Proof of two multivariate q-binomial sums arising in Gromov-Witten theory
aa r X i v : . [ m a t h . C A ] F e b Proof of two multivariate q -binomial sums arising in Gromov–Witten theory C. Krattenthaler
Fakult¨at f¨ur Mathematik, Universit¨at Wien,Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria.WWW:
Abstract.
We prove two multivariate q -binomial identities conjectured by Bousseau, Briniand van Garrel [ Stable maps to Looijenga pairs , ar χ iv:2011.08830 ] which give generatingseries for Gromov–Witten invariants of two specific log Calabi–Yau surfaces. The key identityin all the proofs is Jackson’s q -analogue of the Pfaff–Saalsch¨utz summation formula from thetheory of basic hypergeometric series. The purpose of this note is to prove two conjectured multivariate q -binomial summationidentities from [1]. There, Bousseau, Brini and van Garrel deal with the computation ofGromov–Witten invariants of log Calabi–Yau surfaces (Looijenga pairs). For the two non-tame (but quasi-tame) surfaces dP (0 ,
4) and F (0 , N log( d ,d ) (dP (0 , q ) = [2 d ] q [ d ] q (cid:20) d d (cid:21) q (cid:20) d + d − d − (cid:21) q (1)and (cf. [1, Conjecture A.8]) N log( d ,d ) ( F (0 , q ) = [2 d + d ] q [ d ] q (cid:20) d + d − d − (cid:21) q . (2)Here, the q -integers [ α ] q are defined symmetrically according to physics convention, [ α ] q := q α/ − q − α/ , and the q -binomial coefficients (cid:2) nk (cid:3) q are defined by (cid:20) nk (cid:21) q := ( [ n ] q [ n − q ··· [ n − k +1] q [ k ] q [ k − q ··· [1] q , if 0 ≤ k ≤ n, , otherwise.In relation to the above two conjectures, we prove the following two multivariate sum-mation identities. Mathematics Subject Classification . Primary 33D15; Secondary 05A30 14J32 14N35 53D4557M27.
Key words and phrases.
Looijenga pairs, log Calabi–Yau surfaces, Gromov–Witten invariants, q -bino-mial coefficients, basic hypergeometric series, Pfaff–Saalsch¨utz summation formula.Research partially supported by the Austrian Science Foundation FWF (grant S50-N15) in the frame-work of the Special Research Program “Algorithmic and Enumerative Combinatorics”. heorem 1. For integers d and d with d > d ≥ , we have X m ≥ X k + ··· + k m = d − k n k + ··· + n m k m = d − d k ,...,k m > , k ≥ n > ··· >n m > (cid:20) d k (cid:21) q (cid:20) d − n − n ) k k (cid:21) q · · · (cid:20) d − P m − j =1 ( n j − n m ) k j k m (cid:21) q (cid:20) d k (cid:21) q = [2 d ] q [ d ] q (cid:20) d d (cid:21) q (cid:20) d + d − d (cid:21) q . (3) Theorem 2.
For positive integers d and arbitrary d , we have X m ≥ X n k + ··· + n m k m = d k ,...,k m > n > ··· >n m > (cid:20) d + 2 d k (cid:21) q · · · (cid:20) d + 2 d + 2 P ij =1 ( n i − n j ) k j k i (cid:21) q · · · (cid:20) d + 2 d + 2 P mj =1 ( n m − n j ) k j k m (cid:21) q (cid:20) d P mj =1 k j (cid:21) q = [2 d + d ] q [ d ] q (cid:20) d + d − d (cid:21) q . (4)By [1, Theorem A.6], Theorem 2 implies (1). Similarly, by [1, Theorem A.9], Theorem 3implies (2). As is the case frequently, the identities in Theorems 1 and 2 are difficult (impossible?) toprove directly since the parameters in these identities do not allow for enough flexibility, inparticular if one has an inductive approach in mind (which we do). The key in proving (3)and (4) is to generalise , or, in this case, to refine . By experimenting with the sumsin (3) and (4), I noticed that one can still get closed forms if we fix the sum of the k i ’s, i = 1 , , . . . , m . This leads us to the following key result. Proposition 3.
Let k and d be integers with ≤ k ≤ d . Furthermore, for arbi-trary d set f ( d , d , k ) = X m ≥ X k + ··· + k m = k n k + ··· + n m k m = d k ,...,k m > n > ··· >n m > (cid:20) d k (cid:21) q · (cid:20) d − n − n ) k k (cid:21) q · · · (cid:20) d − P m − j =1 ( n j − n m ) k j k m (cid:21) q . (5) Then f ( d , d , k ) = [2 d ] q [ k ] q (cid:20) d − d + k − k − (cid:21) q (cid:20) d − k − (cid:21) q . For the sake of consistency, in comparison to Theorem A.9 and Conjecture A.10 in [1], here we havereversed the indexing of the k i ’s and the n i ’s, that is, we have replaced k i by k m − i +1 and n i by n m − i +1 , i = 1 , , . . . , m . r +1 φ r (cid:20) a , . . . , a r +1 b , . . . , b r ; q, z (cid:21) = ∞ X ℓ =0 ( a ; q ) ℓ · · · ( a r +1 ; q ) ℓ ( q ; q ) ℓ ( b ; q ) ℓ · · · ( b r ; q ) ℓ z ℓ , (6)where ( a ; q ) = 1 and ( a ; q ) m = Q m − k =0 (1 − aq k ). The “bible” [2] of the theory of basichypergeometric series contains many summation and transformation formulae for suchseries. The formula that we need here is Jackson’s q -analogue of the Pfaff–Saalsch¨utzsummation (see [2, Eq. (1.7.2); Appendix (II.12)]) φ (cid:20) a, b, q − N c, abq − N /c ; q, q (cid:21) = ( c/a ; q ) N ( c/b ; q ) N ( c ; q ) N ( c/ab ; q ) N , (7)where N is a nonnegative integer. Proof of Proposition 3.
We prove the claim by induction on k .First we consider the start of the induction, k = 1. In this case, the summation on theright-hand side of (5) reduces to m = 1, k = 1, n = d , and hence f ( d , d ,
1) = (cid:20) d (cid:21) q , in agreement with our assertion.For the induction step, we rewrite the definition of f ( d , d , k ) in the form f ( d , d , k ) = X m ≥ k X k m =1 ⌊ d /k ⌋ X n m =1 X k + ··· + k m − = k − k m ¯ n k + ··· +¯ n m − k m − = d − n m k k ,...,k m − > n > ··· > ¯ n m − > (cid:20) d k (cid:21) q (cid:20) d − n − n ) k k (cid:21) q · · · (cid:20) d − P m − j =1 ( n j − n m − ) k j k m (cid:21) q (cid:20) d − d + n m k k m (cid:21) q = k X k =1 ⌊ d /k ⌋ X n =1 f ( d , d − nk , k − k ) (cid:20) d − d + 2 nk k (cid:21) q , where ¯ n i = n i − n m , i = 1 , , . . . , m −
1. We may now use the induction hypothesis, and3btain f ( d , d , k ) = k X k =1 ⌊ d /k ⌋ X n =1 [2 d ] q [ k − k ] q (cid:20) d − d + nk + k − k − k − k − (cid:21) q · (cid:20) d − nk − k − k − (cid:21) q (cid:20) d − d + 2 nk k (cid:21) q = ⌊ d /k ⌋ X n =1 k X k =0 [2 d ] q [ k − k ] q (cid:20) d − d + nk + k − k − k − k − (cid:21) q · (cid:20) d − nk − k − k − (cid:21) q (cid:20) d − d + 2 nk k (cid:21) q − ⌊ d /k ⌋ X n =1 [2 d ] q [ k ] q (cid:20) d − d + nk + k − k − (cid:21) q (cid:20) d − nk − k − (cid:21) q . (8)Now we write the sum over k in terms of the standard basic hypergeometric notation (6).Thus, this sum over k becomes[2 d ] q [ k ] q (cid:20) d − d + nk + k − k − (cid:21) q (cid:20) d − nk − k − (cid:21) q × φ (cid:20) q − d +2 d − k n , q − k , q − k q d − k − k n , q − d + d − k − k n ; q, q (cid:21) . The φ -series can be evaluated by means of the q -Pfaff–Saalsch¨utz summation (7). Aftersimplification, we arrive at the expression[2 d ] q [ k ] q (cid:20) d − d + k n − k − (cid:21) q (cid:20) d − k n + k − k − (cid:21) q . If we substitute this in (8), then we get f ( d , d , k ) = ⌊ d /k ⌋ X n =1 [2 d ] q [ k ] q (cid:20) d − d + k n − k − (cid:21) q (cid:20) d − k n + k − k − (cid:21) q − ⌊ d /k ⌋ X n =1 [2 d ] q [ k ] q (cid:20) d − d + nk + k − k − (cid:21) q (cid:20) d − nk − k − (cid:21) q . These are, up to a shift of the index n , essentially the same sums. After cancellation, onlythe term corresponding to n = 1 of the first sum survives, yielding f ( d , d , k ) = [2 d ] q [ k ] q (cid:20) d − d + k − k − (cid:21) q (cid:20) d − k − (cid:21) q . (cid:3) Now we are in the position to prove Theorems 1 and 2.
Proof of Theorem 1.
The left-hand side of (3) can be rewritten as d X k =0 f ( d , d − d , d − k ) (cid:20) d k (cid:21) q . In basic hypergeometric notation (6), this is[2 d ] q [ d ] q (cid:20) d + 2 d − d − (cid:21) q (cid:20) d − d − d − (cid:21) q φ (cid:20) q − d , q − d , q − d q d − d , q − d − d ; q, q (cid:21) . Also this φ -series can be evaluated by means of the q -Pfaff–Saalsch¨utz summation (7).After simplification, one obtains[2 d ] q [ d ] q (cid:20) d d (cid:21) q (cid:20) d + d − d (cid:21) q , as desired. (cid:3) Proof of Theorem 2.
With k = P mj =1 k j , the sum on the left-hand side of (4) canbe rewritten as X k ≥ f ( d + d , d , k ) (cid:20) d k (cid:21) q . In basic hypergeometric notation (6), this is[2 d + d ] q [ d ] q [1] q φ (cid:20) q d + d , q − d , q − d q , q ; q, q (cid:21) . Again, the φ -series can be evaluated by means of the q -Pfaff–Saalsch¨utz summation (7).As a result, we obtain [2 d + d ] q [ d ] q (cid:20) d + d − d (cid:21) q , as desired. (cid:3) We close with the remark that it is highly surprising that in all three proofs the identityfrom the theory of basic hypergeometric series that is required is the q -Pfaff–Saalsch¨utzsummation (7). Usually, one needs the q -Chu–Vandermonde summation here, a transfor-mation formula there, and maybe the q -Pfaff–Saalsch¨utz summation somewhere. However,remarkably, here it is exclusively the q -Pfaff–Saalsch¨utz summation. References
1. P. Bousseau, A. Brini and M. van Garrel,
Stable maps to Looijenga pairs: orbifold examples , preprint; ar χ iv:2011.08830 .2. G. Gasper and M. Rahman, Basic hypergeometric series , second edition, Encyclopedia of Math. AndIts Applications 96, Cambridge University Press, Cambridge, 2004., second edition, Encyclopedia of Math. AndIts Applications 96, Cambridge University Press, Cambridge, 2004.