Analogues of the Ramanujan--Mordell Theorem
aa r X i v : . [ m a t h . N T ] M a y ANALOGUES OF THE RAMANUJAN–MORDELL THEOREM
SHAUN COOPER, BEN KANE AND DONGXI YE
Abstract.
The Ramanujan–Mordell Theorem for sums of an even number of squares is extendedto other quadratic forms and quadratic polynomials. Introduction
One of the classical problems in number theory is to determine exact formulas for the numberof representations of a positive integer n as a sum of 2 k squares, which we denote by r (2 k ; n ). Ifwe set(1.1) z = z ( τ ) := ∞ X m = −∞ ∞ X n = −∞ q m + n where, here and throughout the remainder of this work, τ is a complex number with positiveimaginary part and q = e πiτ , then (considering z as a power series in q ) ∞ X n =0 r (2 k ; n ) q n = z k . The function z k is a modular form and it is well-known that z k ( τ ) = E ∗ k ( τ ) + C k ( τ ) , where E ∗ k ( τ ) is an Eisenstein series and C k ( τ ) is a cusp form. In his remarkable work, Ramanujan[24, Eqs. (145)–(147)] stated without proof explicit formulas for E ∗ k ( τ ) and C k ( τ ), and hencededuced the value of the coefficients r (2 k ; n ). Ramanujan’s result was first proved by Mordell [20].To state it we need Dedekind’s eta function, which is defined by η ( τ ) := q / ∞ Y j =1 (1 − q n ) . Here and throughout, we write η m for η ( mτ ) for any positive integer m . Theorem 1.1 (Ramanujan–Mordell) . Suppose k is a positive integer. Let z be defined by (1.1) .Then (1.2) z k = F k ( τ ) + z k X ≤ j ≤ ( k − c j,k x j Date : November 5, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Ramanujan–Mordell Theorem, theta functions, sums of squares, sums of triangular num-bers, eta functions, modular forms.The research of the second author was supported by grant project numbers 27300314 and 17302515 of the ResearchGrants Council. here c j,k are numerical constants that depend on j and k , x = x ( τ ) := η η η , and F k ( τ ) is an Eisenstein series defined by: F ( τ ) := 1 + 4 ∞ X j =1 q j q j , and for k ≥ , F k ( τ ) := 1 − k ( − k (2 k − B k ∞ X j =1 j k − q j − ( − k + j q j , and F k +1 ( τ ) := 1 + 4( − k E k ∞ X j =1 (cid:18) (2 j ) k q j q j − ( − k + j (2 j − k q j − − q j − (cid:19) . Here B k and E k are the Bernoulli numbers and Euler numbers, respectively, defined by ue u − ∞ X k =0 B k k ! u k and u = ∞ X k =0 E k k ! u k . The reader is referred to [9, p. 2] for a brief account of the history of the study of Theorem 1.1for various k .The goal of this work is to prove the analogues of the Ramanujan–Mordell Theorem for which thequadratic form m + n in (1.1) is replaced with the quadratic form m + pn , or by the quadraticpolynomial m ( m + 1)2 + p n ( n + 1)2 , where p ∈ { , , , } .This work is organized as follows. In Section 2, we set up definitions, state the main results andexplicate several examples. Proofs of the main results are given in Section 3.2. Definitions and Main Results
For k ≥
1, define the normalized Eisenstein series by(2.1) E k ( τ ) := 1 − k B k ∞ X j =1 j k − q j − q j . Let p be an odd prime. The generalized Bernoulli numbers B k,p are defined by(2.2) xe px − p − X j =1 χ p ( j ) e jx = ∞ X k =0 B k,p x k k ! , where χ p ( j ) := (cid:16) jp (cid:17) is the Legendre symbol. Let k be a positive integer which satisfies k ≡ p −
12 (mod 2) . he generalized Eisenstein series E k ( τ ; χ p ) and E ∞ k ( τ ; χ p ) at the cusps 0 and i ∞ , respectively, aredefined by E k ( τ ; χ p ) := δ k, − k B k,p ∞ X j =1 j k − − q pj p − X ℓ =1 (cid:18) ℓp (cid:19) q jℓ and E ∞ k ( τ ; χ p ) := 1 − k B k,p ∞ X j =1 (cid:18) jp (cid:19) j k − q j − q j , where δ m,n is the Kronecker delta function, defined by δ m,n := ( m = n ,0 if m = n .Certain linear combinations of Eisenstein series and generalized Eisenstein series will occur in themain results, and in anticipation of this we define series G k ( τ ; p ), ˜ G k +1 ( τ ; p ), F k ( τ ; p ) and ˜ F ( τ ; p )as follows. For k ≥
1, let G k ( τ ; p ) := E k ( τ ) + ( − p ) k E k ( pτ ) . For k ≥ G k +1 ( τ ; p ) := E ∞ k +1 ( τ ; χ p ) + ( − p ) k E k +1 ( τ ; χ p )and ˜ G k +1 ( τ ; p ) := E ∞ k +1 ( τ ; χ p ) − ( − p ) k E k +1 ( τ ; χ p ) . For p = 3 or 11 and k ≥
0, let(2.3) F k +1 ( τ ; p ) := G k +1 ( τ ; p ) + 2 k +1 G k +1 (4 τ ; p )(2 k + 1)(1 + δ k, )and(2.4) ˜ F k +1 ( τ ; p ) := (2 k +1 −
2) ˜ G k +1 ( τ ; p ) − k +1 G k +1 (2 τ ; p ) + 2 G k +1 ( τ / p )2 k +2 (2 k +1 + 1)(1 + δ k, ) . For p = 7 or 23 and k ≥
0, let(2.5) F k +1 ( τ ; p ) := G k +1 ( τ ; p ) − G k +1 (2 τ ; p ) + 2 k +1 G k +1 (4 τ ; p )(2 k +1 − δ k, )and(2.6) ˜ F k +1 ( τ ; p ) := G k +1 ( τ ; p ) − G k +1 (2 τ ; p )2 k +1 (2 k +1 − δ k, ) . For p = 3, 7, 11 or 23 and any integer k ≥
1, let(2.7) F k ( τ ; p ) := G k ( τ ; p ) − G k (2 τ ; p ) + 2 k G k (4 τ ; p )(2 k − − p ) k )and(2.8) ˜ F k ( τ ; p ) := G k ( τ ; p ) − G k (2 τ ; p )2 k (2 k − − p ) k ) . he observant reader will notice that the series ˜ G k +1 occurs only once in the definitions (2.3)–(2.8), and that is as one of the terms in (2.4). A reason for this will be seen later in Lemma 3.3,in which (3.6) and (3.7) imply that for p = 3 or 11 G k +1 (cid:18) τ + 12 ; p (cid:19) = − G k +1 ( τ ; p ) + (cid:16) − k +1 (cid:17) ˜ G k +1 (2 τ ; p ) + 2 k +1 G k +1 (4 τ ; p ) . For p = 3 ,
7, 11 or 23, let z p and x p be defined by(2.9) z p = z p ( τ ) := ∞ X m = −∞ ∞ X n = −∞ q m + pn , and(2.10) x p = x p ( τ ) := ( η η η p η p ) / ( p +1) ( η η p ) / ( p +1) . The analogue of the Ramanujan–Mordell Theorem, and the main result of this work, is:
Theorem 2.1.
Suppose p = 3 , , or and let k be a positive integer. Then z kp = F k ( τ ; p ) + z kp X ≤ j< ( p +1) k c p,k,j x jp , where c p,k,j are numerical constants that depend only on p , k, and j . There is an analogue of the Ramanujan–Mordell theorem that involves sum of triangular numbersinstead of sums of squares; that is, replace the quadratic form m + n in (1.1) and Theorem 1.1with the quadratic polynomial m ( m + 1) / n ( n + 1) /
2. See [24, pp. 190–191] and [10, Theorems3.5 and 3.6]. To state the corresponding analogue of Theorem 2.1, let ˜ z p and ˜ x p be defined by(2.11) ˜ z p = ˜ z p ( τ ) := q ( p +1) / ∞ X m =0 ∞ X n =0 q m ( m +1)2 + p n ( n +1)2 and(2.12) ˜ x p = ˜ x p ( τ ) := (cid:18) η η p η η p (cid:19) / ( p +1) . Corollary 2.2.
Suppose p = 3 , , or and let k be a positive integer. Then ˜ z kp = ˜ F k ( τ ; p ) + ˜ z kp X ≤ j< ( p +1) k ˜ c p,k,j ˜ x jp , where ˜ c p,k,j are numerical constants that depend only on p , k, and j and are related to the c p,k,j inTheorem . by ˜ c p,k,j = 2 − j/ ( p +1) ( − j c p,k,j . In the remainder of this section we describe some special cases of Theorem 2.1 and Corollary 2.2.Let ϕ ( q ) and ψ ( q ) be Ramanujan’s theta functions defined by ϕ ( q ) = ∞ X n = −∞ q n and ψ ( q ) = ∞ X n =0 q n ( n +1)2 . Because of the occurrence of the argument τ / τ with 2 τ , and hence q with q , to obtain the examples (2.14), (2.18) and (2.24), below. xample 2.3. For k = 1 and p = 3 , , or , Theorem . and Corollary . give ϕ ( q ) ϕ ( q ) = 1 + 2 ∞ X j =1 (cid:18) j (cid:19) (cid:18) q j − q j + 2 q j − q j (cid:19) , (2.13) qψ ( q ) ψ ( q ) = ∞ X j =1 (cid:18) j (cid:19) (cid:18) q j − q j − q j − q j (cid:19) , (2.14) ϕ ( q ) ϕ ( q ) = 1 + 2 ∞ X j =1 (cid:18) j (cid:19) (cid:18) q j − q j − q j − q j + 2 q j − q j (cid:19) , (2.15) qψ ( q ) ψ ( q ) = ∞ X j =1 (cid:18) j (cid:19) (cid:18) q j − q j − q j − q j (cid:19) , (2.16) ϕ ( q ) ϕ ( q ) = 1 + 23 ∞ X j =1 (cid:18) j (cid:19) (cid:18) q j − q j + 2 q j − q j (cid:19) + 43 η η , (2.17) q ψ ( q ) ψ ( q ) = 13 ∞ X j =1 (cid:18) j (cid:19) (cid:18) q j − q j − q j − q j (cid:19) − η η , (2.18) ϕ ( q ) ϕ ( q ) = 1 + 23 ∞ X j =1 (cid:18) j (cid:19) (cid:18) q j − q j − q j − q j + 2 q j − q j (cid:19) (2.19) + 43 η η η η η η − η η ,q ψ ( q ) ψ ( q ) = 13 ∞ X j =1 (cid:18) j (cid:19) (cid:18) q j − q j − q j − q j (cid:19) − η η − η η . (2.20)The identity (2.13) was first stated in an equivalent form by Lorenz [19, p. 420]. Both (2.13) and(2.14) were given by Ramanujan in his second notebook [23, Ch. 19, Entry 3]. See Berndt [8, pp.223–224], Fine [13, p. 73, (31.16), (31.22)] and Hirschhorn [14] for proofs and further information.The identities (2.15) and (2.16) also appear in Ramanujan’s second notebook [23, Ch. 19, Entry17] and proofs have been given by Berndt [8, pp. 302–304]. The identity (2.15) has also beenproved by Pall [21].The identities (2.17) and (2.19) have been recently proved by the third author [25].The identities (2.18) and (2.20) are new. Example 2.4.
For p = 3 , the cases k = 2 , , and of Theorem . and Corollary . give ϕ ( q ) ϕ ( q ) = 1 + 4 ∞ X j =1 (cid:18) jq j − q j − jq j − q j − jq j − q j + 4 jq j − q j + 6 jq j − q j − jq j − q j (cid:19) , (2.21) qψ ( q ) ψ ( q ) = ∞ X j =1 (cid:18) jq j − q j − jq j − q j − jq j − q j + 3 jq j − q j (cid:19) , (2.22) ϕ ( q ) ϕ ( q ) = 1 + 3 ∞ X j =1 (cid:18) j ( q j − q j )1 − q j + 8 j ( q j − q j )1 − q j (cid:19) (2.23) ∞ X j =1 (cid:18) j (cid:19) (cid:18) j q j − q j + 8 j q j − q j (cid:19) + 4 η η ,q ψ ( q ) ψ ( q ) = 332 ∞ X j =1 (cid:18) j ( q j − q j )1 − q j − j ( q j − q j )1 − q j − j ( q j − q j )1 − q j (cid:19) (2.24) − ∞ X j =1 (cid:18) j (cid:19) (cid:18) j q j − q j + 3 j q j − q j − j q j − q j (cid:19) − η η ,ϕ ( q ) ϕ ( q ) = 1 + 85 ∞ X j =1 (cid:18) j q j − q j − j q j − q j + 9 j q j − q j + 16 j q j − q j − j q j − q j + 144 j q j − q j (cid:19) (2.25) + 325 η η η η η η ,q ψ ( q ) ψ ( q ) = 110 ∞ X j =1 (cid:18) j q j − q j − j q j − q j + 9 j q j − q j − j q j − q j (cid:19) − η η η η , (2.26) ϕ ( q ) ϕ ( q ) = 1 + 413 ∞ X j =1 (cid:18) j q j − q j − j q j − q j − j q j − q j + 64 j q j − q j + 54 j q j − q j − j q j − q j (cid:19) (2.27) + 15213 η η η η η η − η η ,q ψ ( q ) ψ ( q ) = 1208 ∞ X j =1 (cid:18) j q j − q j − j q j − q j − j q j − q j + 27 j q j − q j (cid:19) − η η − η η . (2.28)The identity (2.21) was first stated without proof in an equivalent form by Liouville [17, 18]. SeePepin [22], Bachmann [5], Kloosterman [15] and Alaca et al. [2] for proofs. Both (2.21) and (2.22)appear in Ramanujan’s second notebook [23, Ch. 19, Entry 3]. Proofs have been given by Fine[13, (31.4)–(31.43) and (33.2)] and Berndt [8, pp. 223–226].A formula equivalent to (2.23) was proved by Alaca et al. in [3], where it was attributed toBerkovich and Ye´silyurt.The identity (2.25) was proved by Alaca and Williams [4]. A formula similar to (2.25), in whichthe coefficients in η η η η are given as a quadruple sum, has been given by Beridze [6].A formula equivalent to (2.27) was given by Alaca [1]; that formula involves three cusp forms onthe right hand side, while ours involves only two .The identities (2.24), (2.26) and (2.28) are believed to be new. Example 2.5.
For p = 7 , the cases k = 2 and of Theorem . give ϕ ( q ) ϕ ( q ) = 1 + 43 ∞ X j =1 (cid:18) jq j − q j − jq j − q j + 4 jq j − q j − jq j − q j + 14 jq j − q j − jq j − q j (cid:19) (2.29) This is because, if b ( q ) = η η , then b ( q ) + 12 b ( q ) + 64 b ( q ) + b ( − q ) = 0 . η η η η η η ,q ψ ( q ) ψ ( q ) = 13 ∞ X j =1 (cid:18) jq j − q j − jq j − q j − jq j − q j + 7 jq j − q j (cid:19) (2.30) − η η η η ,ϕ ( q ) ϕ ( q ) = 1 + 78 ∞ X j =1 j ( q j + q j − q j + q j − q j − q j )1 − q j (2.31) − ∞ X j =1 j ( q j + q j − q j + q j − q j − q j )1 − q j + 7 ∞ X j =1 j ( q j + q j − q j + q j − q j − q j )1 − q j − ∞ X j =1 (cid:18) j (cid:19) (cid:18) j q j − q j − j q j − q j + 8 j q j − q j (cid:19) + 214 η η η η η η − η η ,q ψ ( q ) ψ ( q ) = 764 ∞ X j =1 j ( q j + q j − q j + q j − q j − q j )1 − q j (2.32) − ∞ X j =1 j ( q j + q j − q j + q j − q j − q j )1 − q j − ∞ X j =1 (cid:18) j (cid:19) (cid:18) j q j − q j − j q j − q j (cid:19) − η η − η η . Identities equivalent to (2.29) and (2.30) have been proved in [12]. The identities (2.31) and(2.32) arise in the theory of 7-cores and were proved by Berkovich and Yesilyurt [7].3.
Proofs
For any positive integer N , let us defineΓ ( N ) = ( (cid:18) a bc d (cid:19) : a, b, c, d ∈ Z , ad − bc = 1 , c ≡ N ) ) . We require the explicit modularity properties of the Eisenstein series on Γ ( p ) as well as the Atkin–Lehner involution W p . Lemma 3.1.
For p = 3 , , or , and any integer k ≥ , and for any (cid:18) a bc d (cid:19) ∈ Γ ( p ) , we have E k +1 (cid:18) aτ + bcτ + d ; χ p (cid:19) = (cid:18) dp (cid:19) ( cτ + d ) k +1 E k +1 ( τ ; χ p ) , (3.1) E ∞ k +1 (cid:18) aτ + bcτ + d ; χ p (cid:19) = (cid:18) dp (cid:19) ( cτ + d ) k +1 E ∞ k +1 ( τ ; χ p ) , (3.2) ∞ k +1 (cid:18) − pτ ; χ p (cid:19) = 1 i √ p ( pτ ) k +1 E k +1 ( τ ; χ p ) , (3.3) E k +1 (cid:18) − pτ ; χ p (cid:19) = √ pi τ k +1 E ∞ k +1 ( τ ; χ p ) . (3.4) Proof.
This is a well-known result, e.g., see Cooper [11] or Kolberg [16]. (cid:3)
Lemma 3.2.
For any positive integer k , we have E k (cid:18) τ + 12 (cid:19) = − E k ( τ ) + (2 k + 2) E k (2 τ ) − k E k (4 τ ) . (3.5) For p = 3 or and any nonnegative integer k , we have E k +1 (cid:18) τ + 12 ; χ p (cid:19) = − E k +1 ( τ ; χ p ) + (2 k +1 − E k +1 (2 τ ; χ p ) + 2 k +1 E k +1 (4 τ ; χ p ) , (3.6) E ∞ k +1 (cid:18) τ + 12 ; χ p (cid:19) = − E ∞ k +1 ( τ ; χ p ) + (2 − k +1 ) E ∞ k +1 (2 τ ; χ p ) + 2 k +1 E ∞ k +1 (4 τ ; χ p ) . (3.7) For p = 7 or and any nonnegative integer k , we have E k +1 (cid:18) τ + 12 ; χ p (cid:19) = − E k +1 ( τ ; χ p ) + (2 k +1 + 2) E k +1 (2 τ ; χ p ) − k +1 E k +1 (4 τ ; χ p ) , (3.8) E ∞ k +1 (cid:18) τ + 12 ; χ p (cid:19) = − E ∞ k +1 ( τ ; χ p ) + (2 k +1 + 2) E ∞ k +1 (2 τ ; χ p ) − k +1 E ∞ k +1 (4 τ ; χ p ) . (3.9) Proof.
First of all, from the definitions of E k ( τ ), E k +1 ( τ ; χ p ) and E ∞ k +1 ( τ ; χ p ), we may deducethe Fourier expansions E k ( τ ) = 1 − k B k ∞ X n =1 σ k − ( n ) q n , (3.10) E k +1 ( τ ; χ p ) = δ k +1 , − k + 2 B k +1 ,p ∞ X n =1 X d | n (cid:18) n/dp (cid:19) d k q n (3.11)and E ∞ k +1 ( τ ; χ p ) = 1 − k + 2 B k +1 ,p ∞ X n =1 X d | n (cid:18) dp (cid:19) d k q n , (3.12)where σ k ( n ) = P d | n d k if n is a positive integer. For (3.5), we first observe that σ k (2 n ) = (cid:16) k (cid:17) σ k ( n ) − k σ k ( n/ , where σ k ( n/
2) is defined to be zero if n/ E k (cid:18) τ + 12 (cid:19) + E k ( τ )= 2 − k B k ∞ X n =1 σ k − (2 n ) q n ! = 2 − k B k ∞ X n =1 (cid:16) (1 + 2 k − ) σ k − ( n ) − k − σ k − ( n/ (cid:17) q n ! (cid:16) k − (cid:17) − k B k ∞ X n =1 σ k − ( n ) q n ! − k − − k B k ∞ X n =1 σ k − ( n ) q n !! = (cid:16) k (cid:17) E k (2 τ ) − k E k (4 τ ) . If we let σ k +1 ( n ) and σ ∞ k +1 ( n ) be defined by σ k +1 ( n ) = X d | n (cid:18) n/dp (cid:19) d k , and σ ∞ k +1 ( n ) = X d | n (cid:18) dp (cid:19) d k , then similarly, we find that σ k +1 (2 n ) = ( (2 k − σ k +1 ( n ) + 2 k σ k +1 ( n/ , for p = 3 or 11,(1 + 2 k ) σ k +1 ( n ) − k σ k +1 ( n/ , for p = 7 or 23,and σ ∞ k +1 (2 n ) = ( (1 − k ) σ ∞ k +1 ( n ) + 2 k σ ∞ k +1 ( n/ , for p = 3 or 11,(1 + 2 k ) σ ∞ k +1 ( n ) − k σ ∞ k +1 ( n/ , for p = 7 or 23.By the above observations, identities (3.6)–(3.9) can be proved in the same fashion, so we omit thedetails. (cid:3) Lemma 3.3.
For any positive integer k , we have (3.13) F k (cid:18) − pτ + 12 ; p (cid:19) = τ k ( − p ) k ( G k (2 τ ) − G k (4 τ ))(2 k − − p ) k ) . For p = 3 or and any non-negative integer k , we have F k +1 (cid:18) − pτ + 12 ; p (cid:19) (3.14) = (2 iτ √ p ) k +1 × (cid:16) k +1 G k +1 (4 τ ; p ) − k −
1) ˜ G k +1 (2 τ ; p ) − G k +1 ( τ ; p ) (cid:17) (2 k +1 + 1)(1 + δ k, ) . For p = 7 or and any non-negative integer k , we have F k +1 (cid:18) − pτ + 12 ; p (cid:19) = (4 iτ √ p ) k +1 × ( G k +1 (4 τ ; p ) − G k +1 (2 τ ; p ))(2 k +1 − δ k, ) . (3.15) Proof.
These follow immediately from the definitions of F k ( τ ; p ) together with Lemmas 3.1 and 3.2and the weight 2 k modularity of E k on SL ( Z ) = Γ (1). (cid:3) Now we are ready for
Proof of Theorem . . Let p = 3, 7, 11 or 23, and let k be a positive integer. Let ℓ be the smallestinteger that satisfies ℓ ≥ ( ( p +1) k − , if p = 3 or 11 and k is odd, ( p +1) k − , otherwise.Consider the functions f ( τ ) = f k,p ( τ ) = F k ( τ ; p ) z p ( τ ) k x p ( τ ) ℓ and g ( τ ) = g p ( τ ) = 1 x p ( τ ) . learly, both f ( τ ) and g ( τ ) are analytic on H . And we may verify that both f ( τ ) and g ( τ ) areinvariant under Γ (4 p ) and W e = (cid:26)(cid:18) ae b pc de (cid:19) : a, b, c, d ∈ Z , the determinant is e (cid:27) for e ∈ { , p, p } . Therefore, both f ( τ ) and g ( τ ) are invariant under Γ (4 p ) + , the group obtainedfrom Γ (4 p ) by adjoining all of its Atkin-Lehner involutions W e . Let us analyze the behavior at τ = i ∞ . By observing the q -expansions, we find that f ( τ ) = 1 + O ( q )(1 + O ( q )) k q ℓ (1 + O ( q )) ℓ = q − ℓ + O ( q − ℓ +1 ) . Therefore f ( τ ) has a pole of order ℓ at i ∞ . Similarly, we note that g ( τ ) has a simple pole at τ = i ∞ . It implies that there exist constants a , . . . , a ℓ ∈ C such that the function h ( τ ) := f ( τ ) − ℓ X j =1 a j g ( τ ) j has no pole at τ = i ∞ , that is, h ( τ ) = a + O ( q ) as τ → i ∞ for some constant a . Let us consider the behavior of h ( τ ) at τ = . By Lemma 3.3 and thetransformation formula for Dedekind’s eta function, we find that f (cid:18) − pτ + 12 (cid:19) = ( C q ℓ − ( p +1) k +1 (1 + O ( q )) , if p = 3 or 11 and k is odd, C q ℓ − ( p +1) k +2 (1 + O ( q )) , otherwise,and g (cid:18) − pτ + 12 (cid:19) = C q (1 + O ( q ))for some constants C , C and C as τ → i ∞ . Therefore, h ( τ ) → a as τ → . Since the only cuspsof Γ (4 p ) + are at i ∞ and , it follows that h ( τ ) is holomorphic on X (Γ (4 p ) + ), and thus h ( τ ) is aconstant, that is, h ( τ ) ≡ a . Therefore, we have f ( τ ) = ℓ X j =0 a j g ( τ ) j , which is equivalent to F k ( τ ; p ) = z kp ℓ X j =0 a j x ℓ − jp = z kp ℓ X j =0 b j x jp , where b j := a ℓ − j . By the choice of ℓ and comparing the constant terms on both sides, we concludethat b = 1 and F k ( τ ; p ) = z kp + z kp X ≤ j< ( p +1) k b j x jp . Now take b j = − c p,k,j to complete the proof. (cid:3) Proof of Corollary . . It follows directly from Theorem 2 . z p ( τ ) = i √ pτ z p (cid:18) − pτ + 12 (cid:19) , x p ( τ ) = − − / ( p +1) ˜ x p (cid:18) − pτ + 12 (cid:19) , ˜ F k ( τ ; p ) = (cid:18) i √ pτ (cid:19) k F k (cid:18) − pτ + 12 (cid:19) . (3.16) (cid:3) References [1] A. Alaca,
On the number of representations of a positive integer by certain quadratic forms in twelve variables,
J. Combin. Number Theory, (2011), 167–177.[2] A. Alaca, S. Alaca, M. F. Lemire and K. S. Williams, Nineteen quaternary quadratic forms,
Acta Arith. (2007), 277–310.[3] A. Alaca, S. Alaca and K. S. Williams,
Some new theta function identities with applications to sextenary quadraticforms,
J. Combin. Number Theory, (2009), 89–98.[4] S. Alaca and K. S. Williams, The number of representations of a positive integer by certain octonary quadraticforms,
Functiones et Approximatio, (2010), 45–54.[5] P. Bachmann, Niedere Zahlentheorie,
Chelsea, New York, 1968.[6] R. I. Beridze,
The representation of numbers by certain quadratic forms in eight variables,
Thbilis. Univ. ˘Srom.A, (1971), 5–16.[7] A. Berkovich and H. Yesilyurt, On the representations of integers by the sextenary quadratic form x + y + z +7( s + t + u ) and -cores, J. Number Theory, (2009), 1366–1378.[8] B. C. Berndt,
Ramanujan’s Notebooks, Part III,
Springer-Verlag, New York, 1991.[9] H. H. Chan and S. Cooper,
Powers of theta functions,
Pacific J. Math., (2008), 1–14.[10] S. Cooper,
On sums of an even number of squares, and an even number of triangular numbers: an elementaryapproach based on Ramanujan’s ψ summation formula. In: q -series with applications to combinatorics, numbertheory, and physics (Urbana, IL, 2000), 115–137, Contemp. Math., , Amer. Math. Soc., Providence, RI, 2001.[11] S. Cooper, Construction of Eisenstein series for Γ ( p ), Int. J. Number Theory (2009), 765–778.[12] S. Cooper and D. Ye, Level 14 and 15 analogues of Ramanujan’s elliptic functions to alternative bases,
Trans.Amer. Math. Soc., DOI: http://dx.doi.org/10.1090/tran6658[13] N. Fine,
Basic Hypergeometric Series and Applications,
AMS, Providence, RI, 1988.[14] M. D. Hirschhorn,
Three classical results on representations of a number,
S´em. Lothar. Combin. (1999), art.B42f, 8 pp.[15] H. D. Kloosterman, On the representation of numbers in the form ax + by + cz + dt , Proc. London Math.Soc. (1926), 143–173.[16] O. Kolberg, Note on the Eisenstein series of Γ ( p ), Arbok Univ. Bergen Mat.-Natur. Ser. (1968), 20 pp.(1969).[17] J. Liouville, Sur la forme x + y + 3( z + t ) , J. Math. Pures Appl. (1860), 147–152.[18] J. Liouville, Remarque nouvelle sur la forme x + y + 3( z + t ), ibid. (1861), 296.[19] L. Lorenz, Oeuvres Scientifiques, Revues et annot´ees par H. Valentiner, Tome second, la Fondation Carlsberg,Librarie Lehmann & Stage, Copenhague, 1904.[20] L. J. Mordell, On the representation of numbers as the sum of r squares, Quart. J. Pure and Appl. Math.,Oxford (1917), 93–104.[21] G. Pall, On the application of a theta formula to representation in binary quadratic forms,
Bull. Amer. Math.Soc., (1931), 863–869.[22] T. Pepin, Sur quelques formes quadratiques quaternaires,
J. Math. Pures Appl. (1890), 5–67.[23] S. Ramanujan, Notebooks (2 volumes) , Tata Institute of Fundamental Research, Bombay, 1957.[24] S. Ramanujan,
Collected Papers,
AMS Chelsea Publishing, Providence, Rhode Island, 2000.[25] D. Ye,
Representations of certain binary quadratic forms as a sum of Lambert series and eta-quotients , Int. J.Number Theory (2015), 1073-1088 Institute of Natural and Mathematical Sciences, Massey University-Albany, Private Bag 102904,North Shore Mail Centre, Auckland, New Zealand, E-mail: [email protected] of Mathematics, University of Hong Kong, Pokfulam, Hong Kong, E-mail address:[email protected] epartment of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin, 53706USA, E-mail: [email protected] of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin, 53706USA, E-mail: [email protected]