aa r X i v : . [ m a t h . N T ] J un SOME NEW POSITIVE OBSERVATIONS
ALEXANDER BERKOVICH
Dedicated to the memory of Dora Bitman
Abstract.
We revisit Bressoud’s generalized Borwein conjecture. Making use of the new positivity-preserving transformations for q -binomial coefficients we establish the truth of infinitely many cases ofthe Bressoud conjecture. In addition, we prove new bounded versions of Lebesgue’s identity and ofEuler’s Pentagonal Number Theorem. Finally, we discuss new companions to Andrews-Gordon mod mod
20 identities. Introduction
Bressoud [10] considered the following polynomials(1.1) G ( N, M, α, β, K, q ) = ∞ X j = −∞ ( − j q Kj ( α + β ) j +( α − β )2 (cid:20) N + MN − Kj (cid:21) q , where(1.2) (cid:20) m + nm (cid:21) q := ( ( q ) m + n ( q ) m ( q ) n , for m, n ∈ N , , otherwise,and ( q ) m = m Y j =1 (1 − q j ) , for m ∈ N , where N denotes the set of nonnegative integers. More generally, for m ∈ N we define( a ) m = ( a ; q ) m = m − Y j =0 (1 − aq j ) , ( a , a , . . . , a k ; q ) m = ( a ; q ) m ( a ; q ) m . . . ( a k ; q ) m . (1.3)Here and throughout we assume that | q | <
1. We note that ( a ) = 1.In 1996, Bressoud [10] conjectured that Conjecture 1.1.
Let K ∈ Z > , N, M, αK, βK ∈ Z ≥ such that ≤ α + β ≤ K − ,β − K ≤ N − M ≤ K − α, (1.4) (strict inequality when K = 2 ). Then G ( N, M, α, β, K, q ) ≥ . Date : June 23, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Bressoud’s conjectures, Positivity-preserving transformations, Rogers-Ramanujan type identi-ties, Terminating q -Series.Research is partly supported by the Simons Foundation, Award ID: 308929. Here, and everywhere, P ( q ) ≥ P ( q ) is a polynomial in q with nonnegative coefficients.We remark that (cid:20) m + nm (cid:21) q ≥ . The famous conjecture of Peter Borwein (Theorem since 2019 [14]) can be stated as A n ( q ) = G ( n, n, , , , q ) ≥ ,B n ( q ) = G ( n − , n + 1 , , , , q ) ≥ ,C n ( q ) = G ( n − , n + 1 , , , , q ) ≥ , (1.5)and(1.6) n Y k =1 (1 − q k − )(1 − q k − ) = A n ( q ) − qB n ( q ) − q C n ( q ) . When α, β ∈ Z , G ( N, M, α, β, K, q ) is a generating function for the so-called partitions with prescribedhook differences [4]. Bressoud’s conjecture is nontrivial when α, β assume fractional values.Many cases of Bressoud’s conjecture were settled in the literature [9], [12], [15], [16], [7], [14].In the next section, we will show how to settle new infinite family of cases.
Theorem 1.2.
For L ∈ N , ν ∈ Z > , s = 0 , , , . . . , ν − G ( L, L + 1 + 2 s, ( ν + 1)(1 + 1 + 2 s ν + 1 ) , ( ν + 1)(1 − s ν + 1 ) , ν + 1 , q ) ≥ . Also, in Section 2, we discuss new bounded versions of Lebesgue’s identity and of Euler’s PentagonalNumber Theorem. In Section 3, we establish and prove some additional isolated positivity results andintroduce new companions to Andrews-Gordon mod
21 and Bressoud mod
20 identities.We conclude this section with a list of seven useful formulas, which can be found in [2]:lim L →∞ (cid:20) Lm (cid:21) q = 1( q ) m , (1.8) lim L,M →∞ (cid:20) L + ML (cid:21) q = 1( q ) ∞ , (1.9)(1.10) (cid:20) n + mn (cid:21) q − = q − nm (cid:20) n + mn (cid:21) q , (1.11) (cid:20) nm (cid:21) q = (cid:20) n − m − (cid:21) q + q m (cid:20) n − m (cid:21) q = (cid:20) n − m (cid:21) q + q n − m (cid:20) n − m − (cid:21) q , (1.12) X n ≥ q ( n ) z n (cid:20) Ln (cid:21) q = ( − z ; q ) L , (1.13) ∞ X j = −∞ ( − j z j q j = (cid:16) q , qz , zq ; q (cid:17) ∞ , (1.14) ∞ X j = −∞ ( − j q j z j (cid:20) L + ML − j (cid:21) q = (cid:16) qz ; q (cid:17) M (cid:0) zq ; q (cid:1) L , with L, M, m, n ∈ N . OME NEW POSITIVE OBSERVATIONS Positivity-preserving Transformations
We start with the following summation formula
Theorem 2.1.
For L ∈ N , a ∈ Z (2.1) X k ≥ C L,k ( q ) (cid:20) k ⌊ k − a ⌋ (cid:21) q = q T ( a ) (cid:20) L + 1 L − a (cid:21) q , where T ( j ) := (cid:18) j + 12 (cid:19) and(2.2) C L,k ( q ) = L X m =0 q T ( m )+ T ( m + k ) (cid:20) Lm, k (cid:21) q , with(2.3) (cid:20) Lm, k (cid:21) q = (cid:20) Lm (cid:21) q (cid:20) L − mk (cid:21) q = (cid:20) Lk (cid:21) q (cid:20) L − km (cid:21) q ≥ . Observe that C L,k ( q ) ≥
0. Using transformation (2.1) it is easy to check that identity of the form(2.4) F ( L, q ) = ∞ X j = −∞ α ( j, q ) (cid:20) L ⌊ L − j ⌋ (cid:21) q , implies that the following identity holds(2.5) X k ≥ C L,k ( q ) F ( k, q ) = ∞ X j = −∞ α ( j, q ) X k ≥ C L,k ( q ) (cid:20) k ⌊ k − j ⌋ (cid:21) q = ∞ X j = −∞ α ( j, q ) q T ( j ) (cid:20) L + 1 L − j (cid:21) q . Hence, if F ( L, q ) ≥ ∞ X j = −∞ α ( j, q ) q T ( j ) (cid:20) L + 1 L − j (cid:21) q ≥ . For that reason, we say that (2.1) is positivity-preserving.Transformation (2.1) is an easy corollary of the theorem proven in [6].
Theorem 2.2 (Berkovich–Uncu) . (2.7) X k ≥ q T ( k ) (cid:20) Lk (cid:21) q (cid:26) T − (cid:18) ka ; q (cid:19) + T − (cid:18) ka + 1 ; q (cid:19)(cid:27) = q T ( a ) (cid:20) L + 1 L − a (cid:21) q . The Andrews-Baxter q -trinomial coefficients [3] can be defined as(2.8) T − (cid:18) ka ; q (cid:19) = X m ≥ ,m ≡ k + a ( mod q T ( m ) (cid:20) km (cid:21) q (cid:20) k − m k − m − a (cid:21) q . It is easy to check that(2.9) T − (cid:18) ka ; q (cid:19) + T − (cid:18) ka + 1 ; q (cid:19) = X m ≥ q T ( m ) (cid:20) km (cid:21) q (cid:20) k − m ⌊ k − m − a ⌋ (cid:21) q . Substituting (2.9) into left hand side of (2.7) and changing k → k + m we complete the proof of (2.1).It is instructive to compare (2.1) with the Corollary (2.6) in [16]. ALEXANDER BERKOVICH
Theorem 2.3 (Warnaar) . For L ∈ N , a ∈ Z (2.10) X k ≥ W L,k ( q ) (cid:20) kk − a (cid:21) q = q a (cid:20) LL − a (cid:21) q , where (2.11) W L,k ( q ) = L X m =0 q ( m + k ) + k (cid:20) Lm, k (cid:21) q ≥ . Observe that unlike (2.10), transformation (2.1) can not be iterated. Interestingly enough, there existsan odd companion to Theorem 2.3.
Theorem 2.4.
For L ∈ N , a ∈ Z (2.12) X k ≥ O L,k ( q ) (cid:20) k + 1 k − a (cid:21) q = q T ( a ) (cid:20) LL − a − (cid:21) q , where (2.13) O L,k ( q ) = L X m =0 q T ( m + k )+2 T ( k ) (cid:20) Lm, k + 1 (cid:21) q ≥ . We remark that while Theorem 2.4 is not explicitly stated in [16], it is a special case of an identity onpage 222 there.Schur’s bounded version of Euler’s Pentagonal Number Theory states(2.14) 1 = ∞ X j = −∞ ( − j q j +12 j (cid:20) L ⌊ L − j ⌋ (cid:21) q . With the aid of (2.5) we can convert (2.14) into(2.15) 0 ≤ L X k =0 C L,k ( q ) = ∞ X j = −∞ ( − j q j (3 j +1) (cid:20) L + 1 L − j (cid:21) q . Hence, G ( L, L + 1 , , , , q ) ≥ . Making use of (1.12), it is easy to check that(2.16) L X k =0 C L,k ( q ) = L X k =0 q T ( k ) (cid:20) Lk (cid:21) q ( − q ) k . And so identity (2.15) can be rewritten as(2.17) L X k =0 q T ( k ) (cid:20) Lk (cid:21) q ( − q ) k = ∞ X j = −∞ ( − j q j (3 j +1) (cid:20) L + 1 L − j (cid:21) q . Letting L → ∞ and using the Jacobi triple product identity (1.13) yields a special case of the Lebesgueidentity [13](2.18) X m ≥ q T ( m ) ( q ) m ( − q ) m = ( q ; q ) ∞ ( q ) ∞ , and so, (2.17) is a new bounded version of the Lebesgue identity. Perform q → q in (2.17) and use (1.10)together with(2.19) ( − q − ; q − ) n = ( − q ) n q − T ( n ) , n ∈ N OME NEW POSITIVE OBSERVATIONS to obtain after simplification a new polynomial version of Euler’s Pentagonal Number Theorem(2.20) L X k =0 ( − q ) L − k q ( L +1) k (cid:20) Lk (cid:21) q = ∞ X j = −∞ ( − j q j + j (cid:20) L + 1 L − j (cid:21) q . It proves that(2.21) G ( L, L + 1 , , , , q ) ≥ . We now move on to prove Theorem 1.2. We start with the finite analogue of the Andrews-Gordonidentity due to Foda-Quano [11].For L ∈ N , ν ∈ Z > , s = 0 , , . . . , ν − X n ,...,n ν ≥ q N + ... + N ν + N ν +1 − s + ... + N ν ν Y i =2 (cid:20) n i + L − P ij =2 N j − E νi,s n i (cid:21) q = ∞ X j = −∞ ( − j q (2 ν +1) j j (1+2 s )2 (cid:20) L ⌊ L − (2 ν +1) j − s ⌋ (cid:21) q . (2.22)Here, N j = n j + n j +1 + . . . + n ν , j = 2 , . . . , ν and E νi,s = max ( i + s − ν, ν = 1 of (2.22).With the aid of (2.5) we obtain0 ≤ X k,n ,...,n ν ≥ C L,k ( q ) q N + ... + N ν + N ν +1 − s + ... + N ν ν Y i =2 (cid:20) n i + k − P ij =2 N j − E νi,s n i (cid:21) q = q T ( s ) ∞ X j = −∞ ( − j q ( ν +1)(2 ν +1) j +( ν +1)(2 s +1) j (cid:20) L + 1 L − s − (2 ν + 1) j (cid:21) q . (2.23)Hence, G ( L, L + 1 + 2 s, ( ν + 1)(1 + 1 + 2 s ν + 1 ) , ( ν + 1)(1 − s ν + 1 ) , ν + 1 , q ) ≥ , for all L ∈ N , ν ∈ Z > , s = 0 , , , . . . , ν −
1. This completes the proof of Theorem 1.2.3.
Further Observations
We replace q by q in (1.14) and then set M = L, L + 1, z = q to find that for L ∈ N (3.1) ∞ X j = −∞ ( − j q T ( j ) (cid:20) LL − j (cid:21) q = δ L, and(3.2) ∞ X j = −∞ ( − j q T ( j ) (cid:20) L + 1 L − j (cid:21) q = 0 , where δ L, = 1 if L = 0 and δ L, = 0 if L >
0. The formulas (3.1) and (3.2) can be combined into(3.3) ∞ X j = −∞ ( − j q T ( j ) (cid:20) L ⌊ L − j ⌋ (cid:21) q = δ L, . Applying Theorem 2.3 to (3.1) yields(3.4) W L, ( q ) = X n ≥ q n (cid:20) Ln (cid:21) q = ∞ X j = −∞ ( − j q j +12 j (cid:20) LL − j (cid:21) q , ALEXANDER BERKOVICH which is Bressoud’s bounded version of the first Rogers-Ramanujan identity [9]. Analogously, applying(2.5) to (3.3) yields(3.5) C L, ( q ) = X n ≥ q n + n (cid:20) Ln (cid:21) q = ∞ X j = −∞ ( − j q j j (cid:20) L + 1 L − j (cid:21) q , which can be recognized as Warnaar’s bounded version of the second Rogers-Ramanujan identity [15].Next, we perform the change of summation variables below ∞ X j = −∞ ( − j q T ( j ) (cid:20) LL − j − (cid:21) q = − ∞ X j = −∞ ( − j q T ( − − j ) (cid:20) LL + 2 j + 1 (cid:21) q =(3.6) − ∞ X j = −∞ ( − j q T ( j ) (cid:20) LL − j − (cid:21) q to conclude that(3.7) q L +1 ∞ X j = −∞ ( − j q T ( j ) (cid:20) LL − j − (cid:21) q = 0 . Adding (3.4) and (3.7) and employing recursion relation (1.11) we obtain(3.8) X n ≥ q n (cid:20) Ln (cid:21) q = ∞ X j = −∞ ( − j q j j (cid:20) L + 1 L − j (cid:21) q . Observe that (3.4) and (3.8) imply that for k ∈ N (3.9) X n ≥ q n (cid:20) ⌊ k ⌋ n (cid:21) q = ∞ X j = −∞ ( − j q j j (cid:20) k ⌊ k − j ⌋ (cid:21) q . Apply Theorem 2.1 to (3.9) to obtain(3.10) X k,n ≥ C L,k q n (cid:20) ⌊ k ⌋ n (cid:21) q = ∞ X j = −∞ ( − j q j j (cid:20) L + 1 L − j (cid:21) q , which proves that G ( L, L + 1 , , , , q ) ≥ . In the limit as L → ∞ (3.10) becomes(3.11) X m,k,n ≥ q T ( m )+ T ( m + k )+ n (cid:2) ⌊ k ⌋ n (cid:3) q ( q ) m ( q ) k = ( q , q , q ; q ) n ( q ) ∞ . This is to be contrasted with Andrews-Gordon identity mod 21 [1](3.12) X n ,n ,...,n ≥ q N + N + ... + N + N + N ( q ) n ( q ) n . . . ( q ) n = ( q , q , q ; q ) ∞ ( q ) ∞ , with N i = n i + . . . + n , i = 1 , . . . ,
9. On the left of (3.11) one has 3-fold sum, while on the left sideof (3.12) one has 9-fold sum. Analogously, applying Theorem 2.3 to (3.4) and Theorem 2.4 to (3.5) and
OME NEW POSITIVE OBSERVATIONS (3.8), we prove that G ( L, L, , , , q ) ≥ ,G ( L − , L + 1 , , , , q ) ≥ ,G ( L − , L + 1 , , , , q ) ≥ , and obtain, as L → ∞ (3.13) X m,k,n ≥ q k +( m + k ) + n ( q ) m ( q ) k (cid:20) kn (cid:21) q = ( q , q , q ; q ) ∞ ( q ) ∞ , (3.14) X m,k,n ≥ q T ( k )+2 T ( m + n )+2 T ( n ) ( q ) m ( q ) k +1 (cid:20) kn (cid:21) q = ( q , q , q ; q ) ∞ ( q ) ∞ , and(3.15) X m,k,n ≥ q T ( k )+2 T ( m + n )+ n ( q ) m ( q ) k +1 (cid:20) kn (cid:21) q = ( q , q , q ; q ) ∞ ( q ) ∞ , respectively.In [7, p. 2332] the following identity was derived(3.16) ∞ X j = −∞ ( − j q j (cid:20) LL − j (cid:21) q = ( − q ; q ) L . We now follow a well-trodden path and check that(3.17) q L +1 X ( − j q j +2 j (cid:20) LL − j − (cid:21) q = 0 . Adding (3.16) and (3.17) we derive, with the aid of (1.11), that(3.18) ∞ X j = −∞ ( − j q j (cid:20) L + 1 L − j (cid:21) q = ( − q ; q ) L . Equations (3.16) and (3.18) imply that for k ∈ N (3.19) ∞ X j = −∞ ( − j q j (cid:20) k ⌊ k − j ⌋ (cid:21) q = ( − q ; q ) ⌊ k ⌋ . Applying Theorem 2.1 to (3.19) and letting L → ∞ we obtain(3.20) X m,k ≥ q T ( m )+ T ( m + k ) ( − q ; q ) ⌊ k ⌋ ( q ) m ( q ) k = ( q , q , q ; q ) ∞ ( q ) ∞ . Compare it with the Bressoud formula in [8](3.21) X n ,...,n ≥ q N + ... + N + N + N ( q ) n . . . ( q ) n ( q ; q ) n = ( q , q , q ; q ) ∞ ( q ) ∞ , where N i = n i + . . . + n , i = 1 , . . . , L → ∞ (3.22) X m,k ≥ q k +( m + k ) ( q ) m ( q ) k ( − q ; q ) k = ( q , q , q ; q ) ∞ ( q ) ∞ ALEXANDER BERKOVICH and(3.23) X m,k ≥ q T ( k )+2 T ( m + k ) ( q ) m ( q ) k +1 ( − q ; q ) k = ( q , q , q ; q ) ∞ ( q ) ∞ , respectively.For our final example, we employ Dyson’s identity [5], [7, p. 2330](3.24) ∞ X j = −∞ ( − j q T (3 j ) (cid:20) L + 1 L − j (cid:21) q = ( q ; q ) L ( q ) L . Applying Theorem 2.4 to (3.24) yields(3.25) ∞ X j = −∞ ( − j q T (3 j ) (cid:20) LL − − j (cid:21) q = X k ≥ O L,k ( q ; q ) k ( q ) k ≥ . This proves that(3.26) G ( L − , L + 1 , , , , q ) ≥ . Letting L → ∞ in (3.25) and using (1.13) we arrive at a new elegant result(3.27) X m,k ≥ q T ( k )+2 T ( m + k ) ( q ) m ( q ) k +1 ( q ; q ) k ( q ) k = ( q ; q ) ∞ ( q ) ∞ . Acknowledgement
I would like to thank George Andrews, James Mc Laughlin and Ali Uncu for their kind interest.
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