
48 ( z + 1) (cid:17) , (4.3) η ( A − z ) = e πi/ / ( − z − / X n ≥ (cid:16) n (cid:17) e (cid:16) n
72 ( z + 1) (cid:17) , (4.4) η ( A − z ) = e πi/ − z − / X n ≥ (cid:16) n (cid:17) e (cid:16) n
96 ( z + 1) (cid:17) , (4.5) η ( A − z ) = e πi/ / ( − z − / X n ≥ (cid:16) n (cid:17) e (cid:16) n
144 ( z + 1) (cid:17) , (4.6) η ( A − z ) = e πi/ / ( − z − / X n ≥ (cid:16) n (cid:17) e (cid:16) n
288 ( z + 1) (cid:17) . (4.7)From (1.4) and (4.2)–(4.7) we obtain the Fourier series expansions of ϕ ( z ) and ϕ (3 z ) at the cusp 1 as ϕ ( A − z ) = η ( A − z ) η ( A − z ) η ( A − z )= e πi/ / ( − z − / (cid:16) X n ≥ (cid:16) n (cid:17) e (cid:0) n
48 ( z + 1) (cid:1)(cid:17) (cid:16) X n ≥ (cid:16) n (cid:17) e (cid:0) n
24 ( z + 1) (cid:1)(cid:17) (cid:16) X n ≥ (cid:16) n (cid:17) e (cid:0) n
96 ( z + 1) (cid:1)(cid:17) , (4.8) ϕ (3 A − z ) = η ( A − z ) η ( A − z ) η ( A − z ) FAMILY OF ETA QUOTIENTS AND THE RAMANUJAN-MORDELL THEOREM 7 = e πi/ / ( − z − / (cid:16) X n ≥ (cid:16) n (cid:17) e (cid:0) n
144 ( z + 1) (cid:1)(cid:17) (cid:16) X n ≥ (cid:16) n (cid:17) e (cid:0) n
72 ( z + 1) (cid:1)(cid:17) (cid:16) X n ≥ (cid:16) n (cid:17) e (cid:0) n
288 ( z + 1) (cid:1)(cid:17) . (4.9)Similarly, by taking A = A in [11, Theorem 1.7 and Proposition 2.1] and L as L := (cid:20) − − − − (cid:21) , L := (cid:20) − − − − (cid:21) , L := (cid:20) − − (cid:21) ,L := (cid:20) − − − − (cid:21) , L := (cid:20) − − (cid:21) , L := (cid:20) − − (cid:21) we obtain the Fourier series expansions of η r ( z ) for r = 1 , , , , ,
12 at the cusp1 / η ( A − z ) = e πi/ ( − z − / X n ≥ (cid:16) n (cid:17) e (cid:16) n
24 ( z − (cid:17) , (4.10) η ( A − z ) = e πi/ / ( − z − / X n ≥ (cid:16) n (cid:17) e (cid:16) n
48 ( z − (cid:17) , (4.11) η ( A − z ) = e πi/ ( − z − / X n ≥ (cid:16) n (cid:17) exp (cid:16) n
24 (3 z + 1) (cid:17) , (4.12) η ( A − z ) = e πi/ − z − / X n ≥ (cid:16) n (cid:17) e (cid:16) n
96 ( z − (cid:17) , (4.13) η ( A − z ) = e πi/ / ( − z − / X n ≥ (cid:16) n (cid:17) e (cid:16) n
48 (3 z + 1) (cid:17) , (4.14) η ( A − z ) = e πi/ − z − / X n ≥ (cid:16) n (cid:17) e (cid:16) n
96 (3 z + 1) (cid:17) . (4.15)From (1.4) and (4.10)–(4.15) we obtain the Fourier series expansions of ϕ ( z ) and ϕ (3 z ) at the cusp 1 / ϕ ( A − z ) = η ( A − z ) η ( A − z ) η ( A − z )= e πi/ / ( − z − / (cid:16) X n ≥ (cid:16) n (cid:17) e (cid:0) n
48 ( z − (cid:1)(cid:17) (cid:16) X n ≥ (cid:16) n (cid:17) e (cid:0) n
24 ( z − (cid:1)(cid:17) (cid:16) X n ≥ (cid:16) n (cid:17) e (cid:0) n
96 ( z − (cid:1)(cid:17) , (4.16) AYS¸E ALACA, S¸ABAN ALACA, ZAFER SELCUK AYGIN ϕ (3 A − z ) = η ( A − z ) η ( A − z ) η ( A − z )= e πi/ / ( − z − / (cid:16) X n ≥ (cid:16) n (cid:17) e (cid:0) n
48 (3 z + 1) (cid:1)(cid:17) (cid:16) X n ≥ (cid:16) n (cid:17) e (cid:0) n
24 (3 z + 1) (cid:1)(cid:17) (cid:16) X n ≥ (cid:16) n (cid:17) e (cid:0) n
96 (3 z + 1) (cid:1)(cid:17) . (4.17)Again by taking A = A in [11, Theorem 1.7 and Proposition 2.1] and L as L := (cid:20) − − (cid:21) , L := (cid:20) − − (cid:21) , L := (cid:20) − − (cid:21) ,L := (cid:20) − − (cid:21) , L := (cid:20) − − (cid:21) , L := (cid:20) − − (cid:21) we obtain the Fourier series expansions of η r ( z ) for r = 1 , , , , ,
12 at the cusp1 / η ( A − z ) = e πi/ ( − z − / X n ≥ (cid:16) n (cid:17) e (cid:16) n
24 ( z + 1) (cid:17) , (4.18) η ( A − z ) = e πi/ ( − z − / X n ≥ (cid:16) n (cid:17) e (cid:16) n
12 ( z + 1) (cid:17) , (4.19) η ( A − z ) = e πi/ / ( − z − / X n ≥ (cid:16) n (cid:17) e (cid:16) n
72 ( z + 1) (cid:17) , (4.20) η ( A − z ) = e πi/ ( − z − / X n ≥ (cid:16) n (cid:17) e (cid:16) n z + 1) (cid:17) , (4.21) η ( A − z ) = e πi/ / ( − z − / X n ≥ (cid:16) n (cid:17) e (cid:16) n
36 ( z + 1) (cid:17) , (4.22) η ( A − z ) = e πi/ / ( − z − / X n ≥ (cid:16) n (cid:17) e (cid:16) n
18 ( z + 1) (cid:17) . (4.23)From (1.4) and (4.18)–(4.23) we obtain the Fourier series expansions of ϕ ( z ) and ϕ (3 z ) at the cusp 1 / ϕ ( A − z ) = η ( A − z ) η ( A − z ) η ( A − z ) FAMILY OF ETA QUOTIENTS AND THE RAMANUJAN-MORDELL THEOREM 9 = e πi/ ( − z − / (cid:16) X n ≥ (cid:16) n (cid:17) e (cid:0) n
12 ( z + 1) (cid:1)(cid:17) (cid:16) X n ≥ (cid:16) n (cid:17) e (cid:0) n
24 ( z + 1) (cid:1)(cid:17) (cid:16) X n ≥ (cid:16) n (cid:17) e (cid:0) n z + 1) (cid:1)(cid:17) , (4.24) ϕ (3 A − z ) = η ( A − z ) η ( A − z ) η ( A − z )= 13 ( − z − / (cid:16) X n ≥ (cid:16) n (cid:17) e (cid:0) n
36 ( z + 1) (cid:1)(cid:17) (cid:16) X n ≥ (cid:16) n (cid:17) e (cid:0) n
72 ( z + 1) (cid:1)(cid:17) (cid:16) X n ≥ (cid:16) n (cid:17) e (cid:0) n
18 ( z + 1) (cid:1)(cid:17) . (4.25)If the Fourier series expansion of a modular form f ( z ) of weight k at the cusp1 /c is of the form f ( A − c z ) = ( − cz − k X n ≥ a n e πinz c , (4.26)where z c ∈ H depends on z and A c , then we refer to ( − cz − k a in (4.26) as the“first term” of the Fourier series expansion of f ( z ) at the cusp 1 /c .For k ≥ i an integer with 0 ≤ i ≤ k we give the first terms ofthe Fourier series expansions of ϕ k − i ( z ) ϕ i (3 z ) at certain cusps in the followingtable. We deduce them (except for the cusps 1 / /
6) from (4.8), (4.9), (4.16),(4.17), (4.24) and (4.25).Table 4.1: First terms of ϕ k − i ( z ) ϕ i (3 z ) at certain cuspscusp ∞ / / / / − z − k ( − k k i − z − k ( − i + k k ( − z − k ( − i i v / ( ϕ k − i ( z ) ϕ i (3 z )) = 3 k − i > ,v / ( ϕ k − i ( z ) ϕ i (3 z )) = k + i > ≤ i ≤ k , that is, the first terms of the Fourier series expansions of ϕ k − i ( z ) ϕ i (3 z ) at cusps 1 / / E k ( tz ). For convenience we set E (2 k,t ) ( z ) = E k ( tz ) for t ∈ N . Theorem 4.1.
Let k ≥ be an integer and t ∈ N . The Fourier series expansionof E (2 k,t ) ( z ) at the cusp /c ∈ Q is E (2 k,t ) ( A − c z ) = (cid:16) gt (cid:17) k ( − cz − k E k (cid:16) g t z + ygt (cid:17) , where g = gcd( t, c ) , y is some integer, and A c is given in (4.1) .Proof. The Fourier series expansion of E (2 k,t ) ( z ) at the cusp 1 /c is given by theFourier series expansion of E (2 k,t ) ( A − c z ) at the cusp ∞ . We have E (2 k,t ) ( A − c z ) = E (2 k,t ) ( − z − cz − E k ( − tz − cz − E k ( γz ) , where γ = (cid:20) − t − c − (cid:21) . As gcd( t/g, c/g ) = 1, there exist y, v ∈ Z such that tg ( − v ) + cg y = 1. Thus L := (cid:20) − t/g y − c/g v (cid:21) ∈ SL ( Z ). Then for k ≥
2, we have E (2 k,t ) ( A − c z ) = E k ( LL − γz )= (cid:16) − c (cid:16) ( − vt + cy ) z + yt (cid:17) + v (cid:17) k E k (cid:16) ( − vt + cy ) z + yt/g (cid:17) = ( g/t ) k (cid:16) c vt − cyg z + vt − cyg (cid:17) k E k (cid:16) g z + ygt (cid:17) = ( g/t ) k ( − cz − k E k (cid:16) g t z + ygt (cid:17) , which completes the proof. (cid:3) It folows from Theorem 4.1 and (1.6) that the first term of the Fourier seriesexpansion of E k ( tz ) at the cusp 1 /c is (cid:16) gt (cid:17) k ( − cz − k (cid:16) − B k k (cid:17) . (4.27) 5. Proofs of Theorems 2.2 and 2.3
Let k ≥ i an integer with 0 ≤ i ≤ k . By (1.4) we have ϕ k − i ( z ) ϕ i (3 z ) = η k − i (2 z ) η k − i ( z ) η k − i (4 z ) · η i (6 z ) η i (3 z ) η i (12 z ) . By Theorem 1.1, we have ϕ k − i ( z ) ϕ i (3 z ) ∈ M k (Γ (12)). By Theorem 2.1(c), wehave ϕ k − i ( z ) ϕ i (3 z ) = X r | b ( r,i,k ) E k ( rz ) + X ≤ j ≤ k − a ( j,i,k ) C j, k ( z )(5.1)for some constants b (1 ,i,k ) , b (2 ,i,k ) , b (3 ,i,k ) , b (4 ,i,k ) , b (6 ,i,k ) , b (12 ,i,k ) , a (1 ,i,k ) , . . . , a (4 k − ,i,k ) .Since C ,k ( z ) , . . . , C k − ,k ( z ) are cusp forms, the first terms of their Fourier se-ries expansions at all cusps are 0. FAMILY OF ETA QUOTIENTS AND THE RAMANUJAN-MORDELL THEOREM 11
By appealing to (4.27) and Table 4.1 we equate the first terms of the Fourierseries expansions of (5.1) in both sides at cusps ∞ , 1, 1 /
2, 1 /
3, 1 /
4, 1 / b (1 ,i,k ) + b (2 ,i,k ) + b (3 ,i,k ) + b (4 ,i,k ) + b (6 ,i,k ) + b (12 ,i,k ) = − kB k ,b (1 ,i,k ) + b (2 ,i,k ) k + b (3 ,i,k ) k + b (4 ,i,k ) k + b (6 ,i,k ) k + b (12 ,i,k ) k = ( − k k i · − kB k ,b (1 ,i,k ) + b (2 ,i,k ) + b (3 ,i,k ) k + b (4 ,i,k ) k + b (6 ,i,k ) k + b (12 ,i,k ) k = 0 ,b (1 ,i,k ) + b (2 ,i,k ) k + b (3 ,i,k ) + b (4 ,i,k ) k + b (6 ,i,k ) k + b (12 ,i,k ) k = ( − i + k k · − kB k ,b (1 ,i,k ) + b (2 ,i,k ) + b (3 ,i,k ) k + b (4 ,i,k ) + b (6 ,i,k ) k + b (12 ,i,k ) k = ( − i i · − kB k ,b (1 ,i,k ) + b (2 ,i,k ) + b (3 ,i,k ) + b (4 ,i,k ) k + b (6 ,i,k ) + b (12 ,i,k ) k = 0 . Solving the above system of linear equations, we obtain the asserted expressionsfor b ( r,i,k ) for r = 1 , , , , ,
12 in (2.3)–(2.8). By (2.1), we have [ j ] C j,k ( z ) = 1 foreach j with 1 ≤ j ≤ k −
5. Equating the coefficients of q j in both sides of (5.1)we obtain N (1 k − i , i ; j ) = X r | b ( r,i,k ) σ k − ( j/r ) + X ≤ l ≤ j − a ( l,i,k ) c l, k ( z ) + a ( j,i,k ) . We isolate a ( j,i,k ) to complete the proof of Theorem 2.2. Finally, Theorem 2.3follows from (1.5), (1.6), (2.1) and Theorem 2.2.6. Examples and Remarks
We now illustrate Theorems 2.2 and 2.3 by some examples.
Example 6.1.
We determine N (1 , ; n ) for all n ∈ N . We take k = 2 and i = 1 in Theorems and . By (2.3)–(2.8) we have (cid:26) b (1 , , = 28 / , b (2 , , = 0 , b (3 , , = − / ,b (4 , , = − / , b (6 , , = 0 , b (12 , , = 1728 / . (6.1) We compute N (1 , ; n ) for n = 1 , , as k − , and obtain N (1 , ; 1) = 12 , N (1 , ; 2) = 60 , N (1 , ; 3) = 164 . (6.2) By (2.9) , (6.1) and (6.2) , we obtain a (1 , , = 32 / , a (2 , , = 48 , a (3 , , = 576 / . Then, by Theorem , for all n ∈ N , we have N (1 , ; n ) = 285 σ ( n ) − σ ( n/ − σ ( n/
4) + 17285 σ ( n/ + 325 c , ( n ) + 48 c , ( n ) + 5765 c , ( n ) , which agrees with the known results, see for example [4] . We note that the last twocofficients in the above expression are different from the ones in [4] since we useda different basis for the space of cusp forms. Example 6.2.
We determine N (1 , ; n ) for all n ∈ N . We take k = 3 and i = 4 in Theorems and . By (2.3)–(2.8) we have (cid:26) b (1 , , = 8 / , b (2 , , = 0 , b (3 , , = 720 / ,b (4 , , = − / , b (6 , , = 0 , b (12 , , = − / . (6.3) We compute N (1 , ; n ) for n = 1 , , , , , , as k − , and obtain N (1 , ; 1) = 8 , N (1 , ; 2) = 24 , N (1 , ; 3) = 48 ,N (1 , ; 4) = 152 , N (1 , ; 5) = 432 ,N (1 , ; 6) = 720 , N (1 , ; 7) = 1344 . (6.4) By (2.9) , (6.3) and (6.4) , we obtain a (1 , , = 720 / , a (2 , , = 14880 / , a (3 , , = 123376 / a (4 , , = 40640 / ,a (5 , , = 1248448 / , a (6 , , = 1551360 / , a (7 , , = 792576 / . Then, by Theorem , for all n ∈ N , we have N (1 , ; n ) = 891 σ ( n ) + 72091 σ ( n/ − σ ( n/ − σ ( n/ c , ( n ) + 1488091 c , ( n ) + 12337691 c , ( n ) + 406407 c , ( n )+ 124844891 c , ( n ) + 155136091 c , ( n ) + 79257691 c , ( n ) , which agrees with the known results, see for example [1] . Example 6.3.
Ramanujan [17] stated a formula for N (1 k , , n ) , which wasproved by Mordell in [14] , see also [7, 3] . By taking i = 0 in Theorems and ,we obtain N (1 k , , n ) = 4 k (2 k − B k (cid:16) ( − k +1 σ k − ( n ) + (1 + ( − k ) σ k − ( n/ − k σ k − ( n/ (cid:17) + X ≤ j ≤ k − a ( j, ,k ) c j, k ( n ) , where a ( j, ,k ) and c j, k ( n ) are given by (2.9) and (2.1) , respectively. The coefficientsof σ -functions in the above formula agree with those in [7, Theroem 1.1] and [3,Theroem 4.1] . Different coefficients in the cusp part are due to the choice of ourbasis for the space S k (cid:0) Γ (12) (cid:1) of cusp forms. FAMILY OF ETA QUOTIENTS AND THE RAMANUJAN-MORDELL THEOREM 13
Remark 6.1.
Throughout the paper we assumed that k ≥ . For k = 1 we have dim (cid:0) S (cid:0) Γ (12) (cid:1)(cid:1) = 0 . A basis for M (cid:0) Γ (12) (cid:1) = E (cid:0) Γ (12) (cid:1) is given in [2] , seealso [19, 5, 6] . Remark 6.2.
Let k ≥ be an integer. Let N ∈ N and χ a Dirichlet character ofmodulus dividing N . We write M k (Γ ( N ) , χ ) to denote the space of modular formsof weight k with multiplier system χ for Γ ( N ) , and E k (Γ ( N ) , χ ) and S k (Γ ( N ) , χ ) to denote the subspaces of Eisenstein forms and cusp forms of M k (Γ ( N ) , χ ) , re-spectively. We deduce from the formulae in [18, Section 6.3, p. 98] that dim (cid:0) S k − (cid:0) Γ (12) , χ (cid:1)(cid:1) = 4 k − , dim (cid:0) S k − (cid:0) Γ (12) , χ (cid:1)(cid:1) = 4 k − , dim (cid:0) S k (cid:0) Γ (12) , χ (cid:1)(cid:1) = 4 k − , where χ ( m ) = (cid:16) − m (cid:17) , χ ( m ) = (cid:16) − m (cid:17) , χ ( m ) = (cid:16) m (cid:17) (6.5) are Legendre-Jacobi-Kronecker symbols. By appealing to a more general version ofTheorem , see for example [16, Theorem 1.64] and [11, Corollary 2.3,p. 37] , we deduce that the families of eta quotients (cid:8) C j, k − ( z ) (cid:9) ≤ j ≤ k − , n η ( z ) η (4 z ) η (12 z ) η (2 z ) η (6 z ) C j, k − ( z ) o ≤ j ≤ k − , n η ( z ) η (4 z ) η (12 z ) η (2 z ) η (6 z ) C j, k ( z ) o ≤ j ≤ k − constitute a basis for S k − (Γ (12) , χ ) , S k − (Γ (12) , χ ) , S k (Γ (12) , χ ) , respec-tively, where χ , χ , χ are given in (6.5) . Acknowledgments
The authors would like to thank Professor Shaun Cooper for bringing this re-search problem to our attention at CNTA XIII (2014). The research of the firsttwo authors was supported by Discovery Grants from the Natural Sciences andEngineering Research Council of Canada (RGPIN-418029-2013 and RGPIN-2015-05208). Zafer Selcuk Aygin’s studies are supported by Turkish Ministry of Edu-cation.
References [1] A. Alaca,
On the number of representations of a positive integer by certain quadratic formsin twelve variables,
J. Comb. Number Theory (2011), 167–177.[2] A. Alaca, S¸. Alaca and Z. S. Aygin, Fourier coefficients of a class of eta quotients of weight , Int. J. Number Theory (2015), 2381–2392. [3] A. Alaca, S¸. Alaca, K. S. Williams, Sums of 4k squares: A polynomial approach,
Journal ofCombinatorics and Number Theory (2009), 33–52.[4] S¸. Alaca, K. S. Williams,
The number of representations of a positive integer by certainoctonary quadratic forms,
Funct. Approx. Comment. Math. (2010), 45–54.[5] A. Alaca, S¸. Alaca, M. F. Lemire and K. S. Williams, Theta function identities and repre-sentations by certain quaternary quadratic forms,
Int. J. Number Theory (2008), 219–239.[6] A. Alaca, S¸. Alaca, M. F. Lemire and K. S. Williams, Nineteen quaternary quadratic forms,
Acta Arith. (2007), 277–310.[7] H. H. Chan, S. Cooper,
Powers of theta functions,
Pacific Journal of Mathematics (2008),1–14.[8] F. Diamond and J. Shurman,
A First Course in Modular Forms,
Graduate Texts in Mathe-matics 228, Springer-Verlag, 2004.[9] B. Gordon and D. Sinor,
Multiplicative properties of η -products, Lecture Notes in Math,vol.1395 Springer-Verlag, New York (1989), 173-200.[10] L. J. P. Kilford,
Modular Forms, A classical and computational introduction,
Imperial Col-lege Press, London, 2008.[11] G. K¨ohler,
Eta Products and Theta Series Identities,
Springer Monographs in Mathematics,Springer, 2011.[12] G. Ligozat,
Courbes modulaires de genre 1,
Bull. Soc. Math. France (1975), 5–80.[13] J. Lovejoy, Ramanujan-type congruences for three colored Frobenius partitions,
J. NumberTheory (2000), 283–290.[14] L. J. Mordell, On the representations of numbers as a sum of 2r squares,
Quart. J. PureAppl. Math. (1917), 93–104.[15] A. Okamoto, On expressions of theta series by η -products, TOKYO J. MATH. (2011),319–326.[16] K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms andq-Series,
Am. Math. Soc., Providence, 2004.[17] S. Ramanujan
On certain arithmetical functions
Trans. Cambridge Philos. Soc. (1916),159–184.[18] W. A. Stein, Modular Forms, A Computational Approach,
Amer. Math. Soc., GraduateStudies in Mathematics (2007).[19] K. S. Williams, Fourier series of a class of eta quotients,
Int. J. Number Theory (2012),993-1004.(2012),993-1004.