80
J. Harnad
G(z) = H q (z) :=
∞
i=0
(1 − q
i z)
−1
=
∞
n=0
z n
(q; q) n
,
(8)
where
(q; q) n := (1 − q)(1 − q
2 ) · · · (1 − q
n )
(9)
for some parameter q, with |q| < 1.
The corresponding rationally weighted (single) Hurwitz numbers are
H
d
G c,d
(μ, ν) :=
1k,l
k+l
μ (1) ,...μ (k) ,ν (1) ,...ν (l) ,
k
i=1 ∗ (μ (i) )+
l
j =1 ∗ (ν (j ) )=d
|μ (i) |=|ν (j ) |=N
W G c,d (μ
(1) , . . . , μ
(k)
; ν
(1) , . . . , ν
(l) )
× H (μ
(1) , . . . , μ
(k) , ν
(1) , . . . , ν
(l) , μ),
where the rational weight factor is
W G c,d (μ
(1) , . . . , μ
(k)
; ν
(1) , . . . , ν
(l) )
:=
(−1)
l
j =1 ∗ (ν (j ) )−l
k!l!
σ ∈S k
σ ∈S l
1≤a 1 <··· 1≤b 1 ···≤b k ≤L
c
∗ (μ (1) )
a σ (1)
· · · c
∗ (μ (k) )
a σ (k)
d
∗ (ν (1) )
b σ (1)
· · · d
∗ (ν (l) )
b σ (l)
.
The quantum weighted (single) Hurwitz numbers are
H
d
H q
(μ) :=
d
k=1
μ ( 1),...μ (k) , |μ (i) |=N
k
i=1 ∗ (μ (i) )=d
W H q (μ
(1) , . . . , μ
(k) )H (μ
(1) , . . . , μ
k) , μ),
(10)
where the quantum weight factor is
W H q (μ
(1) , . . . , μ
(k) ) :=
(−1) d−k
k!
σ ∈S k
k
j =1
1
(1 − q
j
i=1 ∗ (μ (σ (i)) )
.
2 Hypergeometric τ-Functions as Generating Functions
for Weighted Hurwitz Numbers [3–6]
To construct a KP τ -function of hypergeometric type that serves as generating
function for weighted Hurwitz numbers for a given weight generating function G,