A short proof of a symmetry identity for the $(q,\mu,\nu)$-deformed Binomial distribution

We give a short and elementary proof of a $(q, \mu, \nu)$-deformed Binomial distribution identity arising in the study of the $(q, \mu, \nu)$-Boson process and the $(q, \mu, \nu)$-TASEP. This identity found by Corwin in [4] was a key technical step to prove an intertwining relation between the Markov transition matrices of these two classes of discrete-time Markov chains. This was used in turn to derive exact formulas for a large class of observables of both these processes.


Introduction
Zero-range process and exclusion processes are generic stochastic models for transport phenomena on a lattice. Integrability of these models is an important question. In a short letter [5], Evans-Majumdar-Zia considered spatially homogeneous discrete time zero-range processes on periodic domains. They adressed and solved the question of characterizing the jump distributions for which invariant measures are product measures. Povolotsky [6] further examined the precise form of jump distributions allowing solvability by Bethe ansatz, and found the (q, µ, ν)-Boson process and the (q, µ, ν)-TASEP. He also conjectured exact formulas for the model on the infinite lattice. Using a Markov duality between the (q, µ, ν)-Boson process and the (q, µ, ν)-TASEP, Corwin [4] showed a variant of these formulas and provided a method to compute a large class of observables. This can be seen as a generalization of a similar work on q-TASEP and q-Boson process performed in [3,2]. In his proof, the intertwining relation between the two Markov transition matrices essentially boils down to a (q, µ, ν)-deformed Binomial distribution identity [4, Proposition 1.2]. The proof was adapted from [2, Lemma 3.7] which is the ν = 0 case, and required the use of Heine's summation formula for the basic hypergeometric series 2 φ 1 . In the following, we give a short proof of this identity.
A symmetry property for the (q, µ, ν)-deformed Binomial distribution First, we define the three parameter deformation of the Binomial distribution introduced in [6]. Definition 1. For |q| < 1, 0 ν µ < 1 and integers 0 j m, define the function It happens that for each m ∈ N ∪ ∞, this defines a probability distribution on {0, . . . , m}.
Proof. As shown in [4], this equation is equivalent to a specialization of some known summation formula for basic hypergeometric series 2 φ 1 (Heine's q-generalizations of Gauss' summation formula).
This probability distribution can be seen as a q-analogue of the Binomial distribution, depending on two parameters 0 ν µ < 1 and we call it the (q, µ, ν)-Binomial distribution. In [6], various interesting degenerations are studied. We now state and prove the main identity.
Proposition 1 (Proposition 1.2, [4]). Let X (resp. Y ) be a random variable following the (q, µ, ν)-Binomial distribution on {0, . . . , x} (resp. {0, . . . , y}). We have Proof. Let S x,y := x j=0 ϕ q,µ,ν (j|x)q jy . We have to show that S x,y = S y,x for all integers x, y 0. Our proof is based on the fact that S x,y satisfies a recurrence relation which is invariant when exchanging the roles of x and y. First notice that by lemma 1, S x,0 = 1 for all x 0, and by definition S 0,y = 1 for all y 0.
The Pascal identity for q-Binomial coefficients, (see 10.0.3 in [1]), The last equation can be rewritten Thus, the sequence (S x,y ) (x,y)∈N 2 is completely determined by Setting T x,y = S y,x , one notices that the sequence (T x,y ) (x,y)∈N 2 enjoys the same recurrence, which concludes the proof.
Remark. To completely avoid the use of basic hypergeometric series, one would also need a similar proof of lemma 1. One can prove the result by recurrence on m (as in the proof of [2, lemma 1.3]), but the calculations are less elegant when ν = 0.