Matching Tag: constant-term-identities
Let a = ( a 0 , … , a n ) a=(a_0,\dots,a_n) a = ( a 0 , … , a n ) x ( a ) x^{(a)} x ( a ) ∣ a ∣ = a 0 + ⋯ + a n |a|=a_0+\cdots+a_n ∣ a ∣ = a 0 + ⋯ + a n v = ( v 0 , … , v n ) v=(v_0,\dots,v_n) v = ( v 0 , … , v n )
Let p p p D 2 D_2 D 2 t t t Res x 0 , x 1 , x 2 , x 3 , x 4 \operatorname{Res}_{x_0,x_1,x_2,x_3,x_4} Res x 0 , x 1 , x 2 , x 3 , x 4
Let m > 2 m>2 m > 2 p p p D m D_m D m t t t Res x 0 , x 1 , … , x m + 2 \operatorname{Res}_{x_0,x_1,\ldots,x_{m+2}} Res x 0 , x 1 , … , x m + 2
Let m m m p p p A m A_m A m t t t Res x 0 , x 1 , … , x m + 1 \operatorname{Res}_{x_0,x_1,\ldots,x_{m+1}} Res x 0 , x 1 , … , x m + 1
Let h p , m ( t ) h_{p,m}(t) h p , m ( t ) h ~ p , m ( t ) \tilde{h}_{p,m}(t) h ~ p , m ( t ) H p , m ( t ) H_{p,m}(t) H p , m ( t ) H ~ p , m ( t ) \tilde{H}_{p,m}(t) H ~ p , m ( t ) D m D_m D m
Let n n n a , b , k a,b,k a , b , k m , n 0 ≤ n ≤ m + n 0 m,n_0\leq n\leq m+n_0 m , n 0 ≤ n ≤ m + n 0
Let n n n M ⊂ { 1 , 2 , … , n } M\subset\{1,2,\dots,n\} M ⊂ { 1 , 2 , … , n } s ∈ M s\in M s ∈ M r s r_s r s r s ∉ M r_s\notin M r s ∈ / M a = ( a 1 , … , a n ) \boldsymbol{a}=(a_1,\dots,a_n) a = ( a 1 , … , a n )
Let n ≡ ζ ( m o d 4 ) n\equiv\zeta\pmod 4 n ≡ ζ ( mod 4 ) ζ ∈ { 0 , 1 } \zeta\in\{0,1\} ζ ∈ { 0 , 1 } u ∈ C u\in\mathbb C u ∈ C τ i j \tau_{ij} τ ij σ i j \sigma_{ij} σ ij
Adamović–Milas logarithmic constant term conjecture. For odd positive integer n n n k k k m = ( n − 1 ) / 2 m=(n-1)/2 m = ( n − 1 ) /2 K = 2 k + 1 K=2k+1 K = 2 k + 1
Constant-term conjecture. This constant term equals
Let I = { i 1 , … , i m } I=\{i_1,\ldots,i_m\} I = { i 1 , … , i m } { 0 , 1 , … , n } \{0,1,\ldots,n\} { 0 , 1 , … , n } J = { j 1 , … , j m } J=\{j_1,\ldots,j_m\} J = { j 1 , … , j m } { 0 , 1 , … , n } ∖ I \{0,1,\ldots,n\}\setminus I { 0 , 1 , … , n } ∖ I
Let D p ( n 1 , … , n p ; n 0 ; a , b , k ; q ) D_p(n_1,\ldots,n_p;n_0;a,b,k;q) D p ( n 1 , … , n p ; n 0 ; a , b , k ; q ) J ~ α \widetilde J_\alpha J α [ r ] q ! [r]_q! [ r ] q ! q q q
Let D 1 ( n 0 ; n 1 ; a , b ; q ) D_1(n_0;n_1;a,b;q) D 1 ( n 0 ; n 1 ; a , b ; q ) CT \operatorname{CT} CT ( x ; q ) k (x;q)_k ( x ; q ) k q q q Γ q \Gamma_q Γ q
Let k k k m m m x 1 , … , x 2 k + 1 x_1,\ldots,x_{2k+1} x 1 , … , x 2 k + 1 CT x 1 , … , x 2 k + 1 \operatorname{CT}_{x_1,\ldots,x_{2k+1}} CT x 1 , … , x 2 k + 1
