aa r X i v : . [ m a t h . C O ] M a r A POLYNOMIAL IDENTITY IMPLYING SCHUR’S PARTITION THEOREM
ALI KEMAL UNCU
Abstract.
We propose and prove a new polynomial identity that implies Schur’s partition theorem.We give combinatorial interpretations of some of our expressions in the spirit of Kur¸sung¨oz. We alsopresent some related polynomial and q -series identities. Introduction and background
Since the Combinatory Analysis conference in honor of G. E. Andrews’ birthday, in a series of papersKur¸sung¨oz presented his technique of writing generating functions for the number the partition functionswith gap conditions on some classical partition theorems [16–18]. His approach is backed with a combi-natorial construction. This construction can be used to find finite analogs of these generating functions.Berkovich and the author [9] have found finite analogs of the Capparelli’s partition theorem related gen-erating functions presented by Kanade–Russell and Kur¸sung¨oz [15, 16]. Comparing these polynomialswith the earlier found finite analogs of Alladi–Andrews–Gordon and Berkovich and the author’s [1, 8],they listed polynomial identities that directly imply Capparelli’s partition theorems [9]. These polynomialidentities led to many q -series relations involving the q -trinomial coefficients and, with the use of trinomialversion of the Bailey lemma, proven infinite families of q -series identities in the spirit of the Andrews–Gordon Identities [10, 11]. Following the footsteps of [9] and using other combinatorial arguments, theauthor presented other polynomial and q -series identities that are related with the classical partitiontheorems: namely the Euler, the Rogers–Ramanujan, the G¨ollnitz–Gordon, and the little G¨ollnitz the-orems [22]. It should be noted that Kur¸sung¨oz also approached the G¨ollnitz–Gordon theorem [16], andthe comparison of his construction versus the author equivalent formulas are discussed in [22].In this work, we will follow the footsteps of [9, 10, 18, 22] and present a new polynomial identity thatdirectly implies Schur’s partition theorem followed up with the study of some related q -series identities.We define a partition π = ( λ , λ , . . . ) as a non-decreasing finite sequence of positive integers , whichare called parts of the partition π . We will use ν ( π ) and | π | to denote the number of parts and the sum ofall parts (size) of the partition π , respectively. The empty sequence ∅ is the only conventional partitionwith 0 parts and 0 size.We start with an equivalent formulation of the Schur’s partition theorem [21]: Theorem 1.1 (Schur, 1926) . For any non-negative integer n , the number of partitions of n into distinctparts ± modulo is equal to the number of partitions of n , where the gap between parts is at least withthe gap at least 6 if the parts are multiples of 3. This classical example of congruence–gap partition theorem is well studied and there are many proofs[2–6, 12, 13]. Out of this long list of proofs, the first and the only polynomial identity that imply Theo-rem 1.1 should be credited to Alladi–Berkovich [2].Here we prove a new polynomial identity in the spirit of the polynomial identities that yield Capparelli’spartition theorems [9]. We will show that the following new polynomial identity implies Schur’s partitiontheorem:
Date : March 5, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Schur’s Partition Theorem; Integer partitions; q -Trinomial coefficients; q -Series.Research of the author is supported by the Austrian Science Fund FWF, SFB50-07 and SFB50-09 Projects. Theorem 1.2.
For any fixed integer N , let N := N ( m, n , n ) := N − m − n − n , then we have X m,n ,n ≥ q A ( n ,n ,m ) (cid:20) N m (cid:21) q (cid:20) N + (cid:4) n (cid:5)(cid:4) n (cid:5) (cid:21) q (cid:20) N + (cid:4) n (cid:5)(cid:4) n (cid:5) (cid:21) q = N X j = − N q j (3 j − (cid:18) N ; j ; q j (cid:19) , (1.1) where A ( n , n , m ) := (2 m + n + n + 1)(2 m + n + n )2 + m ( n + n ) + ( n + n ) − n . (1.2)The rest of this paper is organized as follows. We start with the necessary definitions that appear inTheorem 1.2 and the rest of the paper in Section 2. A direct proof of Theorem 1.2 is given in Section 3.Section 4 has the combinatorial connection of Theorem 1.2 to Theorem 1.1 showing that Theorem 1.2implies Schur’s Theorem. Some q -series and combinatorial identities of this study is discussed in Section 5.2. Necessary Definitions and Some Useful Formulae
In this work, we will use the standard notations [7,14,23]. For variables a and q with | q | <
1, we definethe q -Pochhammer symbols and a useful abbreviation as:( a ) ∞ := ( a ; q ) ∞ := (1 − a )(1 − aq )(1 − aq )(1 − aq ) . . . , ( a ) n := ( a ; q ) n := ( a ) ∞ ( aq n ; q ) ∞ , ( a , a , . . . , a k ; q ) n := ( a ) n ( a ) n . . . ( a k ) n . We note two well known properties of q -Pochhammer symbols:( a ; q ) n − k = ( a ; q ) n ( q − n a ; q ) k (cid:16) − qa (cid:17) k q ( k ) − nk , (2.1)and ( q − ; q − ) n = ( − n q − ( n +12 )( q ; q ) n . (2.2)Let m, n, a, and b ∈ Z , we define the q -binomial coefficients and the two types of q -trinomial coefficientsas (cid:20) n + mm (cid:21) q := ( q ) n + m ( q ) n ( q ) m , if n ≥ m ≥ , , otherwise , (2.3) (cid:18) m ; b ; qa (cid:19) := X k ≥ q k ( k + b ) (cid:20) mk (cid:21) q (cid:20) m − kk + a (cid:21) q = X k ≥ q k ( k + b ) ( q ) m ( q ) k ( q ) k + a ( q ) m − k − a , (2.4)and T n (cid:18) m ; qa (cid:19) := q m ( m − n ) − ( a ( a − n )2 (cid:18) m ; a − n ; q − a (cid:19) , (2.5)respectively. The following properties of q -binomial coefficients are well known:lim n →∞ (cid:20) nm (cid:21) q = 1( q ) m , (2.6)and (cid:20) n + mm (cid:21) q − = q − mn (cid:20) n + mm (cid:21) q . (2.7) POLYNOMIAL IDENTITY IMPLYING SCHUR’S PARTITION THEOREM Proof of Theorem 1.2
We start by noting that the right-hand side of (1.1) is the S N − (1 , q ) := R N ( q ) function defined in [6]that Andrews originally used to prove Schur’s theorem directly. In his proof, he shows that this objectsatisfies the recurrence relation R N ( q ) = (1 + q N − + q N − ) R N − ( q ) + q N − (1 − q N − ) R N − ( q ) . (3.1)We would like to note that this recurrence can directly be found and automatically proven using SymbolicComputation tools Sigma and qMultiSum [19, 20]. Same is true for the left-hand side sum of (1.1).Using the mentioned implementations, we first prove that the left-hand side summand, to be denoted by F N ( m, n , n ), satisfies the recurrence F N ( m, n , n ) = F N − ( m, n , n ) + q N − F N − ( m, n , n −
1) + q N − F N − ( m, n − , n )+ q N − (1 + q + q ) F N − ( m − , n , n ) + q N − (1 + q + q ) F N − ( m − , n , n )(3.2) − q N − F N − ( m, n − , n −
1) + q N − F N − ( m − , n , n ) , then, by summing (3.2) over m, n , and n from 0 to ∞ , we see that the left-hand side sum L N ( q )satisfies the recurrence L N ( q ) = L N − ( q ) + q N − (1 + q + q + q N − + q N − ) L N − ( q )+ q N − (1 + q + q ) L N − ( q ) + q N − (1 − q N − ) L N − ( q ) . (3.3)Using the recurrence (3.1) in an iterative fashion on its own terms( q N − + q N − ) R N − ( q ) and q N − R N − ( q )is enough to show that R N ( q ) also satisfies (3.3). Now that we established that both sides of (1.1) satisfythe same recurrence (3.3), the last task is to check confirm that the first four initial conditions of bothsides are the same. For that we give the list: L ( q ) = R ( q ) = 1 , L ( q ) = R ( q ) = 1 + q + q , L ( q ) = R ( q ) = 1 + q + q + q + q + 2 q + q + q , L ( q ) = R ( q )= 1 + q + q + q + q + 2 q + 2 q + 3 q + 3 q + 2 q + 2 q + 2 q + 2 q + 2 q + q + q . This proves the identity (1.1) for any non-negative N . For negative values of N both sides of (1.1) is 0.4. Combinatorics of Theorem 1.2
Let n , n , and m be non-negative integers and let the partition π n ,n ,m , to be called minimalconfiguration , be defined as n consecutive 1 modulo 3 parts followed by n consecutive 2 modulo 3 partsfollowed by m parts that are exactly 4 apart from their neighboring parts. For positive n , n , and m ,we have π n ,n ,m := (1 , , , . . . , n −
1) + 1 , n + 2 , n + 5 , . . . , n + n −
1) + 2 , n + n ) + 3 , n + n ) + 7 , n + n ) + 11 , . . . , n + n ) + 3 + 4( m − , (4.1)where we underline the initial chain of the n consecutive 1 mod 3 parts and also underline the following n consecutive 2 mod 3 parts. We do not underline the m singletons .If n , n , or m is 0, in the (4.1) we ignore the related portion of the partition with these numbers. As ALI KEMAL UNCU an example, when n = n = m = 0, we get an empty list (the unique partition of 0) as our minimalconfiguration.It is easy to see that the minimal configuration π n ,n ,m satisfies the gap conditions of the SchurTheorem (Theorem 1.1). Moreover, this partition has n + n + m parts and its size is exactly A ( n , n , m )as in (1.2). The name minimal configuration comes from the fact that π n ,n ,m is the partition with thesmallest size that satisfies the gap conditions of Theorem 1.1 that has n + n − n + n + m parts.We would like to start with such a minimal configurations and build up all partitions that satisfySchur’s gap conditions, bijectively. For that we will define “the forwards motions of the parts” of theminimal configurations first. This will be done in a similar fashion to [9, 16–18, 22], mostly resemblingthe lines of [22].Before presenting the details, we would like to summarize the way we will approach the forwardsmotions. First, we will move the singletons; starting from the largest singleton (greatest as an integer)to the smallest singleton. We will preserve the order of the singletons of π n ,n ,m by moving each partless than or equal to the amount of movement of the previous (greater) part. Then, we will define themotion of the 2 modulo 3 parts as pairs splitting from the end of the 2 modulo 3 initial chain of π n ,n ,m .Once again, this motion will be done starting from the greatest pair (the order with respect to the sumof the pair’s parts) to the smallest pair. We will maintain the ordering of the pairs by letting any pair tomove at most the same amount as the previous pair that moved before it. We will define crossing over asingleton for these 2 mod 3 pairs, as these pairs may come close to a singleton that moved before any oneof the pairs and may violate the Schur’s gap conditions. Finally, we will define the motion of the 1 modulo3 pairs in a similar fashion to the 2 modulo 3 pairs. In this case, we will need the additional treatmentof a 1 modulo 3 pair crossing over consecutive 2 modulo 3 parts of the partition. All the defined motionswill bijective maps and at each step we will make sure the outcome partition satisfies the Theorem 1.1’sgap conditions.Starting from the largest part (the last part) we can move the m -singletons forwards by adding eachelement a non-negative value: r m to the largest part, r m − to the second largest with r m ≥ r m − ... r to the smallest singleton r ≥ r ≥
0. The order 0 ≤ r ≤ r ≤ · · · ≤ r m is enough to ensure that orderof the singletons are preserved after the motions. Such a list ( r , r , . . . , r m ) with 0 ≤ r ≤ r ≤ · · · ≤ r m may not be a partition itself; some r i values might be 0. On the other hand, by ignoring the zero values,it is clear that every such list (used in the forwards motion of the singletons) corresponds to a uniquepartition into ≤ m parts. Therefore, the generating function that is related with the forwards motionsof m singletons is the generating function for the number of partitions into ≤ m parts:(4.2) 1( q ) m . It is clear that the motions of the singletons are bijective and can easily be reversed.After moving the singletons, we start moving the initial chain of the n n + 2 , n + 5 , . . . , n + n −
2) + 2 , n + n −
1) + 2 n + 2 , n + 5 , . . . , n + n −
3) + 2 3( n + n −
2) + 2 , n + n −
1) + 2 | {z } . Later we will start moving these pairs by moving one to the next possible location where the numbersagain become a pair of consecutive 2 modulo 3 parts. Before doing so, note that we are splitting andmoving two parts of an n length initial chain together. Hence, we can at most split and move ⌊ n / ⌋ pairs. In the motion of these pairs, similar to the singletons case, we will move the greatest pair (orderedwith respect to sum of the parts in the pair) forwards the most, then the second largest pair less thanthe motion of the first pair etc. POLYNOMIAL IDENTITY IMPLYING SCHUR’S PARTITION THEOREM For a given pair x, y |{z} of π that satisfies the gap conditions of Schur’s theorem (Theorem 1.1), if π doesnot have a part z such that y + 3 ≤ z < y + 6, we define the motion of this pair as(4.3) x, y |{z} x + 3 , y + 3 | {z } . This forwards motion adds a total of 6 to the size of the partition π , the greater part of the pair moves3 steps forwards, and it does not change the residue class of x and y modulo 3. Moreover, it is clearlybijective and can be undone.There might be a z value that is in 4 or 5 distance to the larger part of the pair that we wouldlike to move. This forwards motions needs us to define particular bijective rules so that the outcomepartition would still satisfy the gap conditions of Schur’s theorem. Given a pair 3 k + 2 , k + 5 | {z } , we definethe following bijective rules for crossing singletons. Similar to adjustments explained in [18], we needto handle different cases differently. These cases will depend on the number of singletons that one pairneeds to cross in a given circumstance: • Case 1: If the pair is crossing a single singleton (that is ≤ r = 0 , ,
2, we have(4.4) 3 k + 2 , k + 5 | {z } , k + 8 + r k + 2 + r, k + 8 , k + 11 | {z } . • Case 2: If the pair is crossing two close singletons (a singleton followed by another singletonthat is ≤ r, s ∈ { , , } with s − r ≤
1, we define the motions:(4.5) 3 k + 2 , k + 5 | {z } , k + 8 + r, k + 12 + s k + 2 + r, k + 6 + s, k + 11 , k + 14 | {z } . Notice here that the ( r, s ) = (2 ,
3) possibility is excluded although this can be considered as twoclose singletons. That is because in this case we can use (4.4) with r = 2 and this would notbreak the Schur’s gap conditions. • Case 3: If the pair to move needs to cross three close singletons, and if employing the motions(4.4) or (4.5) is violating the gap conditions of Theorem 1.1. Let r, s, t ∈ { , } with s − r ≤ t − s ≤ k + 2 , k + 5 | {z } , k +8+ r, k +12+ s, k +16+ t k +2+ r, k +6+ s, k +10+ t, k + 14 , k + 17 | {z } . Observe that a possible part of the partition (if any) that follows the part 3 k + 16 + t in (4.6) isat least of size 3 k + 20 + t . The gap between the largest part of our last motion (4.6) 3 k + 17 hasat least a gap of 3 with this possible part 3 k + 20 + t . Therefore, one can stop the crossing of thepairs over singletons here. This also means one can stop defining particular rules here as well. Ifthey would like to move the pair 3 k + 14 , k + 17 | {z } once again, they can start with checking andemploying the bijective motion rules (4.3)-(4.6).Hence, the list of motions (4.3), (4.4), (4.5), and (4.6) is the full bijective list of motions for the ⌊ n / ⌋ ⌊ n / ⌋ ≤ ⌊ n / ⌋ parts. The generating function for the forwards motions of the 2 modulo 3initial chain is(4.7) 1( q ; q ) ⌊ n / ⌋ . Also, observe that in all these motions the pairs move(4.8) 3 + 3 × “the number of singletons crossed”steps forwards. ALI KEMAL UNCU
Finally, we move on to the motions starting from the initial chain of the n , , , . . . , n −
3) + 1 , n −
2) + 1 , n −
1) + 1 , , , . . . , n −
3) + 1 , n −
2) + 1 , n −
1) + 1 | {z } . Similar to the previous (2 modulo 3 initial chain) case we can split and move at most ⌊ n / ⌋ . Moreover,(4.3) is still valid for this case, and for the rest of the crossing rules all one needs to do is to use the samecases related to (4.4)-(4.6) and subtract 1 from each and every term in these motions. All the size andnumber of forward motion observations that is made for the 2 modulo 3 pairs are still valid for the 1modulo 3 pairs.One new situation in this case appears if a 1 modulo 3 pair comes close to a group of consecutive 2modulo 3 parts of the partition. In this situation, we define the following bijective map. Let l ≥ k + 1 , k + 4 | {z } , k + 8 , k + 11 , . . . , k + 3 l + 5 k + 2 , k + 5 , . . . , k + 3 l − , k + 3 l + 4 , k + 3 l + 7 | {z } . (4.9)Note that l = 1 and 2 cases are covered under the relative versions of (4.4) and (4.5) for the 1 modulo 3pairs. Moreover, note that in this forwards motion the pair makes l extra motions and again the size ofthe overall partition raises only by 6. By the same argument as the previous case now we can see thatthe generating function corresponding to the forwards motion of the 1 modulo 3 initial chain is(4.10) 1( q ; q ) ⌊ n / ⌋ . Combining (4.2), (4.7) and (4.10), it is easy to see that(4.11) q A ( n ,n ,m ) ( q ; q ) ⌊ n / ⌋ ( q ; q ) ⌊ n / ⌋ ( q ) m is the generating function for all the partitions that satisfies the gap conditions of Theorem 1.1 that canbe constructed from the minimal configuration π n ,n ,m defined in (4.1), where A ( n , n , m ) is as definedin (1.2). By summing over all possible n , n , and m we get the following theorem. Theorem 4.1.
Let A ( n , n , m ) be as defined in (1.2) , then (4.12) X n ,n ,m ≥ x n + n + m q A ( n ,n ,m ) ( q ; q ) ⌊ n / ⌋ ( q ; q ) ⌊ n / ⌋ ( q ) m is the generating function for the number of partitions that satisfy the gap conditions of Schur’s theorem(Theorem 1.1), where the exponent of x counts the number of parts of the counted partitions. The triple series (4.12) is the analogue of the double sums presented for the G¨ollnitz–Gordon and littleG¨ollnitz theorems in [22]. This series (as well as the ones in [22]) are inspired by Kur¸sung¨oz’s recentworks [16–18]. Due to the difference in the minimal configuration setups and some of the motions, theauthor and Kur¸sung¨oz gets equivalent but different representations for the same generating functions.Here we present Kur¸sung¨oz’s version of the generating function represented in Theorem 4.1.
Theorem 4.2 (Kur¸sung¨oz, 2018) . Let (4.13) K ( n , n , m ) := 6( n + n ) + 2 m + 6 m ( n + n ) − n + n − m, then (4.14) X n ,n ,m ≥ x n +2 n + m q K ( n ,n ,m ) ( q ; q ) n ( q ; q ) n ( q ) m POLYNOMIAL IDENTITY IMPLYING SCHUR’S PARTITION THEOREM is the generating function for the number of partitions that satisfy the gap conditions of Schur’s theorem(Theorem 1.1), where the exponent of x counts the number of parts of the counted partitions. To avoid any speculative trivial transformation between (4.12) and (4.14) please note that(4.15) A (2 n , n , m ) − K ( n , n , m ) = 2 m. We would also like to present the equality of the series (4.12) and (4.14) after doing even-odd splits forthe variables n and n and regrouping in (4.12). We will also be using (4.15) to write the q -factors inthe summands using the same quadratic K ( n , n , m ). Theorem 4.3.
We have X n ,n ,m ≥ x n +2 n + m q K ( n ,n ,m )+2 m ( q ; q ) n ( q ; q ) n ( q ) m (1 + xq n +6 n +3 m +1 + xq n +6 n +3 m +2 + x q n +12 n +6 m +6 )= X n ,n ,m ≥ x n +2 n + m q K ( n ,n ,m ) ( q ; q ) n ( q ; q ) n ( q ) m , (4.16) where K ( n , n , m ) is as in (4.13) . Now we start finding a finite analogue of (4.12). Let N be a non-negative integer. We would like tofind all the partitions with the largest part ≤ N that are counted by (4.12). For that we need to counthow many times a singleton, a 2 modulo 3 pair and a 1 modulo 3 pair can move forward before exceeding N and change our generating functions from reciprocal of a q -factorials to the necessary q -binomials.The largest singleton of the minimal configuration π n ,n ,m , 3( n + n −
1) + 2, can only move N − [3( n + n ) + 3 + 4( m − ≤ m parts, where each part is ≤ N − [3( n + n −
1) + 2]. The generating function for all such partitions is(4.17) (cid:20) N − [3( n + n ) + 3 + 4( m − mm (cid:21) q Each forwards movement of a 2 modulo 3 pair gets it 3 units closer to the bound N . Then, ignoring thesingletons for a second, the largest 2 modulo 3 pair 3( n + n −
2) + 2 , n + n −
1) + 2 | {z } can move atmost (cid:22) N − [3( n + n −
1) + 2]3 (cid:23) steps forwards before the larger part, 3( n + n −
1) + 2, of the pair goes over the bound on the largestpart N . Recall (4.8): crossing over singletons make these pairs move extra steps forwards. There are m singletons that are greater than the largest pair 3( n + n −
2) + 2 , n + n −
1) + 2 | {z } . Hence beforereaching the bound this pair would need to cross all of those m singletons, and move an extra 3 stepsforwards each time. Therefore, the actual number of steps this pair can take forwards before passing thebound N is (cid:22) N − [3( n + n −
1) + 2]3 (cid:23) − m. This shows us that the bounded forwards motion of the 2 modulo 3 pairs is related with partitionsinto ≤ ⌊ n / ⌋ parts each ≤ ⌊ N − [3( n + n −
1) + 2] / ⌋ − m . This implies that the related generatingfunction for this motion (that changes the size by 6 each time) is(4.18) "j N − [3( n + n − k − m + (cid:4) n (cid:5)(cid:4) n (cid:5) q . Finally, Similar to the previous case, forgetting about the the n m singletons,the largest 1 modulo 3 pair, 3( n −
2) + 1 , n −
1) + 1 | {z } , can move (cid:22) N − [3( n −
1) + 1]3 (cid:23)
ALI KEMAL UNCU forwards before 3( n −
1) + 1 goes over N . Including our observations about the extra steps one pairtakes while crossing over parts, we see that the actual number of steps forwards that the largest pair cantake is (cid:22) N − [3( n −
1) + 1]3 (cid:23) − m − n . With that, similar to the previous case, we see that the generating function related to the forwardsmotions of the ⌊ n / ⌋ "j N − [3( n − k − m − n + (cid:4) n (cid:5)(cid:4) n (cid:5) q . Putting (4.17), (4.18), and (4.19) together, we get that q A ( n ,n ,m ) (cid:20) N − n + n + m ) + 1 m (cid:21) q (4.20) × "j N − [3( n − k − m − n + (cid:4) n (cid:5)(cid:4) n (cid:5) q "j N − [3( n + n − k − m + (cid:4) n (cid:5)(cid:4) n (cid:5) q is the generating function for the number of partitions that satisfies the gap conditions of Theorem 1.1that can be constructed from the minimal configuration π n ,n ,m with the extra bound on the largestpart ≤ N . Summing (4.20) over n , n , and m yields the following theorem. Theorem 4.4.
For any non-negative integer N , the expression X n ,n ,m ≥ x n + n + m q A ( n ,n ,m ) (cid:20) N − n + n + m ) + 1 m (cid:21) q (4.21) × "j N − [3( n − k − m − n + (cid:4) n (cid:5)(cid:4) n (cid:5) q "j N − [3( n + n − k − m + (cid:4) n (cid:5)(cid:4) n (cid:5) q where A ( n , n , m ) is defined as in Theorem 1.2, is the generating function for the number of partitionsthat satisfy the gap conditions of Theorem 1.1 with the extra condition that each part is ≤ N , where theexponent of x counts the number of parts. One direct corollary of Theorem 4.4 is the interpretation of the left-hand side of (1.1) when N N − Corollary 4.5.
For any positive integer N , and N := N − n − n − m , the expression X m,n ,n ≥ q A ( n ,n ,m ) (cid:20) N m (cid:21) q (cid:20) N + (cid:4) n (cid:5)(cid:4) n (cid:5) (cid:21) q (cid:20) N + (cid:4) n (cid:5)(cid:4) n (cid:5) (cid:21) q , where A ( n , n , m ) is defined as in Theorem 1.2, is the generating function for the number of partitionsthat satisfy the gap conditions of Theorem 1.1 with the extra condition that each part is ≤ N − . On the other hand, Andrews [6] interpreted the right-hand side of (1.1) as the same generating functionin the interpretation of Corollary 4.5. This is also proves the validity of Theorem 4.1 for positive values of N , this time using only the combinatorial constructions. In [6, (3.9), pg. 147], Andrews also shows thatthe right-hand side sum converges to the generating function for the number of partitions into distinctparts ± − q, − q ; q ) ∞ . This shows that after taking limits N → ∞ of (1.1), and using (2.6) as needed, we have X n ,n ,m ≥ x n + n + m q A ( n ,n ,m ) ( q ; q ) ⌊ n / ⌋ ( q ; q ) ⌊ n / ⌋ ( q ) m = ( − q, − q ; q ) ∞ , which is the analytic version of the Schur’s theorem (Theorem 1.1). This shows that the polynomialidentity (1.1) (keeping the interpretation, Theorem 4.4 in mind) implies Theorem 1.1. POLYNOMIAL IDENTITY IMPLYING SCHUR’S PARTITION THEOREM Some Implications of Theorem 1.2
We start by sending q /q in (1.1) followed by the use of (2.7) and multiplying both sides with q N / . This yields the equivalent formula(5.1) X m,n ,n ≥ q B ( n ,n ,m,N ) − A ( n ,n ,m ) (cid:20) N m (cid:21) q (cid:20) N + (cid:4) n (cid:5)(cid:4) n (cid:5) (cid:21) q (cid:20) N + (cid:4) n (cid:5)(cid:4) n (cid:5) (cid:21) q = N X j = − N q j T (cid:18) N ; q j (cid:19) , where A ( n , n , m ) is as in (1.2), N = N − n = n = m , and(5.2) B ( n , n , m, N ) = 3 N − (3 N − m ) m − N (cid:16)j n k + j n k(cid:17) . Note that the sides in (5.1) are not polynomials but multiplying both sides with q N/ is enough tomake them polynomials. After multiplying both sides of (5.1) by q N/ , writing the definition of (2.4) infor the right-hand side of (5.1) and using (2.2) multiple times we see that(5.3) ∞ X j = −∞ q N + j T (cid:18) N ; q j (cid:19) = X k,l ≥ q k + l (3 l +1)2 ( q ; q ) N ( q ; q ) N − k − l ( q ; q ) k ( q ; q ) l , after simple changes of variables. We use (2.1) for the term ( q ; q ) ( N − k ) − l to separate the difference ofthe variable l . This way we end up with the expression X k ≥ q k (cid:20) Nk (cid:21) q X l ≥ ( q − N − k ) ; q ) l ( q ; q ) l (cid:16) − q N − k )+2 (cid:17) l . The inner sum can be summed using the q -binomial theorem [14, II.4, p 354], and we get(5.4) ∞ X j = −∞ q N + j T (cid:18) N ; q j (cid:19) = X k ≥ q k (cid:20) Nk (cid:21) q ( − q ; q ) N − k . Not only that, (5.4) with the use of [14, II.1, p 354] on the right-hand side, yields(5.5) lim N →∞ ∞ X j = −∞ q N + j T (cid:18) N ; q j (cid:19) = 1( q ; q ) ∞ ( q ; q ) ∞ . To evaluate the N → ∞ limit on the left-hand side of (5.1) with the extra q N/ , one first needs tomake a change of summation variables and rewrite the q -factor. We would like to use y = N as oursummation variable instead of n , but the parity of N must be kept in check to correctly identify theexponent of the q -factor in this case. Let r ( a, b ) be the remainder of the division a ÷ b , for a, b ∈ N . Afterthe change of variables, the left-hand side of (5.1) multiplied with an extra q N/ becomes X m,n ,y ≥ q Q ( m,n ,y,N ) (cid:20) ym (cid:21) q (cid:20) y + (cid:4) n (cid:5) y (cid:21) q " y + j N − m − n − y k y q , where Q ( m, n , y, N ) = (cid:18) m (cid:19) + y (3 y + 1)2 + n + 3 y r ( N + m + y,
2) + 6 y r ( n , r ( N + m + y + 1 , . Then, by taking the limit N → ∞ for odd and even N and using (5.5) we get the following theorem. Theorem 5.1.
Let t = 1 , , then (5.6) X m,n ,y ≥ q Q t ( m,n ,y ) ( q ; q ) y (cid:20) ym (cid:21) q (cid:20) y + (cid:4) n (cid:5) y (cid:21) q = 1( q ; q ) ∞ ( q ; q ) ∞ , where (5.7) Q t ( m, n , y ) = (cid:18) m (cid:19) + y (3 y + 1)2 + n + 3 y r ( m + y + t,
2) + 6 y r ( n , r ( m + y + 1 + t, Recall that Warnaar [23, (10), pg 2516] proved the following summation formula.(5.8) X i ≥ q i (cid:20) Li (cid:21) q T (cid:18) i ; qa (cid:19) = q a (cid:20) LL − a (cid:21) q . This can be applied to the right-side of (5.1) to get the following theorem.
Theorem 5.2.
Let N = N − m − n − n , for any non-negative integer M we have X N,m,n ,n ≥ q N + B ( n ,n ,m,N ) − A ( n ,n ,m ) (cid:20) N m (cid:21) q (cid:20) MN (cid:21) q (cid:20) N + (cid:4) n (cid:5)(cid:4) n (cid:5) (cid:21) q (cid:20) N + (cid:4) n (cid:5)(cid:4) n (cid:5) (cid:21) q = ( − q, − q ; q ) M , (5.9) where A ( n , n , m ) and B ( n , n , m, N ) are defined as in (1.2) and (5.2) , respectively.Proof. We sum both sides of (5.1) over N from 0 to M after multiplying the summand with q N (cid:20) MN (cid:21) q . This gives the left-hand side of (5.9). For the right-hand side of the formula, we interchange the order ofsummations, use (5.8) followed by the summation formula [7, (3.3.6). p. 36]. This yields q (3 M +1) M ( − q − M ; q ) M , which after basic simplifications is equal to the right-hand side of the equation (5.9). (cid:3) The limit M → ∞ of (5.9) is much more straightforward than the limit n → ∞ . By employing (2.6),we get the following corolary of Theorem 5.2. Corollary 5.3. X N,m,n ,n ≥ q N + B ( n ,n ,m,N ) − A ( n ,n ,m ) ( q ; q ) N (cid:20) N m (cid:21) q (cid:20) N + (cid:4) n (cid:5)(cid:4) n (cid:5) (cid:21) q (cid:20) N + (cid:4) n (cid:5)(cid:4) n (cid:5) (cid:21) q = ( − q, − q ; q ) ∞ where A ( n , n , m ) and B ( n , n , m, N ) are defined as in (1.2) and (5.2) , respectively. Theorem 1.2 (and the equation (5.1)) and Theorem 5.2 also yield some intriguing combinatorial corol-laries at the q = 1 level. Corollary 5.4.
For some non-negative integer M , N := N − n + n − m and M := M − n − n − m ,we have X m,n ,n ≥ (cid:18) M m (cid:19)(cid:18) M + (cid:4) n (cid:5) M (cid:19)(cid:18) M + (cid:4) n (cid:5) M (cid:19) = 3 M , (5.10) and X N,m,n ,n ≥ (cid:18) MN (cid:19)(cid:18) N m (cid:19)(cid:18) N + (cid:4) n (cid:5) N (cid:19)(cid:18) N + (cid:4) n (cid:5) N (cid:19) = 4 M . (5.11) Proof.
The equation (5.11) is a clear consequence of (5.9), or one can get it from (5.10) as it is theclassical binomial theorem. For the equation (5.10), one only needs to recall that N X j = − N x j (cid:18) N ; j ; 1 j (cid:19) = ( x − + 1 + x ) N , and set x to 1. (cid:3) POLYNOMIAL IDENTITY IMPLYING SCHUR’S PARTITION THEOREM Acknowledgments
The author would like to thank Karl Mahlburg for bringing [18] to our attention and for his interest.The author would also like to thank Alexander Berkovich for the stimulating discussion and his suggestionson the manuscript.Research of the author is supported by the Austrian Science Fund FWF, SFB50-07 and SFB50-09Projects.
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