Fermionic representation for basic hypergeometric functions related to Schur polynomials

We present the fermionic representation for the q-deformed hypergeometric functions related to Schur polynomials. For q = 1 it is known that these hypergeometric functions are related to zonal spherical polynomials for GL(N,C)/U(N) symmetric space. We show that multivariate hypergeometric functions are tau-functions of the KP and of the two-dimensional Toda lattice hierarchies. The variables of the hypergeometric functions are the higher times of those hierarchies. The discrete Toda lattice variable shifts parameters of hypergeometric functions. Introduction Hypergeometric functions play an important role both, in physics and in mathematics [3]. Many special functions and polynomials (such as q-Askey-Wilson polynomials, q-Jacobi polynomials, q-Gegenbauer polynomials , q-Racah polynomials , q-Hahn polynomials , expressions for ClebschGordan coefficients) are just certain hypergeometric functions evaluated in special values of parameters. In physics hypergeometric functions and their q-deformed counterparts sometimes play the role of wave functions and correlation functions for quantum integrable systems. In the present paper we shall construct hypergeometric functions as tau-functions of the Kadomtsev-Petviashvili (KP) hierarchy of equations. It is interesting that the KP equation which originally serves in plazma physics [4] now plays a very important role both, in physics (see [5]; see review in [6] for modern applications) and in mathematics. This peculiarity of KadomtsevPetviashvili equation appeared first in the paper V.Dryuma [7] where L-A pair of KP equation was presented, and mainly in the paper of V.E.Zakharov and A.B.Shabat in 1974 where this equation 1 was integrated by the dressing method. Actually it was the paper [1] where so-called hierarchy of higher KP equations appeared. Another very important equation is the two-dimensional Toda lattice integrated first in [8]. In the present paper we use these equations to construct hypergeometric functions which depend on many variables, these variables are KP and Toda lattice higher times. Here we shall use the general approach to integrable hierarchies of Kyoto school [2], see also [10, 9]. Especially a set of papers about Toda lattice [19, 20, 21, 22, 23, 24, 25] are important for us. There are several well-known different multivariate generalizations of hypergeometric series of one variable. Let |q| < 1 and let x1, · · · , xN be indeterminates. The multiple basic hypergeometric series are defined by the formula pΦq ( a1, · · · , ap; b1, · · · , bq; q,x(N) ) = ∑ l(n)≤N (q1 ; q)n · · · (q p; q)n (qb1; q)n · · · (qbq ; q)n q Hn(q) sn (xN) (0.1) where the sum is over all different partitions n defined as: n = (n1, n2, · · · , nr) , n1 ≥ n2 ≥ · · · ≥ nr, r ≤ |n| (0.2) |n| = n1 + n2 + · · ·+ nr, (0.3) Coefficient (q; q)n associated with partition n: (q; q)n = (q ; q)n1(q ; q)n2 · · · (q ; q)nr , (0.4) (q; q)ni = (1− q )(1− q) · · · (1− qi) (0.5) The multiple q defined on partition n: q = q ∑N i=1 (i−1)ni (0.6) and q-deformed ’hook polynomial’ Hn(q) is Hn(q) = ∏ (i,j)∈n ( 1− qij ) (0.7) hij = (ni + n ′ j − i− j + 1) (0.8) where n = (n1 + n ′ 2 + · · ·+ n ′ r′) is the conjugated partition (for the detailed definitions see [15].) Another generalization of hypergeometric series is so-called hypergeometric function of matrix arguments X,Y with indices a and b: pFq (a1, · · · , ap; b1, · · · , bq;X,Y) = ∑ n (a1) (d) n · · · (ap) (d) n (b1) (d) n · · · (bq) (d) n Z|n|(X)Z|n|(Y) |n|!Z|n|(In) (0.9) Here X,Y are N ×N matrix and Z|n|(X), Z|n|(Y) are zonal spherical polynomials, see [14]. There are also hypergeometric functions related to Jack polynomials pFq (d) ( a1, · · · , ap; b1, · · · bq;x(N),y(N) ) = ∑ n (a1) (d) n · · · (ap) (d) n (b1) (d) n · · · (bq) (d) n C (d) |n| (x(N))C (d) |n| (y(N)) |n|!C (d) |n| (1 n) (0.10) where (a) n = l(n) ∏ i=1 ( a− d 2 (i− 1) ) ni (0.11) Here (c)k = c(c+1) · · · (c+ k− 1). It is known that for the special value d = 2 the last expression (0.10) coinsides with (0.1), and with (0.9) for |q| → 1. These last cases we shall consider below. 2 1 A brief introduction to the fermionic description of the KP hierarchy [2] We have fermionic fields: ψ(z) = ∑