Theta Function Identities and Representations by Certain Quaternary Quadratic Forms II

International Mathematical Forum,, 008, no., 59-579 Theta Function Identities and Representations by Certain Quaternary Quadratic Forms II Ayşe Alaca, Şaban Alaca Mathieu F. Lemire and Kenneth S. Williams aalaca@connect.carleton.ca, salaca@connect.carleton.ca mathieul@connect.carleton.ca, kwilliam@connect.carleton.ca Centre for Research in Algebra and Number Theory School of Mathematics and Statistics, Carleton University Ottawa, Ontario, Canada KS 5B6 Abstract Some new theta function identities are proved and used to determine the number of representations of a positive integer n by certain quaternary quadratic forms. Mathematics Subject Classification: E0, E5 Keywords: quaternary quadratic forms, theta functions Introduction Let N, N 0, Z, Q, R and C denote the sets of positive integers, nonnegative integers, integers, rational numbers, real numbers and complex numbers respectively. For q C with q < Ramanujan defined the one-dimensional theta function ϕq by. ϕq := n Zq n. For a, b, c, d N and n N 0 we define. Na, b, c, d; n = card{x, y, z, t Z n = ax + by + cz + dt }. In [] the authors determined Na, b, c, d; n n N for the following quaternary quadratic forms: x + y + z +t

50 A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams x + y +z +6t x +y +z +t x +y +z +6t x +y +z +t x +y +6z +6t x +y +z +6t. For all of these seven forms, Na, b, c, d; n was given in terms of the arithmetic functions An, Bn, Cn and Dn given in Definition.. In this paper we consider the sixteen forms x + y + z +t x + y +z +t x + y +z +t x +y +z +t x +y +z +t x +y +z +t x +y +z +t x +y +z +t x +6y +6z +t x +y +z +t x +y +z +t x +y +z +t x +y +z +t x +y +z +t x +y +6z +6t x +y +z +t and show that Na, b, c, d; n for these forms can be given in terms of An, Bn, Cn, Dn and the two additional arithmetic functions En and F n defined in Definition.. Notation and Preliminary Results For q C with q < we set. ϕq := n Nq n.

Theta function identities 5 As in [5, pp., ] we define. p = pq := ϕ q ϕ q ϕ q and. k = kq := ϕ q ϕq. The representations of ϕq j j {,,,, 6, } in terms of p and k are given in Proposition. see [, Theorem.]. Proposition.. Define p and k by. and. respectively. Then a ϕq =+p / k /, b ϕq = + p / + p / + p / / k /, c ϕq =+p / k /, d ϕq = + p / + p / + p / k /, e ϕq 6 = + p / + p / + p / / k /, f ϕq = + p / + p / + p / k /. Using Proposition. in conjunction with the following well-known basic properties of ϕq..5 ϕq+ϕ q =ϕq, ϕ q+ϕ q =ϕ q, and.6 ϕqϕ q =ϕ q, we obtain ϕ q j j {,,,, 6, } in terms of p and k. Proposition.. a ϕ q = p / + p / k /, b ϕ q =+p /8 p /8 + p /8 k /,

5 A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams c ϕ q = p / + p / k /, d ϕ q = / + p /6 + p /6 p /6 + p / ++p / p / / k /, e ϕ q 6 =+p /8 + p /8 p /8 k /, f ϕ q = / + p /6 + p /6 p /6 + p / ++p / p / / k /. In [, Definition.] we introduced the multiplicative arithmetic functions An, Bn, Cn and Dn. Definition.. For n N we set a An := d = n/d d n d n where D. b c d Bn := d n Cn := d n Dn := d n d d d n/d n d, d = n/d d n = d d n d = d d n n d n/d n d n, d n/d, d, d n/d D k N is the Legendre-Jacobi-Kronecker symbol for discriminant k The following result was given in [, Theorem.]. Proposition.. Let n N. Set n = α β N, where α, β N 0, N N and gcdn, 6 =. Then a An = α β AN, b Bn = α+β α N AN, c Cn = α+β+n / β AN, N d Dn = N / AN = AN. N

Theta function identities 5 Simple consequences of Proposition. are An =Bn, Cn =Dn, if n mod, An = Bn, Cn = Dn, if n mod, An =Cn, Bn =Dn, if n mod, An = Cn, Bn = Dn, if n mod. The next result was deduced from the work of Petr [0], see [, Theorem.]. Proposition.. For q < Anq n = 8 ϕqϕ q + 8 ϕ qϕq n= 8 ϕ qϕ qϕ q 8 ϕ qϕ q ϕ q, Bnq n = 8 ϕqϕ q 8 ϕ qϕq n= + 8 ϕ qϕ qϕ q 8 ϕ qϕ q ϕ q, Cnq n = ϕ qϕ qϕ q ϕ qϕ q ϕ q, n= Dnq n = ϕ qϕ qϕ q ϕ qϕ q ϕ q. n= Solving for the quantities ϕqϕ q, ϕ qϕq, ϕ qϕ qϕ q and ϕ qϕ q ϕ q in Proposition., we obtain the following result, see [, Theorem.]. Proposition.5. For q < a ϕqϕ q =+ b ϕ qϕq =+ An+Bn Cn Dnq n, n= 6An Bn+Cn Dnq n, n= c ϕ qϕ qϕ q =+ Cn Dnq n, n=

5 A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams d ϕ qϕ q ϕ q = Cn+Dnq n. Proposition.6. m / mq m = ϕq ϕ q ϕ q 8. m = m odd Proof. See [, Theorem.]. Proposition.7. m = m odd n= m / mq m = ϕq ϕ q ϕq ϕ q. Proof. Replacing q by q in Proposition.6, and using the identity.6 with q replaced by q, we obtain Proposition.7. Definition.. For n N we set En := i / i and F n := i, j N i, j odd n = i +j Proposition.8. Let n N. Then i, j N i, j odd n = i +j En =F n =0, if n is even. j / j. Proof. If i and j are both odd and n = i +j then n + mod 8, so n is odd. Clearly, as n =±i +±j, i / i = i / i and we have.7 En := i, j Z i, j odd n = i +j j / j = j / j, i / i and F n := i, j Z i, j odd n = i +j j / j.

Theta function identities 55 Theorem.. a Enq n = 6 ϕq ϕ q ϕq ϕ q ϕq ϕ q, b F nq n = 6 ϕq ϕ q ϕq ϕ q ϕq ϕ q. Proof. a Appealing to Proposition.8 and.7, we obtain.8 Enq n = Enq n = n= i = i odd i / iq i j = j odd q j. Now, by., we have.9 n = q n = q n q n = ϕq ϕq = ϕq ϕ q. n= n= Appealing to.8, Proposition.6,.6 with q replaced by q and.9 with q replaced by q we obtain part a. b Appealing to Proposition.8 and.7, we obtain.0 F nq n = F nq n = n= i = i odd q i j = j odd j / jq j. Appealing to.0,.9 and Proposition.7, we obtain part b. Theorem.. a ϕ qϕ q ϕ qϕq = ϕqϕ q ϕ q ϕ qϕ q ϕq. b ϕq ϕ q ϕq ϕ q = ϕq ϕq ϕ q ϕ q. c ϕ q ϕ q ϕ q ϕq = ϕq ϕ q ϕq ϕ q ϕq ϕ q. d ϕqϕ q ϕ qϕ q =ϕ qϕ qϕq ϕ qϕqϕ q.

56 A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams Proof. a We start with the identity + p p = + p p + p. Multiplying both sides by + p / p / + p / k, we obtain + p 9/ p / + p / k + p / p 9/ + p / k = + p 5/ p / + p / k + p / p 5/ + p 7/ k. By Propositions. and., we have ϕ qϕ q = +p 9/ p / + p / k, ϕ qϕq = +p / p 9/ + p / k, ϕqϕ q ϕ q = +p 5/ p / + p / k, ϕ qϕ q ϕq = +p / p 5/ + p 7/ k, and part a follows. b We have ϕ q ϕ q = / p /6 + p /6 + p /6 + p / + p / + p / / k / / p /6 + p /6 + p /6 + p / + p / + p / / k / = / p / + p / + p / +p +p + p + + p + p p / + p / + p / / k = / p / + p / + p / +p + p + p / + p / + p / / k = p / + p / + p / +p +p + p / + p / + p / / k

Theta function identities 57 = p / + p / + p / + p / + p / + p / k = p / + p / + p / k + p / + p / + p / k = ϕqϕ q +ϕ qϕq. Also ϕq ϕq = ϕq+ϕ q ϕq +ϕ q = ϕqϕq +ϕ qϕq +ϕqϕ q +ϕ qϕ q. Hence ϕq ϕq ϕ q ϕ q = ϕqϕq +ϕ qϕq +ϕqϕ q +ϕ qϕ q ϕqϕ q ϕ qϕq = ϕqϕq ϕ qϕq ϕqϕ q +ϕ qϕ q = ϕq ϕ q ϕq ϕ q. c We have ϕq ϕ q ϕq ϕ q ϕq ϕ q = ϕq ϕq ϕ q ϕ q ϕq ϕ q by part b = ϕq ϕ q ϕ q ϕ q ϕq ϕ q = ϕ q ϕ q ϕ q ϕq by part a with q replaced by q as asserted. d We start with the identity + p =+p p + p. Multiplying both sides by + p / p / + p / k, we obtain + p / p / + p 9/ k + p / p / + p / k =+p 7/ p / + p / k + p / p 7/ + p 5/ k. By Propositions. and. we obtain ϕqϕ q ϕ qϕ q =ϕ qϕ qϕq ϕ qϕqϕ q as asserted.

58 A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams Theorem.. a b Enq n = ϕ qϕ qϕq ϕqϕ qϕ q = ϕqϕ q ϕ qϕ q. F nq n = ϕqϕ q ϕ q ϕ qϕ q ϕq = ϕ qϕ q ϕ qϕq. Proof. a By Theorem.a and Theorem.b, we deduce Enq n = 6 ϕq ϕ q ϕq ϕ q ϕq ϕ q = ϕq ϕq ϕ q ϕ q ϕq ϕ q. Replacing q by q, we obtain appealing to Theorem.d Enq n = ϕqϕq ϕ qϕ q ϕqϕ q = ϕ qϕ qϕq ϕqϕ qϕ q = ϕqϕ q ϕ qϕ q, which is part a. b By Theorem.b and Theorem.b, we have F nq n = ϕq ϕq ϕ q ϕ q ϕq ϕ q. Replacing q by q, we obtain appealing to Theorem.a F nq n = ϕqϕq ϕ qϕ q ϕq ϕ q = ϕqϕ q ϕ q ϕ qϕq ϕ q = ϕ qϕ q ϕ qϕq,

Theta function identities 59 which is part b. We conclude this section by giving some arithmetic properties of En and F n. These properties can be used to slightly simplify Theorem 7. and Corollary 7. when n is odd by splitting into subcases modulo and/or modulo. However we do not do this. Theorem.. Let n N. Then a F n =En and b En = F n. Proof. a We have b We have as asserted. F n = En = = i, j N i, j odd n = i +j j, k N j, k odd n = j +k i, j N i, j odd n = i +j = j, k N j, k odd n = j +k j / j = j / j = En. i / i = k,j N k, j odd n =k +j k, j N k, j odd n =k +j k / k = F n, j / j k / k Theorem.5. Let n N. Then En =F n =0, if n mod.

550 A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams Proof. If i, j N is such that n = i +j, then n n i +j i 0, mod, so n mod implies En =F n =0. Theorem.6. Let n N. Then { F n, if n mod, En = F n, if n mod. Proof. Let n N be odd. First we observe that by.7 we have replacing i by j +k. F n = j / j = j / j. i, j = i, j odd n = i +j j, k = j odd n = j + jk + k Secondly, we see that as n = j + jk+ k n = j + j j k+ j k j / j = j / j. j, k = j odd,k even n = j + jk + k j, k = j odd,k odd n = j + jk + k = j, k = j odd n = j + jk + k = F n j / j by.. Thirdly, we see that as n = j + jk + k jk + j + k + mod for j k mod j / k = n+/+k / k that is. j, k = j odd,k odd n = j + jk + k j, k = j, k odd n = j + jk + k j, k = j odd,k odd n = j + jk + k = n+/ j, k = j odd,k odd n = j + jk + k j / k = n / F n, j / j,

Theta function identities 55 by.. Fourthly, we have as n = j + jk+ k n = j + j j k+ j k j, k = j odd,k even n = j + jk + k that is j / k = = j, k = j, k odd n = j + jk + k j, k = j, k odd n = j + jk + k j / j k j / j j, k = j, k odd n = j + jk + k j / k,. j, k = j odd,k even n = j + jk + k j / k = F n+ n / F n, by. and.. Finally En = = i / i = i, j = i, j odd n = i +j j, k = j odd,k even n = j + jk + k + j, k = j odd,k even n = j + jk + k j / j j / k j, k = j odd n = j + jk + k j /+k j +k j, k = j, k odd n = j + jk + k j, k = j, k odd n = j + jk + k j / j j / k = F n F n F n+ n / F n+ n / F n = n / F n, by.7,.,. and., so that as asserted. En = n F n

55 A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams The power series of ϕ qϕ q We see from Proposition.5 that the coefficients of q in the power series expansions of ϕqϕ q, ϕ qϕq, ϕ qϕ qϕ q and ϕ q ϕ qϕ q involve An, Bn, Cn and Dn. In Sections -6 we determine the power series expansions of ϕ qϕ q, ϕ qϕ qϕq, ϕqϕ q ϕ q and ϕqϕ q in powers of q and show that they involve An, Bn, Cn, Dn, En and F n, see Theorems.,., 5. and 6.. These four products of theta functions occur in Theorem. together with those obtained from them by replacing q by q. In this section we determine the power series expansion of ϕ qϕ q in powers of q. Theorem.. ϕ qϕ q = +6 n/ An Bn Cn+Dn q n F nq n. We require a lemma before proving Theorem.. Lemma.. Let n N. Then { 9 N,,, ; n = An Bn, if n mod, An Bn+Cn Dn, if n 0 mod. Proof. If x + y + z +t = n mod then exactly one of x, y and z is even. Thus, by [, Theorem 5.] and Proposition.ab, we have N,,, ; n = N,,, ; n = N,,, 6; n/ = 9An/ + Bn/ = 9 An Bn. If x + y + z +t = n 0 mod then x, y and z are all even and, by [, Theorem.] and Proposition., we have N,,, ; n = N,,, ; n/ = 6An/ Bn/ + Cn/ Dn/

Theta function identities 55 = An Bn+Cn Dn. Proof of Theorem.. By Lemma. we have N,,,, ; nq n = = = n 0mod + n mod An Bn+Cn Dn q n 9 An Bn q n n/ An n/ Bn + + n/ Cn An Bn+ Cn Dn q n On the other hand we have N,,,, ; nq n = = N,,,, ; nq n + n= n=0 + n/ Dn q n n/ An Bn Cn+Dn q n. N,,,, ; n q n n= N,,,, ; nq n + = ϕ qϕq + ϕ qϕq n=0 N,,,, ; n q n

55 A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams = ϕ q+ϕ q ϕq = ϕ q+ϕ q ϕq +ϕ q by. = ϕ qϕq + ϕ qϕ q + ϕ qϕ q +ϕ qϕq = + 6An Bn+Cn Dn q n + + n= + 6An Bn+Cn Dn q n n= ϕ qϕ q +ϕ qϕq by Proposition.5b = + An Bn+ Cn Dn q n ϕ qϕ q +ϕ qϕq. Equating the two expressions for N,,,, ; nq n, we obtain so = n/ An Bn Cn+Dn q n ϕ qϕ q +ϕ qϕq ϕ qϕ q +ϕ qϕq = n/ An Bn Cn+Dn q n. From Theorem.b we have ϕ qϕ q ϕ qϕq = F nq n.

Theta function identities 555 Adding these two equations, and dividing by, we obtain ϕ qϕ q = as asserted. +6 n/ An Bn Cn+Dn q n F nq n, The power series of ϕ qϕ qϕq In this section we determine the power series expansion of ϕ qϕ qϕq in powers of q. Theorem.. ϕ qϕ qϕq = + + n/ An Bn+Cn Dn q n Enq n. We require a number of lemmas. For n N we set Rn := card { x, y, z, t Z n = x + y + z +t, z t mod }. Lemma.. Let n N be such that n 0 mod. Then card { x, y, z, t Z n = x + y + z +t, z t mod, z t mod } = Rn/. Proof. Let n N satisfy n 0 mod. Set and T n = { x, y, z, t Z n = x + y + z +t, z t mod, z t mod } Un = { x, y, z, t Z n/ =x + y + z +t, z t mod }.

556 A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams The mapping f : T n Un given by fx, y, z, t = x/,y/, z +t/, z t/ is a bijection. Thus as asserted. card T n = card Un = Rn/, Lemma.. Let n N be such that n 0 mod. Then N,,, ; n =N,,, ; n/ + 6Rn/. Proof. If x + y + z +t = n 0 mod then Thus x, y, z, t 0, 0, 0, 0, 0, 0,,, 0,, 0, or, 0, 0, mod. N,,, ; n =N,,, ; n/ + card { x, y, z, t Z n = x + y + z +t, z t mod } = N,,, ; n/ + 6 card { x, y, z, t Z n = x + y + z +t, z t mod,z t mod } = N,,, ; n/ + 6Rn/, by Lemma.. Lemma.. Let n N be such that n 0 mod. Then N,,, ; n =N,,, ; n/ + Rn/. Proof. If x + y +z +t = n 0 mod then x + y +z 0 mod so Thus Now x, y, z 0, 0, 0,, 0, or 0,, mod. N,,, ; n =N,,, ; n/ + card { x, y, z, t Z n = x + y +z +t,x mod, y 0 mod, z mod }. card { x, y, z, t Z n = x + y +z +t, x, y, z, 0, mod }

Theta function identities 557 = card { x,y,z,t Z n = x + y + z +t, x,y,z,t 0, 0,, mod } = card { x,y,z,t Z n = x + y + z +t, z,t, mod } = card { x,y,z,t Z n = x + y + z +t, z t mod,z t mod } =Rn/, by Lemma.. The asserted result now follows. Lemma.. Let n N be such that n 0 mod. Then N,,, ; n = N,,, ; n/ + N,,, ; n. Proof. This result follows by eliminating Rn/ from the formulae of Lemmas. and.. Lemma. was stated but not proved by Liouville [, p. 8]. Lemma.5. Let n N. Then N,,, ; n = 9 An Bn+Cn Dn, if n 0 mod. Proof. By [, Theorem.] we have N,,, ; n =6An Bn+Cn Dn, n N. Hence, by Proposition., we obtain for n 0 mod N,,, ; n/ = 6An/ Bn/ + Cn/ Dn/ = An Bn+Cn Dn. Then, by Lemma., we deduce N,,, ; n = An Bn+Cn Dn + 6An Bn+Cn Dn as claimed. = 9 An Bn+Cn Dn,

558 A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams Lemma.6. Let n N. Then N,,, ; n = An Bn, if n mod. Proof. If x + y +z +t = n mod then x + y +z mod so x y mod and z 0 mod. Hence n x + y x y z = + +t +6 so that by [, Theorem 5.] we have N,,, ; n =N,,, 6; n/ = An/ + Bn/ = An Bn, if n mod. Lemma.7. Let n N. Ifn 0 mod then N,,, ; n = + n/ + + n/ Cn Proof. This follows from Lemmas.5 and.6. Proof of Theorem.. We have by Lemma.7 = = N,,, ; nq n An + n/ Bn + n/ Dn. + n/ An + n/ Bn + + n/ Cn + n/ Dn q n An Bn+ Cn Dn q n + n/ An Bn+ Cn Dn q n.

Theta function identities 559 On the other hand we have by. and Proposition.5b N,,, ; nq n = = n=0 N,,, ; nq n + n= N,,, ; nq n + n=0 = ϕ qϕq ϕq + ϕ qϕ q ϕq N,,, ; n q n n= N,,, ; n q n = ϕ qϕq ϕq+ϕ q + ϕ qϕ q ϕq+ϕ q = ϕ qϕq + ϕ qϕ q = + + ϕ qϕ qϕq +ϕ qϕqϕ q n= An Bn+ Cn Dn q n + + + n= An Bn+ Cn Dn q n ϕ qϕ qϕq +ϕqϕ qϕ q = An Bn+ Cn Dn q n + ϕ qϕ qϕq +ϕqϕ qϕ q. Equating the two expressions for N,,, ; nq n, we deduce ϕ qϕ qϕq +ϕqϕ qϕ q = n/ An Bn+ Cn q Dn n,

560 A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams from which we obtain ϕ qϕ qϕq +ϕqϕ qϕ q =+ n/ An Bn+Cn Dn q n. From Theorem.a we have ϕ qϕ qϕq ϕqϕ qϕ q = Adding these two equations, and dividing by, we deduce Enq n. ϕ qϕ qϕq =+ + n/ An Bn+Cn Dn q n Enq n, as asserted. 5 The power series of ϕqϕ q ϕ q In this section we determine the power series expansion of ϕqϕ q ϕ q in powers of q. The method of proof follows that of Theorem.. Theorem 5.. ϕqϕ q ϕ q = + + n/ An+Bn Cn Dn q n F nq n. We require a number of lemmas before giving the proof of Theorem 5.. For n N we set Sn := card { x, y, z, t Z n = x +y +z +t,x y mod }.

Theta function identities 56 Lemma 5.. Let n N be such that n 0 mod. Then card { x, y, z, t Z n = x +y +z +t, x y mod, x y mod } = Sn/. Proof. Let n N satisfy n 0 mod. Set and W n = V n = { x, y, z, t Z n = x +y +z +t, x y mod, x y mod } { x, y, z, t Z n } = x +y +z +t,x ymod. The mapping g : V n W n given by x +y gx, y, z, t =, x y, z,t is a bijection. Thus as claimed. card Cn = card Dn = Sn/, Lemma 5.. Let n N be such that n 0 mod. Then N,,, ; n =N,,, ; n/ + 6Sn/. Proof. If x +y +z +t = n 0 mod then Thus x, y, z, t 0, 0, 0, 0,,, 0, 0,, 0,, 0 or, 0, 0, mod. N,,, ; n =N,,, ; n/ + card { x, y, z, t Z n = x +y +z +t,x y mod } = N,,, ; n/ + 6 card { x, y, z, t Z n = x +y +z +t, x y mod,x y mod } = N,,, ; n/ + 6Sn/, by Lemma 5.. Lemma 5.. Let n N be such that n 0 mod. Then N,,, ; n =N,,, ; n/ + Sn/.

56 A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams Proof. If x +y +z +t = n 0 mod then x +y +z 0 mod so Thus x, y, z 0, 0, 0,,, 0 or, 0, mod. N,,, ; n =N,,, ; n/ + card { x, y, z, t Z n = x +y +z +t,x mod, y mod, z 0 mod }. Now card { x, y, z, t Z n = x +y +z +t, x, y, z,, 0 mod } = card { x, y, z, t Z n = x +y +z +t, x, y, mod } = card { x, y, z, t Z n = x +y +z +t, x y mod, x y mod } =Sn/, by Lemma 5.. Lemma 5.. Let n N be such that n 0 mod. Then N,,, ; n = N,,, ; n/ + N,,, ; n. Proof. From Lemmas 5. and 5. we obtain N,,, ; n = N,,, ; n/ + Sn/ = N,,, ; n/ + N,,, ; n N,,, ; n/ 6 as asserted. = N,,, ; n/ + N,,, ; n, Although Lemma 5. is an analogue of Lemma., it was not stated by Liouville. Lemma 5.5. Let n N. Then N,,, ; n = An+ Bn Cn Dn, if n 0 mod.

Theta function identities 56 Proof. By [, Theorem 8.] we have N,,, ; n =An+Bn Cn Dn, n N. Hence for n 0 mod we have by Proposition. N,,, ; n/ = An/ + Bn/ Cn/ Dn/ = An+ Bn Cn Dn. Then, by Lemma 5., we obtain N,,, ; n = An+ Bn Cn Dn + An+Bn Cn Dn for n 0 mod. Lemma 5.6. Let n N. Then = An+ Bn Cn Dn N,,, ; n = An+ Bn, if n mod. Proof. If x +y +z +t = n mod then x +y +z mod so x, y, z 0,, mod. Hence n = x + y + z y z + +6t. Then, by [, Theorem 0.] and Proposition., we have for n mod N,,, ; n =N,,, 6; n/ = An/ Bn/ = An+ Bn, as asserted. Lemma 5.7. Let n N. Then N,,, ; n = + n/ An+ + n/ Bn + n/ Cn + n/ Dn, if n 0 mod.

56 A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams Proof. This follows from Lemmas 5.5 and 5.6. Proof of Theorem 5.. By Lemma 5.7 we have N,,, ; nq n = = + + n/ An+ + n/ Bn + n/ Cn An+Bn Cn Dn q n + n/ Dn q n An+Bn n/ Cn Dn q n. On the other hand we have by. and Proposition.5a N,,, ; nq n = N,,, ; nq n + N,,, ; n q n = n=0 n= N,,, ; nq n + n=0 = ϕqϕ q ϕq + ϕ qϕ q ϕq n= N,,, ; n q n = ϕqϕ q ϕq +ϕ q + ϕ qϕ q ϕq +ϕ q = ϕqϕ q + ϕ qϕ q + ϕqϕ q ϕ q +ϕ qϕ q ϕq = + An+Bn Cn Dn q n n= + + An+Bn Cn Dn q n n=

Theta function identities 565 + ϕqϕ q ϕ q +ϕ qϕq ϕ q = + An+Bn Cn Dn q n + ϕqϕq ϕ q +ϕ qϕq ϕ q. Equating the two expressions for N,,, ; nq n, we obtain ϕqϕ q ϕ q +ϕ qϕq ϕ q =+ n/ An+Bn Cn Dn q n. From Theorem.b we have ϕqϕ q ϕ q ϕ qϕq ϕ q = F nq n. Adding these two identities, and dividing by, we obtain ϕqϕ q ϕ q =+ n/ An+Bn Cn Dn q n as asserted. + F nq n, 6 The power series of ϕqϕ q In this section we determine the power series expansion of ϕqϕ q in powers of q. The proof is similar to that of Theorem.. Theorem 6.. ϕqϕ q = n/ An+Bn+Cn+Dn q n

566 A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams We need a lemma. + Enq n. Lemma 6.. Let n N. Then { N,,, ; n = An+ Bn Cn Dn, if n 0 mod, An+ Bn, if n mod. Proof. If x +y +z +t = n 0 mod then x +y +z 0 mod so x y z 0 mod. Thus N,,, ; n = N,,, ; n/ = An/ + Bn/ Cn/ Dn/ = An+ Bn Cn Dn by [, eq. 8.] and Proposition.. If x +y +z +t = n mod then x + y + z mod so x, y, z,, 0,, 0, or 0,, mod. Hence, for n mod, we have N,,, ; n = card { x, y, z, t Z n =x +y +z +t, x y mod, z 0 mod } = card { x, y, z, t Z n =x +y +z +t, x y mod } { = card x, y, z, t Z n x + y x y } = + +t +6z = card {x, y, z, t Z n } =x +y +z +6t = An/ Bn/ = An+ Bn by [, Theorem 0.] and Proposition.. Proof of Theorem 6.. We have by. and Propostion.5a n =0 N,,, ; nq n

Theta function identities 567 = N,,, ; nq n + n=0 N,,, ; n q n n=0 = ϕ q ϕq + ϕ q ϕq = ϕ q ϕq+ϕ q + ϕ q ϕq+ϕ q = ϕqϕ q +ϕ qϕ q + ϕqϕ q +ϕ qϕ q = ϕqϕ q +ϕ qϕ q + + An+Bn Cn Dn q n + + = + n= An+Bn Cn Dn q n n= ϕqϕ q +ϕ qϕ q + An+Bn Cn Dn q n. On the other hand we have by Lemma 6. n =0 N,,, ; nq n =+ + =+ n 0mod n mod An+ Bn Cn Dn q n An+ Bn q n An+Bn Cn Dn q n

568 A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams n/ An+Bn+Cn+Dn q n. Thus, equating the two expressions for n =0 N,,, ; nq n, we have ϕqϕ q +ϕ qϕ q = n/ An+Bn+Cn+Dn q n. By Theorem.a we have ϕqϕ q ϕ qϕ q = Enq n. Adding these two identities, and dividing by, we obtain ϕqϕ q = as asserted. + n/ An+Bn+Cn+Dn q n Enq n, 7 Sixteen quaternary forms Our first theorem of this section shows that the generating functions of the sixteen forms listed in Section, namely, 7. ϕ qϕq, ϕ qϕq ϕq, ϕ qϕq ϕq, ϕqϕ q ϕq, ϕqϕ q ϕq, ϕqϕq ϕ q, ϕqϕq ϕ q, ϕqϕ q ϕq, ϕqϕ q 6 ϕq, ϕqϕ q,

Theta function identities 569 ϕ q ϕq ϕq, ϕ q ϕq, ϕ q ϕq ϕq, ϕq ϕ q, ϕq ϕq ϕ q 6, ϕq ϕq ϕ q, can all be expressed as linear combinations of the eight products 7. ϕqϕ q, ϕ qϕq, ϕ qϕ qϕ q, ϕ qϕ q ϕ q, ϕ qϕ q, ϕ qϕ qϕq, ϕqϕ q, ϕqϕ q ϕ q, and the eight products formed from them by replacing q by q. Theorem 7.. a ϕ qϕq = ϕ qϕq + ϕ qϕ q. b ϕ qϕq ϕq = ϕ qϕq + ϕ qϕ qϕq. c ϕ qϕq ϕq = ϕ qϕq + ϕ qϕ qϕq + ϕ qϕ q + ϕ qϕ qϕ q. d ϕqϕ q ϕq = ϕ qϕq + ϕ qϕ q + ϕqϕ qϕq + ϕqϕ qϕ q. e ϕqϕ q ϕq = ϕqϕ q + ϕqϕ q ϕ q. f ϕqϕq ϕ q = ϕ qϕq + ϕqϕ qϕq + ϕ qϕ qϕq. g ϕqϕq ϕ q = ϕqϕ q + ϕqϕq ϕ q + ϕqϕ q ϕ q.

570 A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams h ϕqϕ q ϕq = 8 ϕ qϕq + 8 ϕ qϕ q + 8 ϕqϕ qϕq + 8 ϕqϕ qϕ q + ϕ qϕ qϕq + ϕ qϕ qϕ q. i ϕqϕ q 6 ϕq = ϕqϕ q + ϕqϕ q ϕ q + ϕqϕ q ϕq + ϕqϕ q. j ϕqϕ q = 8 ϕqϕ q + 8 ϕqϕ q ϕ q + 8 ϕqϕq ϕ q + 8 ϕqϕ q. k ϕ q ϕq ϕq = ϕ qϕq + ϕ qϕ qϕq + ϕqϕ qϕq + ϕq ϕ q. l ϕ q ϕq = ϕqϕ q + ϕ qϕ q. m ϕ q ϕq ϕq = ϕqϕ q + ϕqϕ q ϕ q + ϕ qϕ q + ϕ qϕ q ϕ q. n ϕq ϕ q = 8 ϕ qϕq + 8 ϕ qϕq + 8 ϕqϕ qϕq + 8 ϕ qϕ qϕq. o ϕq ϕq ϕ q 6 = ϕqϕ q + ϕ qϕ q + ϕqϕq ϕ q + ϕ qϕq ϕ q. p ϕq ϕq ϕ q = 8 ϕqϕ q + 8 ϕ qϕ q + 8 ϕqϕq ϕ q + 8 ϕ qϕq ϕ q + ϕqϕ q ϕ q + ϕ qϕ q ϕ q. Proof. We just give the proof of formula n as the rest can be proved similarly. We have by. ϕq ϕ q = ϕq ϕq+ ϕ q

Theta function identities 57 = 8 ϕ qϕq + 8 ϕ qϕ qϕq + 8 ϕqϕ qϕq + 8 ϕ qϕq as claimed. The power series expansions of the products listed in 7. are given in Proposition.5 and Theorems.,., 5. and 6.. Using these in Theorem 7. we obtain our main result. Theorem 7.. Let n N. Then a N,,, ; n = b N,,, ; n = c N,,, ; n = d N,,, ; n = e N,,, ; n = An Bn+ Cn Dn+F n, if n mod, 9 An Bn, if n mod, An Bn+Cn Dn, if n 0 mod. An Bn+ Cn Dn+En, if n mod, An Bn, if n mod, 9 An Bn+Cn Dn, if n 0 mod. An Bn+ Cn Dn+En+ F n, if n mod, An Bn, if n mod, An Bn+Cn Dn, if n 0 mod. An Bn En+ F n, if n mod, An Bn, if n mod, An Bn+Cn Dn, if n 0 mod. An+Bn Cn Dn+F n, if n mod, An+ Bn, if n mod, An+ Bn Cn Dn, if n 0 mod.

57 A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams f N,,, ; n = g N,,, ; n = An Bn+En, if n mod, 0, if n mod, An Bn+Cn Dn, if n 0 mod. An+ Bn+F n, if n mod, 0, if n mod, An+Bn Cn Dn, if n 0 mod. An Bn+Cn Dn+En+F n, if n mod, h N,,, ; n = 0, if n mod, An Bn+Cn Dn, if n 0 mod. An+Bn+En+F n, if n mod, i N, 6, 6, ; n = An+ Bn, if n mod, An+ Bn Cn Dn, if n 0 mod. An+Bn+Cn +Dn+En+F n, if n mod, j N,,, ; n = 0, if n mod, An+ Bn Cn Dn, if n 0 mod. k N,,, ; n = l N,,, ; n = An Bn+ En F n, if n mod, An Bn, if n mod, An Bn+Cn Dn, if n 0 mod. An+Bn Cn Dn En, if n mod, An+ Bn, if n mod, An+ Bn Cn Dn, if n 0 mod.

Theta function identities 57 m N,,, ; n = An +Bn Cn Dn En+F n, if n mod, An+ Bn, if n mod, An+ Bn Cn Dn, if n 0 mod. An Bn Cn +Dn+En F n, if n mod, n N,,, ; n = 0, if n mod, An Bn+Cn Dn, if n 0 mod. o N,, 6, 6; n = An+Bn En F n, if n mod, An+ Bn, if n mod, An+ Bn Cn Dn, if n 0 mod. An +Bn Cn Dn En+F n, if n mod, p N,,, ; n = 0, if n mod, An+ Bn Cn Dn, if n 0 mod. Appealing to Theorem 7. and Proposition., and using the identity An+sBn+tCn+stDn N = α + t α+β+n / β + s α+β we obtain the following corollary, see Liouville [6]-[9]. AN, Corollary 7.. Let n N. Set n = α β N, where α N 0, β N 0, N N and gcdn, 6 =. Then a N,,, ; n + β+ N N β+ β AN+F n, if α =0, N = β+ + β AN, if α =, α N α+β+ N + β+ α+β AN, if α.

57 A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams b N,,, ; n N β+ N + β+ β AN+En, if α =0, N = β+ + β AN, if α =, α N α+β+ N + β+ α+β AN, if α. c N,,, ; n + N β+ β AN F n, if α =0, N β+ + En+ = N β+ + β AN, if α =, α N α+β+ N + β+ α+β AN, if α. d N,,, ; n N β+ β AN En+ F n, if α =0, N = β+ + β AN, if α =, α N α+β+ N + β+ α+β AN, if α. e N,,, ; n N β+ N β + β AN+F n, if α =0, N = β β AN, if α =, α N α+β+ N β + α+β AN, if α. f N,,, ; n = N β+ β AN+En, if α =0, 0, if α =, α + N α+β+ β+ α+β N AN, if α.

Theta function identities 575 g N,,, ; n = N β + β AN+F n, if α =0, 0, if α =, α h N,,, ; n + N α+β+ N β+ β + α+β N AN, if α. N β+ β AN F n, if α =0, + En+ = 0, if α =, α N α+β+ N + β+ α+β AN, if α. i N, 6, 6, ; n N β + β AN+ En+ F n, if α =0, N = β β AN, if α =, α N α+β+ N β + α+β AN, if α. j N,,, ; n + N β + β AN F n, if α =0, N β+ + En+ = 0, if α =, α N α+β+ AN, if α. β + α+β N k N,,, ; n N β+ β AN+ En F n, if α =0, N = β+ + β AN, if α =, α N α+β+ N + β+ α+β AN, if α.

576 A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams l N,,, ; n N β+ N β + β AN En, if α =0, N = β β AN, if α =, α N α+β+ N β + α+β AN, if α. m N,,, ; n N β + β AN F n, if α =0, N β+ En+ = N β β AN, if α =, α N α+β+ N β + α+β AN, if α. n N,,, ; n N β+ β AN F n, if α =0, N β+ + En = 0, if α =, α N α+β+ + AN, if α. β+ α+β N o N,, 6, 6; n N β + β AN En F n, if α =0, N = β β AN, if α =, α N α+β+ N β + α+β AN, if α. p N,,, ; n N β + β AN F n, if α =0, N β+ En+ = 0, if α =, α N α+β+ AN, if α. β + α+β N

Theta function identities 577 8 Conclusion We conclude by noting that the quantities Rn and Sn used in Sections and 5 respectively can be determined explicitly. Replacing n by n in Lemma. we obtain From [, Theorem.] we have Rn = N,,, ; n N,,, ; n. 6 N,,, ; n =6An Bn+Cn Dn. Thus, by Proposition., we have N,,, ; n =An 8Bn+Cn Dn. Hence Rn =An Bn = α β+ α+β N AN, which is Theorem. of []. In a similar way starting from Lemma 5. and appealing to Theorem 8. of [], we can obtain an explicit formula for Sn. We leave this to the reader. The methods of this paper can be used to determine explicit formulae for Na, b, c, d; n valid for all n N for other forms ax + by + cz + dt, see for example [], []. Acknowledgements The fourth author was supported by research grant A-7 from the Natural Sciences and Engineering Research Council of Canada. References [] A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams, Theta function identities and representations by certain quaternary quadratic forms, Int. J. Number Theory, to appear. [] A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams, Jacobi s identity and representation of integers by certain quaternary quadratic forms, Int. J. Modern Math. 007, -76.

578 A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams [] A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams, Nineteen quaternary quadratic forms, Acta Arith., to appear. [] A. Alaca, Ş. Alaca, M. F. Lemire and K. S. Williams, The number of representations of a positive integer by certain quaternary quadratic forms, Int. J. Number Theory, to appear. [5] A. Alaca, Ş. Alaca and K. S. Williams, Evaluation of the convolution sums σlσm and σlσm, Advances in Theoretical and l+m=n l+m=n Applied Mathematics 006, 7-8. [6] J. Liouville, Sur la forme x + y + z +t, J. Math. Pures Appl. 8 86, 6-68. [7] J. Liouville, Sur la forme x +y +z +t, J. Math. Pures Appl. 8 86, 69-7. [8] J. Liouville, Sur la forme x + y +z +t, J. Math. Pures Appl. 8 86, 7-76. [9] J. Liouville, Sur la forme x +y +z +t, J. Math. Pures Appl. 8 86, 77-78. [0] J. Liouville, Sur la forme x +y +z +t, J. Math. Pures Appl. 8 86, 79-8. [] J. Liouville, Sur la forme x + y +z +t, J. Math. Pures Appl. 8 86, 8-8. [] J. Liouville, Sur la forme x +y +z +t, J. Math. Pures Appl. 8 86, 85-88. [] J. Liouville, Sur la forme x +y +z +t, J. Math. Pures Appl. 8 86, 89-9. [] J. Liouville, Sur la forme x +y +z +t, J. Math. Pures Appl. 8 86, 9-8. [5] J. Liouville, Sur la forme x +y +z +t, J. Pures Appl. Math. 8 86, 9-0. [6] J. Liouville, Sur la forme x +y +z +t, J. Pures Appl. Math. 8 86, -. [7] J. Liouville, Sur la forme x +y +z +t, J. Pures Appl. Math. 8 86, -8.

Theta function identities 579 [8] J. Liouville, Sur la forme x +y +z +t, J. Pures Appl. Math. 8 86, 9-5. [9] J. Liouville, Sur la forme x +y +z +t, J. Pures Appl. Math. 8 86, 5-5. [0] K. Petr, O počtu třid forem kvadratických záporného diskriminantu, Rozpravy Ceské Akademie Císare Frantiska Josefa I 0 90, -. Received: July 0, 007