Logarithmic and complex constant term identities
aa r X i v : . [ m a t h . C O ] M a r LOGARITHMIC AND COMPLEXCONSTANT TERM IDENTITIES
TOM CHAPPELL, ALAIN LASCOUX, S. OLE WARNAAR, AND WADIM ZUDILIN
To Jon
Abstract.
In recent work on the representation theory of vertex algebras re-lated to the Virasoro minimal models M (2 , p ), Adamovi´c and Milas discoveredlogarithmic analogues of (special cases of) the famous Dyson and Morris con-stant term identities. In this paper we show how the identities of Adamovi´cand Milas arise naturally by differentiating as-yet-conjectural complex ana-logues of the constant term identities of Dyson and Morris. We also discussthe existence of complex and logarithmic constant term identities for arbitraryroot systems, and in particular prove complex and logarithmic constant termidentities for the root system G . Keywords:
Constant term identities; perfect matchings; Pfaffians; root sys-tems; Jon’s birthday. Jonathan Borwein
Jon Borwein is known for his love of mathematical constants . We hope this paperwill spark his interest in constant terms .1.
Constant term identities
The study of constant term identities originated in Dyson’s famous 1962 paper
Statistical theory of the energy levels of complex systems [9]. In this paper Dysonconjectured that for a , . . . , a n nonnegative integers(1.1) CT Y ≤ i = j ≤ n (cid:16) − x i x j (cid:17) a i = ( a + a + · · · + a n )! a ! a ! · · · a n ! , where CT f ( X ) stands for the constant term of the Laurent polynomial (or possi-bly Laurent series) f ( X ) = f ( x , . . . , x n ). Dyson’s conjecture was almost instantlyproved by Gunson and Wilson [14, 36]. In a very elegant proof, published sev-eral years later [13], Good showed that (1.1) is a direct consequence of Lagrangeinterpolation applied to f ( X ) = 1.In 1982 Macdonald generalised the equal-parameter case of Dyson’s ex-conjecture,i.e.,(1.2) CT Y ≤ i = j ≤ n (cid:16) − x i x j (cid:17) k = ( kn )!( k !) n , Mathematics Subject Classification. to all irreducible, reduced root systems; here (1.2) corresponds to the root systemA n − . Adopting standard notation and terminology—see [17] or the next section—Macdonald conjectured that [25](1.3) CT Y α ∈ Φ (1 − e α ) k = r Y i =1 (cid:18) kd i k (cid:19) , where Φ is one of the root systems A n − , B n , C n , D n , E , E , E , F , G of rank r and d , . . . , d r are the degrees of its fundamental invariants. For k = 1 the Macdonaldconjectures are an easy consequence of Weyl’s denominator formula X w ∈ W sgn( w ) e w ( ρ ) − ρ = Y α> (cid:0) − e − α (cid:1) (where W is the Weyl group of Φ and ρ the Weyl vector), and for B n , C n , D n but k general they follow from the Selberg integral. The first uniform proof of(1.3)—based on hypergeometric shift operators—was given by Opdam in 1989 [24].In his PhD thesis [27] Morris used the Selberg integral to prove a generalisationof (1.2), now commonly referred to as the Morris or Macdonald–Morris constantterm identity:(1.4) CT (cid:20) n Y i =1 (1 − x i ) a (cid:16) − x i (cid:17) b Y ≤ i = j ≤ n (cid:16) − x i x j (cid:17) k (cid:21) = n − Y i =0 ( a + b + ik )!(( i + 1) k )!( a + ik )!( b + ik )! k ! , where a and b are arbitrary nonnegative integers.In their recent representation-theoretic work on W -algebra extensions of the M (2 , p ) minimal models of conformal field theory [1, 2], Adamovi´c and Milas dis-covered a novel type of constant term identities, which they termed logarithmicconstant term identities . Before stating the results of Adamovi´c and Milas, somemore notation is needed.Let ( a ) m = a ( a +1) · · · ( a + m −
1) denote the usual Pochhammer symbol or risingfactorial, and let u be either a formal or complex variable. Then the (generalised)binomial coefficient (cid:0) um (cid:1) is defined as (cid:18) um (cid:19) = ( − m ( − u ) m m ! . We now interpret (1 − x ) u and log(1 − x ) as the (formal) power series(1.5) (1 − x ) u = ∞ X m =0 ( − x ) m (cid:18) um (cid:19) and log(1 − x ) = − ∞ X m =1 x m m = dd u (1 − x ) u (cid:12)(cid:12)(cid:12) u =0 . Finally, for X = ( x , . . . , x n ) we define the Vandermonde product∆( X ) = Y ≤ i One of the discoveries of Adamovi´c and Milas is the following beautiful logarith-mic analogue of the equal-parameter case (1.2) of Dyson’s identity. Conjecture 1.1 ( [1, Conjecture A.12]) . For n an odd positive integer and k anonnegative integer define m := ( n − / and K := 2 k + 1 . Then (1.6) CT (cid:20) ∆( X ) n Y i =1 x − mi m Y i =1 log (cid:16) − x i x i − (cid:17) Y ≤ i = j ≤ n (cid:16) − x i x j (cid:17) k (cid:21) = ( nK )!! n !!( K !!) n . We remark that the kernel on the left is a Laurent series in X of (total) degree0. Moreover, in the absence of the term Q mi =1 log(1 − x i /x i − ) the kernel isa skew-symmetric Laurent polynomial which therefore has a vanishing constantterm. Using identities for harmonic numbers, Adamovi´c and Milas proved (1.6) for n = 3, see [1, Corollary 11.11].Another result of Adamovi´c and Milas, first conjectured in [1, Conjecture 10.3](and proved for n = 3 in (the second) Theorem 1.1 of that paper, see page 3925) andsubsequently proved in [2, Theorem 1.4], is the following Morris-type logarithmicconstant term identity. Theorem 1.2. With the same notation as above, (1.7)CT (cid:20) ∆( X ) n Y i =1 x − ( k +1)( n +1) i (1 − x i ) a m Y i =1 log (cid:16) − x i x i − (cid:17) Y ≤ i = j ≤ n (cid:16) − x i x j (cid:17) k (cid:21) = c nk n − Y i =0 (cid:18) a + Ki/ m + 1) K − (cid:19) , where a is an indeterminate, c nk a nonzero constant, and (1.8) c ,k = (3 K )!( k !) k + 1)!( K !) (cid:18) K − K − (cid:19) − (cid:18) K/ − K − (cid:19) − . As we shall see later, the above can be generalised to include an additional freeparameter resulting in a logarithmic constant term identity more closely resemblingMorris’ identity, see (1.9) below.The work of Adamovi´c and Milas raises the following obvious questions:(1) Can any of the methods of proof of the classical constant term identities, seee.g., [7,8,11–15,19–21,24,29–31,36–40], be utilised to prove the logarithmiccounterparts?(2) Do more of Macdonald’s identities (1.3) admit logarithmic analogues?(3) All of the classical constant term identities have q -analogues [16, 18, 25, 27].Do such q -analogues also exist in the logarithmic setting?As to the first and third questions, we can be disappointingly short; we have notbeen able to successfully apply any of the known methods of proof of constant termidentities to also prove Conjecture 1.1, and attempts to find q -analogues have beenequally unsuccessful. (In fact, we now believe q -analogues do not exist.)As to the second question, we have found a very appealing explanation—itselfbased on further conjectures!—of the logarithmic constant term identities of Adamo-vi´c and Milas. They arise by differentiating a complex version of Morris’ constantterm identity. Although such complex constant term identities are conjectured to TOM CHAPPELL, ALAIN LASCOUX, S. OLE WARNAAR, AND WADIM ZUDILIN exist for other root systems as well—this is actually proved in the case G —itseems that only for A n and G these complex identities imply elegant logarithmicidentities.The remainder of this paper is organised as follows. In the next section we intro-duce some standard notation related to root systems. Then, in Section 3, we studycertain sign functions and prove a related Pfaffian identity needed subsequently.In Section 4, we conjecture a complex analogue of the Morris constant term iden-tity 1.4 for n odd, and prove this for n = 3 using Zeilberger’s method of creativetelescoping [4, 28]. In Section 5 we show that the complex Morris identity impliesthe following logarithmic analogue of (1.4). Theorem 1.3 ( Logarithmic Morris constant term identity ) . With the samenotation as in Conjecture 1.1 and conditional on the complex Morris constant termidentity (4.5) to hold, we have (1.9)CT (cid:20) ∆( X ) n Y i =1 x − mi (1 − x i ) a (cid:16) − x i (cid:17) b m Y i =1 log (cid:16) − x i x i − (cid:17) Y ≤ i = j ≤ n (cid:16) − x i x j (cid:17) k (cid:21) = 1 n !! n − Y i =0 (2 a + 2 b + iK )!!(( i + 1) K )!!(2 a + iK )!!(2 b + iK )!! K !! , where a, b are nonnegative integers. In Section 6 we prove complex as well as logarithmic analogues of (1.3) for theroot system G , and finally, in Section 7 we briefly discuss the classical roots systemsB n , C n and D n .2. Preliminaries on root systems and constant terms In the final two sections of this paper we consider root systems of types otherthan A, and below we briefly recall some standard notation concerning root systemsand constant term identities. For more details we refer the reader to [17, 25]. α α α α Figure 1. The root systems A (left) and G (right) with ∆ = { α , α } .Let Φ be an irreducible, reduced root system in a real Euclidean space E withbilinear symmetric form ( · , · ). Fix a base ∆ of Φ and denote by Φ + the set ofpositive roots. Write α > α ∈ Φ + . The Weyl vector ρ is defined as half thesum of the positive roots: ρ = P α> α . The height ht( β ) of the root β is givenby ht( β ) = ( β, ρ ). Let r be the rank of Φ (that is, the dimension of E ). Then the OGARITHMIC AND COMPLEX CONSTANT TERM IDENTITIES 5 degrees 1 < d ≤ d ≤ · · · ≤ d r of the fundamental invariants of Φ are uniquelydetermined by Y i ≥ − t d i − t = Y α> − t ht( α )+1 − t ht( α ) . For example, in the standard representation of the root system A n − , E = { ( x , . . . , x n ) ∈ R n : x + · · · + x n = 0 } , (2.1) Φ = { ǫ i − ǫ j : 1 ≤ i = j ≤ n } and ∆ = { α , . . . , α n − } = { ǫ i − ǫ i +1 : 1 ≤ i ≤ n − } , where ǫ i denotes the i th standard unit vector in R n . Since ht( ǫ i − ǫ j ) = j − i , Y α> − t ht( α )+1 − t ht( α ) = Y ≤ i Let Φ s and Φ l denote the set of short and long roots of G respec-tively. Then (2.4) CT Y α ∈ Φ l (1 − e α ) k Y α ∈ Φ s (1 − e α ) m = (3 k + 3 m )!(3 k )!(2 k )!(2 m )!(3 k + 2 m )!(2 k + m )!( k + m )! k ! k ! m ! . Note that for k = 0 or m = 0 this yields (1.2) for n = 3. As we shall see inSection 6, it is the above identity, not it equal-parameter case (2.2), that admits alogarithmic analogue. 3. The signatures τ ij In our discussion of complex and logarithmic constant term identities in Sec-tions 4–7, an important role is played by certain signatures τ ij . For the convenienceof the reader, in this section we have collected all relevant facts about the τ ij .For a fixed odd positive integer n and m := ( n − / τ ij for 1 ≤ i < j ≤ n by(3.1) τ ij = ( j ≤ m + i, − j > m + i, and extend this to all 1 ≤ i, j ≤ n by setting τ ij = − τ ji . Assuming that 1 ≤ i < n we have τ in = χ ( n ≤ m + i ) − χ ( n > m + i ) , where χ (true) = 1 and χ (false) = 0. Since n − m = m + 1, this is the same as τ in = χ ( i > m ) − χ ( i ≤ m ) = − τ ,i +1 = τ i +1 , . For 1 ≤ i, j < n we clearly also have τ ij = τ i +1 ,j +1 . Hence the matrix(3.2) T := ( τ ij ) ≤ i,j ≤ n is a skew-symmetric circulant matrix . For example, for n = 5,T = − − − − − − − − − − . We note that all of the row-sums (and column-sums) of the above matrix are zero.Because T is a circulant matrix, to verify this property holds for all (odd) n weonly needs to verify this for the first row: n X j =1 τ j = m +1 X j =2 − n X j = m +2 m − ( n − m − 1) = m − m = 0 . By the skew symmetry, the vanishing of the row sums may also be stated as follows. OGARITHMIC AND COMPLEX CONSTANT TERM IDENTITIES 7 Lemma 3.1. For ≤ i ≤ n , i − X j =1 τ ji = n X j = i +1 τ ij . A property of the signatures τ ij , which will be important in our later discussions,can be neatly clarified by having recourse to Pfaffians.By a perfect matching (or 1-factor) on [ n + 1] := { , , . . . , n + 1 } we mean agraph on the vertex set [ n + 1] such that each vertex has degree one, see e.g., [6,35].If in a perfect matching π the vertices i < j are connected by an edge we say that( i, j ) ∈ π . Two edges ( i, j ) and ( k, l ) of π are said to be crossing if i < k < j < l or k < i < l < j . The crossing number c ( i, j ) of the edge ( i, j ) ∈ π is the number ofedges crossed by ( i, j ), and the crossing number c ( π ) is the total number of pairsof crossing edges: c ( π ) = P ( i,j ) ∈ π c ( i, j ). We can embed perfect matching in the xy -plane, such that (i) the vertex labelled i occurs at the point ( i, i, j ) and ( k, l ) intersect exactly once if they are crossing and do not intersect ifthey are non-crossing. For example, the perfect matching { (1 , , (2 , , (4 , , (6 , } corresponds to 1 2 3 4 5 6 7 8and has crossing number 2 ( c (4 , 5) = 0, c (1 , 3) = c (6 , 8) = 1 and c (2 , 7) = 2).The Pfaffian of a (2 N ) × (2 N ) skew-symmetric matrix A is defined as [6,22,23,35]:(3.3) Pf( A ) := X π ( − c ( π ) Y ( i,j ) ∈ π A ij . After these preliminaries on perfect matching and Pfaffians we now form asecond skew-symmetric matrix, closely related to T. First we extend the τ ij to1 ≤ i, j ≤ n + 1 by setting τ i,n +1 = b i . We then define the ( n + 1) × ( n + 1)skew-symmetric matrix Q ( a, b ) = ( Q ij ( a, b )) ≤ i,j ≤ n +1 , where a = ( a , . . . , a n +1 )and b = ( b , . . . , b n ), as follows:(3.4) Q ij ( a, b ) = τ ij a i a j for 1 ≤ i < j ≤ n + 1 . For example, for n = 5, Q ( a, b ) = a a a a − a a − a a a a b − a a a a a a − a a a a b − a a − a a a a a a a a b a a − a a − a a a a a a b a a a a − a a − a a a a b − a a b − a a b − a a b − a a b − a a b . Note that T is the submatrix of Q (cid:0) (1 n +1 ) , b (cid:1) obtained by deleting the last row andcolumn. Proposition 3.2. We have Pf (cid:0) Q ( a, b ) (cid:1) = ( − m ) a a · · · a n +1 ( b + b + · · · + b n ) . TOM CHAPPELL, ALAIN LASCOUX, S. OLE WARNAAR, AND WADIM ZUDILIN Proof. The main point of our proof below is to exploit a cyclic symmetry of theterms contributing to Pf (cid:0) Q ( a, b ) (cid:1) . This reduces the computation of the Pfaffian tothat of a sub-Pfaffian of lower order.Let S ( π ; a, b ) denote the summand of Pf (cid:0) Q ( a, b ) (cid:1) , that is,Pf (cid:0) Q ( a, b ) (cid:1) = X π S ( π ; a, b ) with S ( π ; a, b ) = ( − c ( π ) Y ( i,j ) ∈ π Q ij ( a, b ) . From the definition (3.4) of Q ij ( a, b ) and the fact that π is a perfect matching on[ n + 1],(3.5) S ( π ; a, b ) = ( − c ( π ) Y ( i,j ) ∈ π a i a j τ ij = ( − c ( π ) a · · · a n +1 Y ( i,j ) ∈ π τ ij . We now observe that S ( π ; a, b ) is, up to a cyclic permutation of b , invariant underthe permutation w given by (1 , , , . . . , n, n + 1) ( n, , , . . . , n − , n + 1). To seethis, denote by π ′ the image of π under w . For example, the image of the perfectmatching given on the previous page is1 2 3 4 5 6 7 8Under the permutation w , all edges not containing the vertices 1 or n +1 are shiftedone unit to the left: ( i, j ) ( i − , j − , j ) containing vertex 1we have:(i) If j ≤ n then (1 , j ) ( j − , n ). This also implies that the edge ( j ′ , n + 1)( j ′ ≥ 2) containing vertex n + 1 maps to ( j ′ − , n + 1).(ii) If j = n + 1 then (1 , j ) = (1 , n + 1) ( n, n + 1) = ( j − , n + 1).First we consider (i). If we remove the edge (1 , j ) from π and carry out w , thenthe number of crossings of its image is exactly that of π . Hence we only need tofocus on the edge (1 , j ) and its image under w . In π the edge (1 , j ) has crossingnumber c (1 , j ) ≡ j (mod 2), while the edge ( j − , n ) in π ′ has crossing number c ( j − , n ) ≡ n − j ≡ j + 1 (mod 2). Hence ( − c ( π ) = − ( − c ( π ′ ) . Since τ ij = τ i − ,j − (for 2 ≤ i < j ≤ n ) and τ ,j = − τ j − ,n it thus follows that π and π ′ havethe same sign. Finally we note that under w , b i = τ i,n +1 τ i − ,n +1 = b i − (since i = 1). We thus conclude that(3.6) S (cid:0) π ; a, ( b , . . . , b n ) (cid:1) S (cid:0) π ′ ; a, ( b , . . . , b n , b ) (cid:1) , where we note that both sides depend on a single b i ( = b ) only. For example, theperfect matching in the above two figures correspond to S (cid:0) (1 , , (2 , , (4 , , (6 , a, ( b , . . . , b ) (cid:1) = ( − · a a · ( − a a ) · a a · a a b = − a · · · a b and S (cid:0) (1 , , (2 , , (3 , , (5 , a, ( b , . . . , b ) (cid:1) = ( − · ( − a a ) · ( − a a ) · a a × a a b = − a · · · a b . OGARITHMIC AND COMPLEX CONSTANT TERM IDENTITIES 9 The case (ii) is even simpler; the edge (1 , n + 1) in π and its image ( n, n + 1) in π ′ both have crossing number 0. The crossing numbers of all other edges do notchange by a global shift of one unit to the right, so that c ( π ) = c ( π ′ ): w Moreover, τ ij = τ i − ,j − (for 2 ≤ i < j ≤ n ) so that π and π ′ again have the samesign. Finally, from b = τ ,n +1 τ n,n +1 = b n it follows that once again (3.6) holds,where this time both sides depend only on b .From (3.6) it follows that the Pfaffian Pf (cid:0) Q ( a, b ) (cid:1) is symmetric under cyclicpermutations of the b i . But since the Pfaffian, viewed as a function of b , has degree1 it thus follows (see also (3.5)) thatPf (cid:0) Q ( a, b ) (cid:1) = Ca · · · a n +1 ( b + · · · + b n )for some yet-unknown constant C . We shall determine C by computing the coef-ficient of b n of Pf (cid:0) Q ((1 n +1 ) , b (cid:1) , which is equal to the Pfaffian of the (2 m ) × (2 m )submatrix M of T obtained by deleting its last row and column.We recall the property Pf( M ) = Pf( U t M U ) of Pfaffians, where U is a unipotenttriangular matrix [35]. Choosing the non-zero entries of the (2 m ) × (2 m ) matrix U to be U ii = 1 for i = 1 , . . . , m , and U i,i + m = 1 for i = 1 , . . . , m , one transforms M into M ′ I − I ∅ ! , where M ′ is the upper-left m × m submatrix of M and I is the m × m identitymatrix. The Pfaffian of the above matrix, and hence that of M , is exactly (cf. [35])( − m ) det( I ) = ( − m ). This, in turn, implies that C = ( − m ), and therequired formula follows. (cid:3) Remark . By a slight modification of the above proof the following more generalPfaffian results. Let Q ij ( X, a, b ) := τ ij a i a j ( x i + x j ) for 1 ≤ i < j ≤ n and Q i,n +1 ( X, a, b ) := τ i,n +1 a i a n +1 = a i a n +1 b i for 1 ≤ i ≤ n, and use this to form the ( n + 1) × ( n + 1) skew-symmetric matrix Q ( X, a, b ). ThenPf (cid:0) Q ( X, a, b ) (cid:1) = 2 m − ( − m ) a a · · · a n +1 n X i =1 b i ( x i +1 · · · x i + m + x i + m +1 · · · x i + n − ) , where x i + n := x i for i > n . For X = (1 / , . . . , / 2) this yields Proposition 3.2. The complex Morris constant term identity Thanks to Lemma 3.1, Y ≤ i = j ≤ n (cid:16) − x i x j (cid:17) = Y ≤ i For a, b nonnegative integers and Re(1 + u ) > , (4.3) CT (cid:20) (1 − x ) a (1 − y ) a (1 − z ) a (cid:16) − x (cid:17) b (cid:16) − y (cid:17) b (cid:16) − z (cid:17) b × (cid:16) − xy (cid:17) u (cid:16) − yz (cid:17) u (cid:16) − zx (cid:17) u (cid:21) = cos (cid:0) πu (cid:1) Γ(1 + u )Γ (1 + u ) Y i =0 (1 + iu ) a + b (1 + iu ) a (1 + iu ) b . As follows from its proof, a slightly more general result in fact holds. Using( z ) n + m = ( z ) n ( z + n ) m and (1 − x ) a (1 − x − ) b = ( − x ) − b (1 − x ) a + b , then replacing a a − b , and finally using ( z − b ) b = ( − b (1 − z ) b , the identity (4.3) can also be OGARITHMIC AND COMPLEX CONSTANT TERM IDENTITIES 11 stated as(4.4) (cid:2) x b y b z b (cid:3)(cid:20) (1 − x ) a (1 − y ) a (1 − z ) a (cid:16) − xy (cid:17) u (cid:16) − yz (cid:17) u (cid:16) − zx (cid:17) u (cid:21) = cos (cid:0) πu (cid:1) Γ(1 + u )Γ (1 + u ) Y i =0 ( − a − iu ) b (1 + iu ) b , where (cid:2) X c ] f ( X ) (with X c = x c · · · x c n n ) denotes the coefficient of X c in f ( X ). Thisalternative form of (4.3) is true for all a, u ∈ C such that Re(1 + u ) > Conjecture 4.2 ( Complex Morris constant term identity ) . Let n be an oddpositive integer, a, b nonnegative integers and u ∈ C such that Re(1 + nu ) > .Then there exists a polynomial P n ( x ) , independent of a and b , such that P n (0) =1 / ( n − , P n (1) = 1 , and (4.5) CT (cid:20) n Y i =1 (1 − x i ) a (cid:16) − x i (cid:17) b Y ≤ i Instead of proving (4.3) we establish the slightly moregeneral (4.4).By a six-fold use of the binomial expansion (1.5), the constant term identity(4.4) can be written as the following combinatorial sum: ∞ X m ,m ,m =0 2 Y i =0 ( − m i (cid:18) um i (cid:19)(cid:18) ab + m i − m i +1 (cid:19) = cos (cid:0) πu (cid:1) Γ(1 + u )Γ (1 + u ) Y i =0 ( − a − iu ) b (1 + iu ) b , where m := m and where a, u ∈ C such that Re(1+ u ) > b is a nonnegativeinteger. If we denote the summand of this identity by f b ( u, − − a ; m ) where m := ( m , m , m ), then we need to prove that(4.6) F b ( u, v ) := X m ∈ Z f b ( u, v ; m ) = cos (cid:0) πu (cid:1) Γ(1 + 3 u )Γ (1 + u ) Y i =0 (1 + v − iu ) b (1 + iu ) b , for Re(1 + 3 u ) > u and v . In particular we write F b and f b ( m ) for F b ( u, v ) and f b ( u, v ; m ).The function f ( m ) vanishes unless m = m = m . Hence F = ∞ X m =0 ( − m (cid:18) um (cid:19) = F (cid:20) − u, − u, − u , (cid:21) , where we adopt standard notation for (generalised) hypergeometric series, see e.g.,[3, 5]. The F series is summable by the 2 a = b = c = − u case of Dixon’ssum [3, Eq. (2.2.11)](4.7) F (cid:20) a, b, c a − b, a − c ; 1 (cid:21) = Γ(1 + a )Γ(1 + 2 a − b )Γ(1 + 2 a − c )Γ(1 + a − b − c )Γ(1 + 2 a )Γ(1 + a − b )Γ(1 + a − c )Γ(1 + 2 a − b − c ) OGARITHMIC AND COMPLEX CONSTANT TERM IDENTITIES 13 for Re(1 + a − b − c ) > 0. As a result, F = Γ(1 − u )Γ(1 + u )Γ(1 − u )Γ(1 + 2 u ) · Γ(1 + 3 u )Γ (1 + u ) = cos( πu ) Γ(1 + 3 u )Γ (1 + u ) , proving the b = 0 instance of (4.6).In the remainder we assume that b ≥ C be the generator of the cyclic group C acting on m as C ( m ) = ( m , m , m ).With the help of the multivariable Zeilberger algorithm [4], one discovers the (hu-manly verifiable) rational function identity(4.8) t b ( m ) Y i =0 ( b + iu ) + Y i =0 ( b + v − iu )= X i =0 (cid:16) r b (cid:0) e + C i ( m ) (cid:1) s b (cid:0) C i ( m ) (cid:1) + r b (cid:0) C i ( m ) (cid:1)(cid:17) , where r b ( m ) = − m ( b + v + m − m )6( b + m − m )( b + m − m ) × (cid:0) (2 b + v )(3 b + 3 bv + 2 uv ) + 2( m − m )(3 b + 3 bv + v − uv ) (cid:1) ,s b ( m ) = − f b − ( e + m ) f b − ( m ) = (2 u − m )( b + v + m − m )( b + m − m − m )( b + m − m )( b + v + m − m − ,t b ( m ) = − f b ( m ) f b − ( m ) = Y i =0 b + v + m i − m i +1 b + m i − m i +1 , and e + m := (1 + m , m , m ). If we multiply (4.8) by − f b − ( m ) and use that f b ( m ) = f b ( C i ( m )) we find that f b ( m ) Y i =0 ( b + iu ) − f b − ( m ) Y i =0 ( b + v − iu )= X i =0 h r b (cid:0) e + C i ( m ) (cid:1) f b − (cid:0) e + C i ( m ) (cid:1) − r b (cid:0) C i ( m ) (cid:1) f b − (cid:0) C i ( m ) (cid:1)i , Summing this over m ∈ Z the right-hand side telescopes to zero, resulting in F b = F b − Y i =0 ( b + v − iu )( b + iu ) . By b -fold iteration this yields F b = F Y i =0 (1 + v − iu ) b (1 + iu ) b . (cid:3) The logarithmic Morris constant term identity This section contains three parts. In the first very short part, we present anintegral analogue of the logarithmic Morris constant term identity. This integralmay be viewed as a logarithmic version of the well-known Morris integral. The sec-ond and third, more substantial parts, contain respectively a proof of Theorem 1.3 and, exploiting some further results of Adamovi´c and Milas, a strengthening of thistheorem.5.1. A logarithmic Morris integral. By a repeated use of Cauchy’s integralformula, constant term identities such as (1.4) or (1.9) may be recast in the formof multiple integral evaluations. In the case of (1.4) this leads to the well-knownMorris integral [10, 27] Z [ − π, π ] n n Y i =1 e i( a − b ) θ i sin a + b ( θ i ) Y ≤ i In this subsection we prove that the logarithmicMorris constant term identity (1.9) is nothing but the m th derivative of the complexMorris constant term identity (4.5) evaluated at u = K := 2 k + 1.To set things up we first prepare a technical lemma. For S n the symmetric groupon n letters and w ∈ S n , we denote by sgn( w ) the signature of the permutation w ,see e.g., [26]. The identity permutation in S n will be written as 1I. Lemma 5.1. For n an odd integer, set m := ( n − / . Let t ij for ≤ i < j ≤ n +1 be a collection of signatures ( i.e., each t ij is either +1 or − such that t i,n +1 = 1 ,and ˜ Q a skew-symmetric matrix with entries ˜ Q ij = t ij for ≤ i < j ≤ n + 1 .If f ( X ) is a skew-symmetric polynomial in X = ( x , . . . , x n ) , g ( z ) a Laurentpolynomial or Laurent series in the scalar variable z , and g ij ( X ) := g (( x i /x j ) t ij ) ,then the following statements hold. (1) For w ∈ S n , denote g ( w ; X ) := Q mk =1 g ( x w k − /x w k ) . Then CT (cid:2) f ( X ) g ( w ; X ) (cid:3) = sgn( w ) CT (cid:2) f ( X ) g (1I; X ) (cid:3) . (2) For π a perfect matching on [ n + 1] , (5.1) X π CT (cid:20) f ( X ) Y ( i,j ) ∈ πj = n +1 g ij ( X ) (cid:21) = Pf( ˜ Q ) CT (cid:2) f ( X ) g (1I; X ) (cid:3) . OGARITHMIC AND COMPLEX CONSTANT TERM IDENTITIES 15 We will be needing a special case of this lemma corresponding to t ij = τ ij for1 ≤ i, j ≤ n , with the τ ij defined in (3.1). Then the matrix ˜ Q is coincides with Q (cid:0) (1 n +1 ) , (1 n ) (cid:1) of (3.4), so that by Lemma 3.2, Pf( ˜ Q ) = ( − m ) n . We summarisethis in the following corollary. Corollary 5.2. If in Lemma 5.1 we specialise t ij = τ ij for ≤ i < j ≤ n , then (5.2) X π CT (cid:20) f ( X ) Y ( i,j ) ∈ πj = n +1 g ij ( X ) (cid:21) = ( − m ) n CT (cid:2) f ( X ) g (1I; X ) (cid:3) . Proof of Lemma 5.1. (1) According to the “Stanton–Stembridge trick” [33, 34, 39],CT (cid:2) h ( X ) (cid:3) = CT (cid:2) w (cid:0) h ( X ) (cid:1)(cid:3) for w ∈ S n , where w ( h ( X )) is shorthand for h ( x w , . . . , x w n ).For our particular choice of h , the skew-symmetric factor f ( X ) produces theclaimed sign.(2) A permutation w ∈ S n may be interpreted as a signed perfect matching( − d ( w ) ( w , w ) · · · ( w n − , w n − )( w n , w n +1 ), where d ( w ) counts the number |{ k ≤ m : w k − > w k }| . By claim (1), the left hand-side of (5.1) is a multiple ofCT (cid:2) f ( X ) g (1I; X ) (cid:3) ; the factor is exactly the sum P π ( − c ( π ) Q t ij , in which onerecognises the Pfaffian of ˜ Q . (cid:3) Conditional proof of (1.9) . Suppressing the a and b dependence, denote the leftand right-hand sides of (4.5) by L n ( u ) and R n ( u ) respectively. We then wish toshow that (1.9) is identical to L ( m ) n ( K ) = R ( m ) n ( K ) . Let us first consider the right hand side, which we write as R n ( u ) = p n ( u ) r n ( u ),where p n ( u ) = x m P n ( x ) , x = x ( u ) = cos (cid:0) πu (cid:1) and(5.3) r n ( u ) = Γ(1 + nu )Γ n (1 + u ) n − Y i =0 (1 + iu ) a + b (1 + iu ) a (1 + iu ) b . Since x ( K ) = 0, it follows that for 0 ≤ j ≤ m ,(5.4) p ( j ) n ( K ) = ( − km + m (cid:16) π (cid:17) m m !( n − δ jm . Therefore, since r n ( u ) is m times differentiable at u = K ,(5.5) R ( m ) n ( K ) = p ( m ) n ( K ) r n ( K ) . By the functional equation for the gamma function(5.6) Γ(1 + N ) = N !! 2 − N/ p π/ N > ,N !! 2 − N/ if N ≥ , and, consequently,(5.7) (1 + N ) a = ( N + 2 a )!!2 a N !!6 TOM CHAPPELL, ALAIN LASCOUX, S. OLE WARNAAR, AND WADIM ZUDILIN If in Lemma 5.1 we specialise t ij = τ ij for ≤ i < j ≤ n , then (5.2) X π CT (cid:20) f ( X ) Y ( i,j ) ∈ πj = n +1 g ij ( X ) (cid:21) = ( − m ) n CT (cid:2) f ( X ) g (1I; X ) (cid:3) . Proof of Lemma 5.1. (1) According to the “Stanton–Stembridge trick” [33, 34, 39],CT (cid:2) h ( X ) (cid:3) = CT (cid:2) w (cid:0) h ( X ) (cid:1)(cid:3) for w ∈ S n , where w ( h ( X )) is shorthand for h ( x w , . . . , x w n ).For our particular choice of h , the skew-symmetric factor f ( X ) produces theclaimed sign.(2) A permutation w ∈ S n may be interpreted as a signed perfect matching( − d ( w ) ( w , w ) · · · ( w n − , w n − )( w n , w n +1 ), where d ( w ) counts the number |{ k ≤ m : w k − > w k }| . By claim (1), the left hand-side of (5.1) is a multiple ofCT (cid:2) f ( X ) g (1I; X ) (cid:3) ; the factor is exactly the sum P π ( − c ( π ) Q t ij , in which onerecognises the Pfaffian of ˜ Q . (cid:3) Conditional proof of (1.9) . Suppressing the a and b dependence, denote the leftand right-hand sides of (4.5) by L n ( u ) and R n ( u ) respectively. We then wish toshow that (1.9) is identical to L ( m ) n ( K ) = R ( m ) n ( K ) . Let us first consider the right hand side, which we write as R n ( u ) = p n ( u ) r n ( u ),where p n ( u ) = x m P n ( x ) , x = x ( u ) = cos (cid:0) πu (cid:1) and(5.3) r n ( u ) = Γ(1 + nu )Γ n (1 + u ) n − Y i =0 (1 + iu ) a + b (1 + iu ) a (1 + iu ) b . Since x ( K ) = 0, it follows that for 0 ≤ j ≤ m ,(5.4) p ( j ) n ( K ) = ( − km + m (cid:16) π (cid:17) m m !( n − δ jm . Therefore, since r n ( u ) is m times differentiable at u = K ,(5.5) R ( m ) n ( K ) = p ( m ) n ( K ) r n ( K ) . By the functional equation for the gamma function(5.6) Γ(1 + N ) = N !! 2 − N/ p π/ N > ,N !! 2 − N/ if N ≥ , and, consequently,(5.7) (1 + N ) a = ( N + 2 a )!!2 a N !!6 TOM CHAPPELL, ALAIN LASCOUX, S. OLE WARNAAR, AND WADIM ZUDILIN for any nonnegative integer N . Applying these formulae to (5.3) with u = K , wefind that(5.8) r n ( K ) = (cid:16) π (cid:17) m n − Y i =0 (2 a + 2 b + iK )!!(( i + 1) K )!!(2 a + iK )!!(2 b + iK )!! K !! . Combined with (5.4) and (5.5) this implies(5.9) R ( m ) n ( K ) = ( − ( k +1) m m !( n − n − Y i =0 (2 a + 2 b + iK )!!(( i + 1) K )!!(2 a + iK )!!(2 b + iK )!! K !! . Next we focus on the calculation of L ( m ) n ( K ). To keep all equations in check wedefine f ab ( X ) := n Y i =1 (1 − x i ) a (cid:16) − x i (cid:17) b . and(5.10) F ab ( X ) := ∆( X ) n Y i =1 x − mi (1 − x i ) a (cid:16) − x i (cid:17) b Y ≤ i = j ≤ n (cid:16) − x i x j (cid:17) k . Let i := ( i , . . . , i m ) and j := ( j , . . . , j m ). Then, by a straightforward applica-tion of the product rule,(5.11) L ( m ) n ( u ) = X ≤ i A strengthening of Theorem 1.3. As will be described in more detailbelow, using some further results of Adamovi´c and Milas, it follows that the loga-rithmic Morris constant term identity (1.9) holds provided it holds for a = b = 0,i.e., provided the logarithmic analogue (1.2) of Dyson’s identity holds. The proofof Theorem 1.3 given in the previous subsection implies that the latter follows fromwhat could be termed the complex analogue of Dyson’s identity, i.e., the a = b = 0case of (4.5):(5.12) CT (cid:20) n Y i =1 Y ≤ i Theorem 5.3 ( Logarithmic Morris constant term identity, strong ver-sion ) . The complex Dyson constant term identity (5.12) implies the logarithmicMorris constant term identity. To justify this claim, let e r ( X ) for r = 0 , , . . . , n denote the r th elementarysymmetric function. The e r ( X ) have generating function [26](5.13) n X r =0 z r e r ( X ) = n Y i =1 (1 + zx i ) . Recalling definition (5.10) of F ab , we now define f r ( a ) = f r ( a, b, k, n ) by f r ( a ) = CT (cid:2) ( − r e r ( X ) G ab ( X ) (cid:3) , where G ab ( X ) = F ab ( X ) m Y i =1 log (cid:16) − x i x i − (cid:17) . In the following b may be viewed as a formal or complex variable, but a must betaken to be an integer.From (5.13) with z = − n X r =0 f r ( a ) = CT (cid:2) G a +1 ,b ( X ) (cid:3) = f ( a + 1) . According to [2, Theorem 7.1] (translated into the notation of this paper) we alsohave(5.15) ( n − r )(2 b + rK ) f r ( a ) = ( r + 1)(2 a + 2 + ( n − r − K ) f r +1 ( a ) , where we recall that K := 2 k + 1. Iterating this recursion yields f r ( a ) = f ( a ) (cid:18) nr (cid:19) r − Y i =0 b + iK a + 2 + ( n − i − K . Summing both sides over r and using (5.14) leads to f ( a + 1) = f ( a ) F (cid:20) − n, b/K − n − (2 a + 2) /K ; 1 (cid:21) . The F series sums to ((2 a + 2 b + 2) /K ) n / ((2 a + 2) K ) n by the Chu–Vandermondesum [3, Corollary 2.2.3]. Therefore, f ( a + 1) = f ( a ) n − Y i =0 a + 2 b + 2 + iK a + 2 + iK . This functional equation can be solved to finally yield f ( a ) = f (0) n − Y i =0 (2 a + 2 b + iK )!!( iK )!!(2 b + iK )!!(2 a + iK )!! . To summarise the above computations, we have established thatCT (cid:2) G ab ( X ) (cid:3) = CT (cid:2) G ,b ( X ) (cid:3) n − Y i =0 (2 a + 2 b + iK )!!( iK )!!(2 b + iK )!!(2 a + iK )!! . But since G , ( X ) is homogeneous of degree 0 it follows thatCT (cid:2) G ,b ( X ) (cid:3) = CT (cid:2) G , ( X ) (cid:3) , so that indeed the logarithmic Morris constant term identity is implied by its a = b = 0 case.We finally remark that it seems highly plausible that the recurrence (5.15) hasthe following analogue for the complex Morris identity (enhanced by the term( − r e r ( X ) in the kernel):( n − r )(2 b + ru ) f r ( a ) = ( r + 1)(2 a + 2 + ( n − r − u ) f r +1 ( a ) . However, the fact that for general complex u the kernel is not a skew-symmetricfunction seems to prevent the proof of [2, Theorem 7.1] carrying over to the complexcase in a straightforward manner. OGARITHMIC AND COMPLEX CONSTANT TERM IDENTITIES 19 The root system G In this section we prove complex and logarithmic analogues of the Habsieger–Zeilberger identity (2.4). Theorem 6.1 ( Complex G constant term identity ) . For u, v ∈ C such that Re(1 + u ) > and Re(1 + ( u + v )) > , (6.1) CT (cid:20)(cid:16) − yzx (cid:17) u (cid:16) − xzy (cid:17) u (cid:16) − xyz (cid:17) u (cid:16) − xy (cid:17) v (cid:16) − yz (cid:17) v (cid:16) − zx (cid:17) v (cid:21) = cos (cid:0) πu (cid:1) cos (cid:0) πv (cid:1) Γ(1 + ( u + v ))Γ(1 + u )Γ(1 + u )Γ(1 + v )Γ(1 + u + v )Γ(1 + u + v )Γ(1 + ( u + v ))Γ (1 + u )Γ(1 + v ) . Proof. We adopt the method of proof employed by Habsieger and Zeilberger [15,38]in their proof of Theorem 2.1.If A ( x, y, z ; a, u ) denotes the kernel on the left of the complex Morris identity(4.4) for n = 3, and if and G ( x, y, z ; u, v ) denotes the kernel on the left of (6.1),then G ( x, y, z ; u, v ) = A ( x/y, y/z, z/x, v, u ) . Therefore,CT G ( x, y, z ; u, v ) = CT A ( x/y, y/z, z/x ; v, u )= CT A ( x, y, z ; v, u ) (cid:12)(cid:12) xyz =1 = ∞ X b =0 (cid:2) x b y b z b (cid:3) A ( x, y, z ; v, u )= cos (cid:0) πu (cid:1) Γ(1 + u )Γ (1 + u ) F (cid:20) − v, − u − v, − u − v u, u ; 1 (cid:21) , where the last equality follows from (4.4). Summing the F series by Dixon’s sum(4.7) with (2 a, b, c ) ( − v, − u − v, − u − v ) completes the proof. (cid:3) Just as for the root system A n − , we can use the complex G constant termidentity to proof a logarithmic analogue of (2.4). Theorem 6.2. Assume the representation of the G root system as given in Sec-tion 2, and let Φ + s and Φ + l denote the set of positive short and positive long rootsrespectively. Define G ( K, M ) = 13 (3 K + 3 M )!!(3 K )!!(2 K )!!(2 M )!!(3 K + 2 M )!!(2 K + M )!!( K + M )!! K !! K !! M !! . Then for k, m nonnegative integers, CT (cid:20) e − α − α log(1 − e α ) Y α ∈ Φ + l (1 − e α ) Y α ∈ Φ l (1 − e α ) k Y α ∈ Φ s (1 − e α ) m (cid:21) = G ( K, M ) , where ( K, M ) := (2 k + 1 , m ) , and CT (cid:20) e − α − α log(1 − e α ) Y α ∈ Φ + s (1 − e α ) Y α ∈ Φ l (1 − e α ) k Y α ∈ Φ s (1 − e α ) m (cid:21) = G ( K, M ) , where ( K, M ) := (2 k, m + 1) . We can more explicitly write the kernels of the two logarithmic G constant termidentities as z xy (cid:16) − x yz (cid:17)(cid:16) − y xz (cid:17)(cid:16) − xyz (cid:17) log (cid:16) − y xz (cid:17) × (cid:18)(cid:16) − x yz (cid:17)(cid:16) − y xz (cid:17)(cid:16) − z xy (cid:17)(cid:16) − yzx (cid:17)(cid:16) − xzy (cid:17)(cid:16) − xyz (cid:17)(cid:19) k × (cid:18)(cid:16) − xy (cid:17)(cid:16) − xz (cid:17)(cid:16) − yx (cid:17)(cid:16) − yz (cid:17)(cid:16) − zx (cid:17)(cid:16) − zy (cid:17)(cid:19) m and zx (cid:16) − xy (cid:17)(cid:16) − yz (cid:17)(cid:16) − xz (cid:17) log (cid:16) − xy (cid:17) × (cid:18)(cid:16) − x yz (cid:17)(cid:16) − y xz (cid:17)(cid:16) − z xy (cid:17)(cid:16) − yzx (cid:17)(cid:16) − xzy (cid:17)(cid:16) − xyz (cid:17)(cid:19) k × (cid:18)(cid:16) − xy (cid:17)(cid:16) − xz (cid:17)(cid:16) − yx (cid:17)(cid:16) − yz (cid:17)(cid:16) − zx (cid:17)(cid:16) − zy (cid:17)(cid:19) m respectively. Proof of Theorem 6.2. If we differentiate (6.1) with respect to u , use the cyclicsymmetry in ( x, y, z ) of the kernel on the left, and finally set u = 2 k + 1 = K , weget3 CT (cid:20) log (cid:16) − xzy (cid:17)(cid:16) − yzx (cid:17) K (cid:16) − xzy (cid:17) K (cid:16) − xyz (cid:17) K (cid:16) − xy (cid:17) v (cid:16) − yz (cid:17) v (cid:16) − zx (cid:17) v (cid:21) = ( − k +1 π (cid:0) πv (cid:1) Γ(1 + ( K + v ))Γ(1 + K )Γ(1 + K )Γ(1 + v )Γ(1 + K + v )Γ(1 + K + v )Γ(1 + ( K + v ))Γ (1 + K )Γ(1 + v ) . Setting v = 2 m = M and carrying out some simplifications using (5.6) and (5.7)completes the proof of the first claim.In much the same way, if we differentiate (6.1) with respect to v , use the cyclicsymmetry in ( x, y, z ) and set v = 2 m + 1 = M , we get3 CT (cid:20) log (cid:16) − xy (cid:17)(cid:16) − yzx (cid:17) u (cid:16) − xzy (cid:17) u (cid:16) − xyz (cid:17) u (cid:16) − xy (cid:17) M (cid:16) − yz (cid:17) M (cid:16) − zx (cid:17) M (cid:21) = ( − m +1 π (cid:0) πu (cid:1) Γ(1 + ( u + M ))Γ(1 + u )Γ(1 + u )Γ(1 + M )Γ(1 + u + M )Γ(1 + u + M )Γ(1 + ( u + M ))Γ (1 + u )Γ(1 + M ) . Setting u = 2 k = K yields the second claim. (cid:3) Other root systems Although further root systems admit complex analogues of the Macdonald con-stant term identities (1.3) or (2.3), it seems the existence of elegant logarithmicidentities is restricted to A n and G . To see why this is so, we will discuss theroot systems B n , C n and D n . In order to treat all three simultaneously, it will beconvenient to consider the more general non-reduced root system BC n . With ǫ i again denoting the i th standard unit vector in R n , this root system is given byΦ = {± ǫ i : 1 ≤ i ≤ n } ∪ {± ǫ i : 1 ≤ i ≤ n } ∪ {± ǫ i ± ǫ j : 1 ≤ i < j ≤ n } . OGARITHMIC AND COMPLEX CONSTANT TERM IDENTITIES 21 Using the Selberg integral, Macdonald proved that [25](7.1) CT (cid:20) n Y i =1 (1 − x ± i ) a (1 − x ± i ) b Y ≤ i For n ≡ define m := ( n − / and p := m/ . If we choose τ ij as in (3.1) and σ ij , ≤ i < j ≤ n , as (7.5) σ ij = ( − if p < j − i ≤ p, otherwise , then (7.4) , and thus (7.2) , is satisfied. We can extend the definition of σ ij to all 1 ≤ i, j ≤ n by setting σ ij = − σ ji . Thenthe matrix Σ = ( σ ij ) ≤ i,j ≤ n is a skew-symmetric Toeplitz matrix. For example, for n = 5 the above choice for the σ ij generatesΣ = − − − − − − − 11 1 − − − . Proof of Lemma 7.1. Note that by Lemma 3.1 we only need to prove that n X j = i +1 σ ij + i − X j =1 σ ji = 0 . If for 1 ≤ j ≤ i − σ i,n + j := − σ ij = σ ji then this becomes(7.6) n + i − X j = i +1 σ ij = 0 . We now observe that σ i +1 ,j +1 = σ i,j . For j < n or j > n this follows immediatelyfrom (7.5). For j = n it follows from σ ,i +1 = σ i,n , which again follows from (7.5)since p < n − i ≤ p is equivalent to p < i ≤ p . Thanks to σ i +1 ,j +1 = σ i,j we onlyneed to check (7.6) for i = 1. Then n X j =2 σ ij = p +1 X j =2 − p +1 X j = p +2 n X j =3 p +2 n − p − . (cid:3) Lemma 7.2. For n ≡ define m := ( n − / . If we choose τ ij as in (3.1) and σ ij as σ ij = ( if i + j is even or i + j = m + 2 , − if i + j is odd and i + j = m + 2 , then (7.4) , and thus (7.2) , is satisfied.Proof. By a simple modification of Lemma 3.1 it follows that for n even and m =( n − / n X j = i +1 τ ij − i − X j =1 τ ji = ( − ≤ i ≤ m + 1 , m + 1 < i < n. Hence we must show that n X j = i +1 σ ij + i − X j =1 σ ji = ( ≤ i ≤ m + 1 , − m + 1 < i < n. OGARITHMIC AND COMPLEX CONSTANT TERM IDENTITIES 23 But this is obvious. The sum on the left is over n − i + j then even i + j . Hence, without the exceptional condition on i + j = m + 2,the sum would always be − 1. To have i + j = m + 2 as part of one of the two sumswe must have i ≤ m + 1, in which case one − − (cid:3) Lemmas 7.1 and 7.2 backed up with numerical data for n = 4 and n = 5 suggestthe following generalisation of (7.3). Conjecture 7.3 ( Complex BC n constant term identity ) . Let n ≡ ζ (mod 4) where ζ = 0 , , and let u ∈ C such that min { Re(1+2 b +( n − u ) , Re(1+ nu ) } > .Assume that τ ij and σ ij for ≤ i < j ≤ n are signatures satisfying (7.4) . 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