Binomial coefficients and skew interpolation functions

Proposition 7.1.2. For parameters satisfying the balancing q^mtⁿ⁻¹ab= 1we have

P_λ^∗(m,n)(. . . aq^µitⁿ⁻ⁱ. . .;a, b;q;t;p)

=δλµC⁰mⁿ−λ(pqtⁿ⁻¹;q;t;p)C⁺mⁿ−λ( 1

q^2ma²;q;t;p)C⁻λ(pq, t;q;t;p)C⁺λ(t²ⁿ⁻²a²;q;t;p) C⁰_λ(tⁿ;q;t;p) . Whenλ= 0, the value of the theta interpolation function is very nice:

P₀^∗(m,n)(. . . xi. . .;a, b;q;t;p) = Y

1≤i≤n

θp(bx^±1_i ;q)m.

We now define the abelian interpolation functions (abelian here will stand for elliptic in the variables cf. [Rai06]) as renormalized theta interpolation functions:

R^∗(n)_λ (;a, b;q;t;p) = P_λ^∗(m,n)(;a, q^−mb;q;t;p) P₀^∗(m,n)(;a, q^−mb;q;t;p) ,

form≥λ1.

Remark 7.1.3. These functions are elliptic in the variables. It can be shown the formula given above is independent of m (hence the notation; see [Rai06]). Moreover, they functions have poles whenever the denominator has zeros: at points of the form x_i = b^±1. ever the denominator has zeros: at points of the formx_i=b^±1.

Remark 7.1.4. For n = 1, the abelian interpolation functions are just explicit ratios of theta Pochhammer symbols, and were introduced in Section6.2. To wit:

R^∗(1)_l (x;a, b;q;p) = θ_p(ax^±1;q)_l θp(bq^−lx^±1;q)l

.

We begin with theelliptic binomial coefficients. They are defined as follows:

λ µ

[a,b];q;t;p

:= ∆µ(a

b|tⁿ,1/b;q;t;p)R^∗(n)_µ (. . .√

aq^λit¹⁻ⁱ. . .;t¹⁻ⁿ√ a, b/√

a;q;t;p), (7.2.1)

for any integern≥`(λ), `(µ).

Remark 7.2.1. The binomial coefficient thus described is elliptic inaand b, as well as symmetric under (a, b, t, q)7→(1/a,1/b,1/t,1/q).

We list the following evaluations (and transformation) we will find useful:

λ 0

[a,b];q;t;p

= 1, λ

λ

[a,b];q;t;p

= C⁺_λ(a;q;t;p)∆⁰_λ(a/b|1/b;q;t;p) C⁺λ(^a_b;q;t;p)∆⁰_λ(a|b;q;t;p) , mⁿ

λ

[a,b];q;t;p

= ∆_λ(a

b|tⁿ, q^−m, t¹⁻ⁿq^ma,1/b;q;t;p),

mⁿ+λ mⁿ+µ

[a,b];q;t;p mⁿ

mⁿ

[a,b];q;t;p

= ∆⁰_λ(q^2ma|b, pqaq^m/b, pqtⁿ⁻¹q^m, t¹⁻ⁿq^2ma;q;t;p)

∆⁰_µ(q^2ma/b|1/b, pqaq^m, pqtⁿ⁻¹q^m, t¹⁻ⁿq^2ma/b;q;t;p) λ

µ

[q^2ma,b];q;t;p

.

(7.2.2)

Because we want to specialize b at negative powers of q (b = 1/q) and the round parenthe- ses binomial coefficients just described have poles there, we will need the following angle bracket renormalization of the binomial coefficients:

λ µ

[a,b](v1,...,vk);q;t;p

:= ∆⁰_λ(a|b, v1, . . . , vk;q;t;p)

∆⁰_µ(a/b|1/b, v1, . . . , vk;q;t;p) λ

µ

[a,b];q;t;p

.

If both λ = l and µ = m consist of only one part, we have the following expression for the generalized angle-bracket binomial coefficient:

l m

[a,b];q;p

:= Γp,q(_a^bq^−l−m, q^{m a}_b, aq^l+m, bq^l−m) Γp,q(_a^bq^−2m, q^{2m a}_b, aq^l, b)

m

Y

j=1

θp(q^j−l−1) θ_p(q^−j) . The last product of theta functions can be written as

m

Y

j=1

θp(q^j−l−1)

θ_p(q^−j) =q^m(m−l) θp(q;q)l

θ_p(q;q)_l−mθ_p(q;q)_m.

Taking the limit p → 0, this degenerates to a version of the q-binomial coefficient q^m(m−l) _m^l

q. Takingq→1 yields the usual binomial coefficient.

Whenb= 1, the binomial angle-bracket coefficients reduce to Kronecker delta symbols. We also have the following important proposition (see [Rai06]).

Proposition 7.2.2. Ifb= 1/q, the binomial coefficientλ µ

[a,1/q];q;t;p vanishes unlessλ_i−1≤µ_i≤ λ_i (µ is obtained fromλby removing a vertical strip).

Whent=qthe formula for the renormalized (angle bracket) binomial coefficient becomes deter- minantal by an application of Warnaar’s formula from Section2.4and using the fact that interpo- lation functions can themselves be written as determinants.

Proposition 7.2.3.

λ µ

[a,b];q;p

= Y

1≤i<j≤n

q^−µiθ(q^{µi−i−µj+j}, q^{µi+µj+2−i−j}a/b;p) q^−λiθ(q^{λi−i−λj+j}, q^{λi+λj+2−i−j}a;p) det

1≤i,j≤n





λi+n−i µ_j+n−j

[ ^a

q2n−2,b];q;p



.

Remark 7.2.4. The reader should compare the above formula with the formula for transition prob- abilities for the Markov chains introduced in Chapter4. To be more precise, the binomial coefficient with t =q and b-parameter 1/q is indeed a transition probability like the ones defined in Section 4.1. That is, such a binomial coefficient is a coefficient of an appropriate difference operator from Section 4.2 where z_i ∝ q^µi+n−i. We discuss the appropriate specializations of parameters corre- sponding to the transition probabilities of Section4.1in Theorems7.5.1and7.5.5. This observation allows us to prove the quasi-commutation for difference operators (Lemma 4.2.3) in the following way. In4.2.3 we first specialize thezi’s to be proportional toq^λi+n−i. We express the coefficients of difference operators as determinants and sum everything using the Cauchy-Binet determinantal identity (written forM, N arbitrary square matrices we can compose):

X

Z=(...,zi,...)

1≤i,j≤ndet M(xi, zj) det

1≤i,j≤nN(zi, yj) = det

1≤i,j≤n(M N)(xi, yj).

The multivariate quasi-commutation relation reduces then to the univariate commutation which is proven directly. This proves it forzi∝q^λi+n−i but for genericqthis is sufficient as all such powers are dense (in other words, we analytically continue inq^λi+n−i).

Remark 7.2.5. Ifb= 1/q thenλi−νi∈ {0,1} and the determinant is of a block diagonal matrix with each block either upper or lower triangular (2-diagonal even). Hence in this case the determinant evaluates to the product of the diagonal elements.

The following identity allows us to flip the top and bottom of binomial coefficients whenλis of length at mostn. It is also useful in computing binomial coefficients explicitly.

1ⁿ+λ µ

[a,q⁻¹];q;t;p

= C₁⁰n(q⁻¹;q;t;p) C₁⁰n(pqaq;q;t;p)

∆_µ(qa|tⁿ, t¹⁻ⁿqa;q;t;p)

∆λ(q²a|tⁿ, t¹⁻ⁿq^a;q;t;p) µ

λ

[qa,q⁻¹];q;t;p

. (7.2.3)

We are now in a position to introduce theelliptic skew interpolation functions. They are defined

by the following hypergeometric formula:

R^∗_λ/κ([v0, . . . , v2n−1];a, b;q;t;p) := X

κ⊂µ⊂λ

λ µ

[a/b,ab/pq];q;t;p

µ κ

[pq/b²,pqQ

0≤r<2nvr/ab];q;t;p

∆⁰_µ(pq/b²|pq/bv0, pq/bv1, . . . , pq/bv_2n−1;q;t;p). (7.2.4) Remark 7.2.6. The skew interpolation functions are invariant under permutations of the v_i’s as well as under insertion/deletion of (v⁰,1/v⁰) pairs among thev parameters.

With 0, 2 and 4 varguments, the interpolation functions have the following simple expressions:

R^∗_λ/κ([];a, b;q;t;p) =δ_λκ, R^∗_λ/κ([v₀, v₁];a, b;q;t;p) =

λ κ

[a/b,v₀v₁](a/v₀,a/v₁);q;t;p

, R^∗_λ/κ([v0, v1, v2, v3];a, b;q;t;p) =

X

µ

λ µ

[a/b,v₀v₁](a/v₀,a/v₁);q;t;p

µ κ

[a/v₀v₁b,v₂v₃](a/v₀v₁v₂,a/v₀v₁v₃);q;t;p

.

We will, in some sense, only make use of such interpolation functions, though at first they will appear with an arbitrary number ofvparameters. When the bottom partition is equal to 0, the skew interpolation functions are indeed a version of the usual interpolation functions defined in Section 7.1:

R^∗(n)_λ (z₁, . . . , z_n;a, b;q;t;p) =

∆⁰_λ(tⁿ⁻¹a/b|pqa/tb;q;t;p)R^∗_λ/0([t^1/2z₁^±1, . . . , t^1/2z_n^±1];t^n−1/2a, t^1/2b;q;t;p).

We can combine the last equality of (7.2.2) with (7.2.3) and rewrite in terms of skew interpolation functions to obtain (following [Rai11]):

Lemma 7.2.7. Let v0, . . . , v_2k−1 be such that v2rv2r+1 =q^−mr with mr nonnegative integers, for 0≤r < k. Assumem,m⁰,nare also nonnegative integers satisfyingm⁰ =m+P

rmr and letλ, κ be partitions of length at mostn. Then

∆⁰_mn+κ(a/bQ

0≤r<2kvr|ab/pqQ

0≤r<2kvr;q;t;p)

∆⁰_m0n+λ(a/b|ab/pq;q;t;p) R_m^∗0n+λ/mⁿ+κ([v0, . . . , v_2k−1];a, b;q;t;p) Q

0≤r<k∆⁰_mn

r(Q⁰a/Qrb|Q⁰a/Qrv2r, Q⁰a/Qrv2r+1;q;t;p)

∆⁰_m0−mⁿ(Qa/b|ab/pq, pqQQ⁰a/b;q;t;p)

×∆κ(aQQ⁰/b|tⁿ, atQ⁰/tⁿb, abQ⁰/pq, pqQ/Q⁰ab;q;t;p)

∆λ(aQ⁰²/b|tⁿ, atQ⁰/tⁿb, abQ⁰/pq, pq/ab;q;t;p)

×∆⁰_κ(a⁰/b⁰Q

0≤r<2kvr|a⁰b⁰/pqQ

0≤r<2kvr;q;t;p)

∆⁰_λ(a⁰/b⁰|a⁰b⁰/pq;q;t;p) R^∗_κ/λ([v₀, . . . , v_2k−1];a⁰, b⁰;q;t;p),

wherea⁰=pqQ/b,b⁰=pq/Q⁰b andQ=q^m,Q⁰=q^m0.

We end this section by listing a couple of vanishing properties of skew interpolation functions fol- lowing [Rai11]. They will be used in the next section to ensure termination of certain hypergeometric sums. We first list the general results and then specialize in the case of interest to us.

Theorem 7.2.8. If the2k v parameters of the skew interpolation functions can be arranged in such a way that v2iv2i+1=tⁿⁱq^−mi for positive integers mi, ni0 and0≤i < kthen for any partitionκ,

R^∗_λ/κ([v0, v1, . . . , v_2k−1];a, b;q;t;p) = 0, unless

κi ≤λi≤κ_i−N +M, whereM =P

im_i,N=P

in_i and by definitionκ_i= 0fori≤0.

Remark 7.2.9. This result is of interest to us for κ= 0. Note forN = 0, κ= 0 we have that all parts ofλfor which the function is nonzero are bounded by M. Alternatively, forM = 0 we have that partitionsλfor which the function is nonzero have at mostN parts.

Theorem 7.2.10. Fix notation as in Theorem 7.2.8 and assume that thev parameters have the property thatv2iv2i+1=tⁿⁱq^−mi for1≤i < k and that

a/ Y

1≤i<2k

vi =tⁿ⁰q^−m0.

If κn₀+1≤m0, then

R^∗_λ/κ([v0, v1, . . . , v2k−1];a, b;q;t;p) = 0, unless λN+1≤M.

Remark 7.2.11. Forκ= 0 (our interest), this coincides with Theorem7.2.8. The reason for stating both is that we will use both whenκ= 0 (or rather use the common version twice). If in one we take M = 0, in the otherN = 0 (and in bothκ= 0), then skew interpolation functions will vanish unlessλ∈M^N.