Euler's partition theorem and the combinatorics of -sequences

A coin-flipping interpretation of the l-nomial
For real r, let C r
(l) be a weighted coin for which the probability of
tails is ul r /∆(l) r and the probability of heads is vl r /∆(l) r .
The probability of getting exactly k heads when tossing the n coins
C (l)
−(n−1)/2 ,
C (l)
1−(n−1)/2 ,
C (l)
2−(n−1)/2 , . . . , C (l)
(n−1)/2 :
( n
k) (l)
∆ (l)
−(n−1)/2 ∆(l) 1−(n−1)/2 · · · ∆(l) (n−1)/2−1 ∆(l) (n−1)/2
∆ (l)
j/2
may not be an integer, but ∆(l)
−j/2 ∆(l) j/2 = ∆(l) j
+ 2 is.