Product Identities for Theta Functions
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- Τρύφαινα Ελευθερίου
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1 Product Identities for Theta Functions Zhu Cao April 7, 008
2 Defining Products of Theta Functions n 1 (a; q) 0 = 1, (a; q) n = (1 aq k ), n 1, k=0 (a; q) = lim (a; q) n, q < 1. n Jacobi s triple product identity z n q n = ( qz; q ) ( q/z; q ) (q ; q ), q < 1. n= Ramanujan s general theta function For ab < 1, f(a, b) = n= a n(n+1)/ b n(n 1)/ = ( a; ab) ( b; ab) (ab; ab). 1
3 Ramanujan For k Z +, f(a, b) = Define ϕ(q) : = f(q; q) = n= ψ(q) : = f(q, q 3 ) = 1 f(1, q) = f( q) : = f( q, q ) = (q; q). n= a n(n+1)/ b n(n 1)/ = q n = ( q; q ) (q ; q ), k 1 r=0 n=0 n= q n(n+1)/ = (q ; q ) (q; q ), a (kn+r)(kn+r+1)/ b (kn+r)(kn+r 1)/. U k = a k(k+1)/ b k(k 1)/, V k = a k(k 1)/ b k(k+1)/. f(a, b) = f(u 1, V 1 ) = k 1 r=0 U r f( U k+r, V k r ). U r U r
4 Products of Two or more theta functions Problem: Conditions under which the product of two or more theta functions can be written as linear combinations of other products of theta functions. Motivation M.D.Hirschhorn s generalization of Winquist s identity, 1987 Ramanujan, 1919 p(11n + 6) 0 (mod 11), where p(n) is the number of partitions of the positive integer n. Theorem 1 (Winquist, 1969). For any nonzero complex numbers a, b and for q < 1, ( 1) m+n q 3m +3n +3m+n m= n= (a 3m b 3n a 3m b 3n+1 a 3n+1 b 3m 1 + a 3n+ b 3m 1 ) =(q; q) (a; q) (a 1 q; q) (b; q) (b 1 q; q) (ab; q) (a 1 b 1 q; q) (ab 1 ; q) (a 1 bq; q). 3
5 Main idea Let z i 0, l i Z +, h i Z, (i = 1,,..., n). n S : = ( z i q l i h i ; q l i ) ( z 1 i q h i ; q l i ) (q l i ; q l i ) = i=1 x 1 = x = x n = z 1 x 1 z x z n x n q n j=1 ( 1 l j h j )x j + 1 l jx j. y = Ax, A is an integral matrix, det A 0. x 1 y 1 x = x., y = y.. x = A 1 y = 1 det A A y, k = det A, B = sgn(det A)A. x n x = 1 k By. y n 4
6 By 0 (mod k). If y c i (mod k) (i = 1,,..., m) is the solution set of the above system of congruences, we substitute y with ky + c i in x = 1 k By. Then we have x = By + 1 k Bc i (i = 1,,..., m). A system of congruences a i (mod n i ) with 1 i k is called a covering system if every integer y satisfies y a i (mod n i ) for at least one value of i. A covering system in which each integer is covered by just one congruence is called an exact covering system. We can take {By + 1 k Bc 1,, By + 1 k Bc m} as an exact covering system of Z n. We need the coefficients of all y i y j = 0 (i j, i, j = 1,,..., n) in order to separate y 1, y,..., y n. Generalized Orthogonal Relation l 1 b 11 b 1 + l b 1 b + + l n b n1 b n = 0, l 1 b 11 b l b 1 b l n b n1 b n3 = 0,. l 1 b 1(n 1) b 1n + l b (n 1) b n + + l n b n(n 1) b nn = 0. 5
7 Procedure for obtaining series-product identities 1. For fixed positive integers l 1, l,..., l n, find n n matrices B satisfying The generalized orthogonal relation, det B = ±k n 1, kb 1 is an integral matrix, where k N.. Solve system of congruences By 0 (mod k). Suppose we have m solutions to it. By computing the contribution of each solution, we can write each product of n theta functions as a linear combination of m products of n theta functions. 6
8 Products of Two Theta Functions The generalized orthogonal relation is l 1 b 11 b 1 + l b 1 b = 0. Without lose of generality, we can assume three entries of matrix B are positive, one is negative. We also assume det B = k. Choose ( ) k1 k B =, 1 1 Corollary 1. If ab < 1, (cd) = (ab) k 1k, where both k 1 and k are positive integers, then f(a, b)f(c, d) = k 1 +k 1 i=0 a i +i b i i f(a k 1 +k 1 +k 1 i b k 1 k 1 +k 1 i d, a k 1 k 1 k 1 i b k 1 +k 1 k 1 i c) f(a k +k +k i b k k +k i c, a k k k i b k +k k i d). 7
9 Proof. det B = k = k 1 + k. B = y 1 y 0 ( k1 k 1 1 ), (mod k). k cases: y 1 y 0 (mod k),..., y 1 y k 1 (mod k). For y 1 y i (mod k), replace y 1 by ky 1 + i and y by ky + i in x = 1 k By. We find that ( ) ( ) ( ) ( ) x1 k1 k = y1 i +. x 1 1 y 0 Summing up the k parts in the sum S, we obtain Corollary 1. 8
10 Special case For ab = cd, f(a, b)f(c, d) = f(ad, bc)f(ac, bd) + af( c a, a c abcd)f(d a, a d abcd). Gasper, three term relation for sigma functions. Corollary. b(ad, q/ad, bc, q/bc, d/a, qa/d, c/b, qb/c; q) +c(ab, q/ab, cd, q/cd, b/a, qa/b, d/c, qc/d; q) +d(ac, q/ac, bd, q/bd, c/a, qa/c, b/d, qd/b; q) = 0 Hirschhorn s generalization of the quintuple product identity For cd = (ab), f(a, b)f(c, d) =f(ac, bd)f(a 3 bd, ab 3 c) + af( d a, a d abcd)f(bc a, a bc a b c d ) + bf( c b, b c abcd)f(ad b, b ad a b c d ). 9
11 Göllnitz-Gordon functions ( q; q ) n S(q) : = q n = (q ; q ) n T (q) : = n=0 n=0 f(1, q)f(1, q 7 ) = ( q; q ) n (q ; q ) n q n +n = 7 i=0 1 (q; q 8 ) (q 4 ; q 8 ) (q 7 ; q 8 ), 1 (q 3 ; q 8 ) (q 4 ; q 8 ) (q 5 ; q 8 ). q i i f(q 7+i, q 1 i )f(q 1+7i, q 35 7i ), ψ(q)ψ(q 7 ) =f(q, q 7 )f(q 1, q 35 ) + q 3 f(q 3, q 5 )f(q 49, q 7 ) + qψ(q )ψ(q 14 ) + ψ(q 8 )ϕ(q 8 ) + q 6 ϕ(q 4 )ψ(q 56 ). f( 1, q)f( 1, q 7 ) = m i=0 ( 1) i q i i f(q m+i, q 1 i )f(q m m +mi, q m +3m mi ) = 0, f(q, q 7 )f(q 1, q 35 ) + q 3 f(q 3, q 5 )f(q 49, q 7 ) = ψ(q 8 )ϕ(q 8 ) + q 6 ϕ(q 4 )ψ(q 56 ) + qψ(q )ψ(q 14 ). 10
12 ψ(q)ψ(q 7 ) = ψ(q 8 )ϕ(q 8 ) + q 6 ϕ(q 4 )ψ(q 56 ) + qψ(q )ψ(q 14 ), ψ(q)ψ(q 7 ) = f(q, q 7 )f(q 1, q 35 ) + q 3 f(q 3, q 5 )f(q 49, q 7 ), ψ(q )ϕ( q 3 ) = f( q 3, q 5 )f( q 9, q 15 ) + q f( q, q 7 )f( q 3, q 1 ), ψ(q)ϕ( q ) = f ( q 3, q 5 ) + qf ( q, q 7 ), ψ(q 6 )ϕ(q) = f(q, q 7 )f(q 9, q 15 ) + qf(q 3, q 5 )f(q 3, q 1 ). Corollary 3 (S.-S. Huang, 00). S(q 7 )T (q) q 3 S(q)T (q 7 ) = 1, S(q 3 )S(q) + q T (q 3 )T (q) = (q3 ; q 3 ) (q 4 ; q 4 ) (q; q) (q 1 ; q 1 ), S (q) + qt (q) = (q ; q ) 6, (q; q) 3 (q 4 ; q 4 ) 3 S(q 3 )T (q) qs(q)t (q 3 ) = (q; q) (q 1 ; q 1 ) (q 3 ; q 3 ) (q 4 ; q 4 ). 11
13 Definition 1. Let P (q) denote any power series in q. Then the t-dissection of P is given by m-dissection of ϕ(q)ϕ(q m ) ϕ(q)ϕ(q m ) = m-dissection of ψ(q)ϕ(q m ) ψ(q)ϕ(q m ) = P (q) =: m 1 i=0 4-dissection of ψ ( q )f ( q) m 1 i=0 t 1 k=0 q k P k (q t ). q i f (q m +mi, q m mi ). q i i f (q 4m m+4mi, q 4m +m 4mi ). ψ ( q )f ( q) = ϕ ( q 8 )f( q 8 ) qψ(q 8 )f ( q 4 ). 1
14 B = ( 1 1 ) Corollary 4. For ab = cd, f(a, b)f(c, d) =f(a bc, ab d )f(ab 3 c, a 3 bd) + af(a 3 b c, bd )f(bc/a, a 5 b 3 d) + cf(a 4 b 3 c, d /a)f(b c 3 d, a d) + acf(a 5 b 4 c, b/c )f(b c, a 4 b d) + adf(ac, a b 3 d )f(c/a, a 6 b 4 d). Special case The septuple product identity (z; q ) (z 1 q ; q ) (z ; q ) (z q ; q ) (q ; q ) =(q 4 ; q 10 ) (q 6 ; q 10 ) (q 10 ; q 10 ) {z3 (z 5 q 8 ; q 10 ) (z 5 q ; q 10 ) + (z 5 q ; q 10 ) (z 5 q 8 ; q 10 ) } (q ; q 10 ) (q 8 ; q 10 ) (q 10 ; q 10 ) {z(z 5 q 4 ; q 10 ) (z 5 q 6 ; q 10 ) + z (z 5 q 6 ; q 10 ) (z 5 q 4 ; q 10 ) }. 13
15 Products of Three or More Theta Functions B = The set of the 3 columns of B is an orthogonal set. det B = 196 = B 1 is an integer matrix. k = 36. y 1 + y + y 3 0 (mod 6), y 1 y 0 (mod 6), y 1 + y y 3 0 (mod 6). 14
16 S :=(z 1 ; q) (z 1 1 q; q) (z ; q) (z 1 q; q) (z 3 ; q) (z 1 3 q; q) (q; q) 3 =( z 1 z z 3 q; q 6 ) ( z 1 1 z z 1 3 q 5 ; q 6 ) (q 6, q 6 ) (z 1 z 1 z 3 q; q 3 ) (z 1 1 z z 1 3 q ; q 3 ) (q 3, q 3 ) ( z 1 z 1 3 q; q ) ( z 1 1 z 3 q; q ) (q, q ) + B = = ( 1) + 1 = ( 1) = z 1 z z 3 z 1 z 1 z 3 1 z 1 z 3. 15
17 Corollary 5. For a, b 0, and q < 1, (a; q) (a 1 q; q) (b; q) (b 1 q; q) (ab; q) (a 1 b 1 q; q) (q; q) =( ab 1 q; q ) ( a 1 bq; q ) (q, q ) {( a 3 b 3 q; q 6 ) ( a 3 b 3 q 5 ; q 6 ) (q 6, q 6 ) a b ( a 3 b 3 q 5 ; q 6 ) ( a 3 b 3 q; q 6 ) (q 6, q 6 ) } + ( ab 1 q ; q ) ( a 1 b; q ) (q, q ) {a b( a 3 b 3 q 4 ; q 6 ) ( a 3 b 3 q ; q 6 ) (q 6, q 6 ) a( a 3 b 3 q ; q 6 ) ( a 3 b 3 q 4 ; q 6 ) (q 6, q 6 ) }. Corollary 6. 16(q; q) 8 = m,n= {(n 6m + 1)(n + 6m ) q m (n 6m 3)(n + 6m + ) q m + (n 6m 1)(n + 6m + ) q n+m (n 6m + 1)(n + 6m) q n m }q n +3m. 16
18 l 1 = l = 1, l 3 = B = Corollary 7. For a 1 b 1 = a b = q, a 3 b 3 = q, q < 1, f(a 1, b 1 )f(a, b )f(a 3, b 3 ) =f(a 1 a a 3, b 1 b b 3 )f(a 1 b, b 1 a )f(a 1 a b 3, b 1 b a 3 ) + a 1 f(a 1 a a 3 q, b 1 b b 3 /q)f(a 1 b q, b 1 a /q)f(a 1 a b 3 q, b 1 b a 3 /q) + a 1 a f(a 1 a a 3 q, b 1 b b 3 /q )f(a 1 b, b 1 a )f(a 1 a b 3 q, b 1 b a 3 /q ) + a 1 a 3 f(a 1 a a 3 q 3, b 1 b b 3 /q 3 )f(a 1 b q, b 1 a /q)f(a 1 a b 3 /q, b 1 b a 3 q)., 17
19 l 1 = l = l 3 = l 4 = 1 B = For q = a 1 b 1 = a b = a 3 b 3 = a 4 b 4, q < 1,. f(a 1, b 1 )f(a, b )f(a 3, b 3 )f(a 4, b 4 ) is a linear combination of eight products of four theta functions. Corollary 7 is a special case of the identity generated by this matrix. 18
20 Generalized Schröter Formula T (x, q) = n= xn q n, where x 0, q < 1. Theorem (Schröter s Formula). For positive integers a, b, T (x, q a )T (y, q b ) = a+b 1 n=0 y n q bn T (xyq bn ; q a+b )T (x b y a q abn, q ab +a b ). Theorem 3. For positive integers a, b, k i (i = 1, ), k k 1 b and k a, T (x, q a )T (y, q b ) = a/k +k 1 b/k 1 n=0 y n q bn T (xy k 1 q k 1bn ; q a+bk 1 ) T (x k 1b/k y a/k q abn/k, q ab k 1 /k +a b/k ). 19
21 Special case: W. Chu and Q. Yan, 007 Corollary 8. Let α, β, γ be three natural integers with gcd(α, γ) = 1 and λ = 1 + αβ γ. For two indeterminate x and y with x 0 and y 0, there holds the algebraic identity αβ γ x; q α x βγ y; q γ = ( 1) l q (l )α x l ( 1) αβ x λ y αβ q (αβ )γ+lα ; q λα l=0 where x; q = (q; q) (x; q) (q/x; q). ( 1) βγ yq (βγ+1 )γ lαβγ ; q λγ, 0
22 Corollary 9. For positive integer a, b,and k, with k a, T (x, q a )T (y, q b ) = a k +bk 1 n=0 y n q bn T (xy k q bkn, q a+bk )T (x b y a k q ab k n, q ab + a b k ). Corollary 10 (Blecksmith-Brillhart-Gerst Theorem, 1988). Define f 0 (a, b) = f(a, b) and f 1 (a, b) = f( a, b). Let a, b, c and d denote complex numbers such that ab, cd < 1, and suppose that there exist positive integer α, β, and m such that Let ε 1, ε {0, 1}. Then f ε1 (a, b)f ε (c, d) = r R (ab) β = (cd) α(m αβ) ( 1) ε r c r(r+1)/ d r(r 1)/ f δ1 ( a(cd) α(α+1 n)/ c α ), b(cd)α(α+1+n)/ d α ( (a/b) β/ (cd) (m αβ)(m+1+n)/ f δ, (b/a)β/ (cd) (m αβ)(m+1 n)/ d m αβ c m αβ ), 1
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