B Proofs for the Lagrange interpolation polynomials of type A

z=ξ^λ_λ⁰⁺¹^,(n)

= (−1)^n+s−j q^αb1b2· · ·bst^n−j −a⁻¹₁ a⁻¹₂ · · ·a⁻¹_s t^−(n−j)

if 1≤j≤λ0+ 1. Using (A.11), (A.12), (A.15) and the above equation, we have

δ→∞lim

Aφ(z)

z^s+n−1₁ z^s+n−2₂ · · ·z_λ^s+n−λ⁰⁻¹

0+1

z=ξ^λ_λ⁰⁺¹^,(n)

= (n−1)!

λ0+1

j=1

(−1)^j lim

δ→∞

F_j^ε(z)−G^ε_j(z) z1z2· · ·zj−1z_j^s+n−j

× E^λ_λ⁰^,(n−1)(z

bj)∆⁽ⁿ⁻¹⁾(z

bj)

z₁^s+n−2z₂^s+n−3· · ·z_j−1^s+n−jz^s+n−j−1_j+1 · · ·z^s+n−λ_λ ⁰⁻¹

0+1

z=ξ_λ^λ⁰⁺¹^,(n)

= (n−1)!(−1)^n+s

λ0+1

j=1

q^αb₁b₂· · ·b_st^n−j−a⁻¹₁ a⁻¹₂ · · ·a⁻¹_s t^−(n−j)

×∆^(n−λ⁰⁻¹⁾ ξ_λ^0,(n−λ⁰⁻¹⁾

= (n−1)!(−1)^n+s−1 1−q^αa1a2· · ·asb1b2· · ·bst^2n−2−λ⁰

1−t^λ⁰⁺¹ a₁a₂· · ·a_stⁿ⁻¹(1−t)

×∆^(n−λ⁰⁻¹⁾ ξ_λ^0,(n−λ⁰⁻¹⁾

. (A.21)

Comparing (A.10) with (A.21), we therefore obtain the explicit expression ofc^ε₀ =c^ε_(λ

0+1,λ1,...,λs)

as given by (5.12).

B Proofs for the Lagrange interpolation polynomials of type A

In this section we provide proofs for the propositions in Section 4.

B.1 Construction of the interpolation polynomials

Let H^z_s,n be the C-linear space specified by (4.1). For x = (x1, . . . , xs) ∈ (C^∗)^s and µ = (µ₁, . . . , µ_s)∈N^s with

|µ|=µ1+· · ·+µs≤n, we set

x^n−|µ|_µ = z1, z2, . . . , zn−|µ|, x1, x1t, . . . , x1t^µ¹⁻¹

| {z }

µ1

, . . . , xs, xst, . . . , xst^µ^s⁻¹

| {z }

µs

∈(C^∗)ⁿ, (B.1)

leaving the first n− |µ|variables unspecialized.

Theorem B.1. For genericx= (x₁, . . . , x_s)∈(C^∗)^sthere exists a uniqueC-basis{E_µ(x;z)|µ∈ N^s,|µ| ≤n} of the C-linear spaceH^z_s,n satisfying

(1) |λ|=|µ| =⇒ E_λ x;x^n−|µ|_µ

=δ_λµQn−|λ|

i=1

j=1 z_i−x_jt^λ^j ,

(2) |λ|<|µ| =⇒ E_λ x;x^n−|µ|µ

= 0.

Remark. The functions Eµ(x, z) for µ = (µ1, . . . , µs) ∈ N^s, where |µ| ≤ n, in the above theorem are the same as E^µµ⁰1,...,µs(x, z), (µ₀, µ₁, . . . , µ_s) ∈Z_s+1,n, in Theorem 4.1, because the map (µ₀, µ₁, . . . , µ_s)7→(µ₁, . . . , µ_s) defines a bijection Z_s+1,n→ {µ∈N^s| |µ| ≤n}. Throughout this section, for simplicity we use the notationEµ(x, z) (µ∈N^s,|µ| ≤n) instead ofE^µµ⁰1,...,µs(x, z), µ∈Z_s+1,n.

We provide a proof of this theorem in this subsection.

Example B.2. The case n= 1. Forx = (x1, . . . , xs) the symbols x^n−|µ|µ of the casen= 1 are written as

x¹₀=z∈C^∗ and x⁰_i =xi∈C^∗ for i= 1, . . . , s, where0= (0, . . . ,0)∈N^sand i = (0, . . . ,0,

1,0, . . . ,0)∈N^s. Then it immediately follows that the functions

E0(x;z) =

i=1

(z−xi) and Ei(x;z) = Y

1≤j≤s j6=i

z−xj

xi−xj

(B.2)

satisfy the conditions of Theorem B.1.

In the same way as (4.1) we define H^w_n,s=

f(w)∈C[w₁, . . . , w_s]^S^s|deg_w_if(w)≤nfori= 1, . . . , s , which satisfies dim_CH^w_n,s = ^n+s_s

. For x = (x1, . . . , xs) ∈(C^∗)^s, w= (w1, . . . , ws) ∈(C^∗)^s and µ= (µ₁, . . . , µ_s)∈N^s satisfying |µ| ≤n, we set

F_µ(x;w) =

i=1 s

j=1 µi

k=1

x_it^k−1−w_j

, (B.3)

whose degree with respect to w is |µ|, where |µ| ≤ n. By definition we have F_µ(x;w) ∈ H^w_n,s. For example, we have

F0(x;w) = 1 and Fi(x;w) =

j=1

(xi−wj). (B.4)

By definition we have the relation Fµ(x;w)Fν xt^µ;w

=Fµ+ν(x;w), (B.5)

where xt^µ is given by (2.4). This relation is the fundamental tool for studying E_λ(x;z) in the succeeding arguments.

Lemma B.3. For each µ, ν ∈N^s, Fµ(x;xt^ν) = 0 unless µi ≤νi for i= 1, . . . , s. In particular, F_µ(x;xt^ν) = 0 for µν. Moreover, if x∈(C^∗)^s is generic, then F_ν(x;xt^ν)6= 0 for all ν ∈N^s.

This lemma implies that the matrix F = Fµ(x;xt^ν)

µ,ν∈Ns

|µ|,|ν|≤n is upper triangular, and also invertible if x∈C^s is generic.

Proof . By definition, for µ, ν ∈ N^s we have Fµ(x;xt^ν) = Qs i=1

Qs j=1

Qµi

k=1 xit^k−1 −xjt^ν^j . Thus if there exists i∈ {1, . . . , s} such thatν_i < µ_i, then F_µ(x;xt^ν) = 0. If ν ≺µ, then ν_i< µ_i for some i∈ {1,2, . . . , s} by definition, and hence we obtain F_µ(x;xt^ν) = 0 ifν ≺µ. Moreover, from (B.3), we obtain

F_ν(x;xt^ν) =

i=1

x^ν_iⁱt(^νi₂)(t;t)_ν_i Y

1≤j≤s j6=i

νi

k=1

x_it^k−1−x_jt^ν^j 6= 0,

if we impose an appropriate genericity condition on x∈(C^∗)^s. Lemma B.4 (duality). For z = (z1, . . . , zn) ∈(C^∗)ⁿ, w= (w1, . . . , ws) ∈ (C^∗)^s let Ψ(z;w) be the function defined by

Ψ(z;w) =

i=1 s

j=1

(z_i−w_j). (B.6)

Then for x = (x1, . . . , xs) ∈ (C^∗)^s the function Ψ(z;w) expands in terms of Fµ(x;w), |µ| ≤ n, as

Ψ(z;w) = X

µ∈N^s

|µ|≤n

Eµ(x;z)Fµ(x;w), (B.7)

where Eµ(x;z), |µ| ≤ n, are the coefficients independent of w, and satisfy Eµ(x;z) ∈ H^z_s,n,

|µ| ≤n, as functions of z.

Example B.5. The casen= 1. From (B.2) and (B.4), the identity (B.7) follows from

j=1

(z−w_j) =

i=1

(z−x_i) +

i=1





j=1

(x_i−w_j)



 Y

1≤j≤s j6=i

z−x_j xi−xj

=E0(x;z)F0(x;w) +

i=1

Ei(x;z)Fi(x;w). (B.8)

Proof . From LemmaB.3the set{F_µ(x;w)|µ∈N^s,|µ| ≤n} ⊂H^w_n,sis linearly independent and

|{F_µ(x;w)|µ∈N^s,|µ| ≤n}|= ^n+s_n

. This indicates that {F_µ(x;w)|µ∈N^s,|µ| ≤n}is a basis of theClinear space H^w_n,s. SinceΨ(z;w)∈H^w_n,s, the functionΨ(z;w) ofwis expanded in terms of Fµ(x;w),µ∈N^s,|µ| ≤n, as the form (B.7).

Next we proveE_λ(x;z)∈H^z_s,n. From (B.7) we have Ψ(z;xt^ν) = X

µ∈N^s

|µ|≤n

Eµ(x;z)Fµ x;xt^ν

. (B.9)

We denote by G = G_µν(x)

µ,ν∈Ns

|µ|,|ν|≤n

the inverse matrix of F = F_µ(x;xt^ν)

µ,ν∈Ns

|µ|,|ν|≤n

. Then, by Lemma B.3, (B.9) can be rewritten as

E_λ(x;z) = X

ν∈N^s

|ν|≤n

Ψ(z;xt^ν)G_νλ(x), where |λ| ≤n.

Since Ψ(z;xt^ν)∈H^z_s,n, we obtainE_λ(x;z)∈H^z_s,n.

Lemma B.6. Forz= (z1, . . . , zn)∈(C^∗)ⁿ, the explicit form of E0(x;z) is given by E0(x;z) =

i=1 s

j=1

(zi−xj). (B.10)

Proof . From the expansion (B.7) of the casew=x, we have Ψ(z;x) = X

µ∈N^s

|µ|≤n

Eµ(x;z)Fµ(x;x).

Since we have Fµ(x;x) = δµ0 by definition, we obtain E0(x;z) = Ψ(z;x), which coincides

with (B.10).

Lemma B.7. For z = (z1, . . . , zn) ∈ (C^∗)ⁿ, the functions Eµ(x;z), |µ| ≤ n, defined by the relation (B.7) satisfy the following:

(1) |λ|=|µ| =⇒ Eλ x;x^n−|µ|µ

=δλµQn−|λ|

i=1

j=1 zi−xjt^λ^j , (2) |λ|<|µ| =⇒ E_λ x;x^n−|µ|µ

= 0.

As a consequence of the above, the set {E_µ(x;z)| |µ| ≤ n} ⊂H^z_s,n is linearly independent, i.e., {E_µ(x;z)| |µ| ≤n} is a basis of H^z_s,n.

Proof . We calculate Ψ(z;w) of the case z = x^n−|µ|_µ in two ways. Using the identity (B.7) partially forΨ x^n−|µ|µ ;w

, we have the expansion Ψ(x^n−|µ|_µ ;w) =Fµ(x;w)

n−|µ|

i=1 s

j=1

(zi−wj) =Fµ(x;w) X

ν∈N^s

|ν|≤n−|µ|

Eν(xt^µ;z⁰)Fν(xt^µ;w), (B.11)

where z⁰ = (z₁, . . . , z_n−|µ|) ∈ (C^∗)^n−|µ|. By property (B.5) of F_µ(x;w), the expression (B.11) may be rewritten as

Ψ(x^n−|µ|_µ ;w) = X

ν∈N^s

|ν+µ|≤n

E_ν(xt^µ;z⁰)F_ν+µ(x;w) = X

λ−µ∈N^s

|λ|≤n

E_λ−µ(xt^µ;z⁰)F_λ(x;w)

= X

λ∈N^s

|λ|≤n

Eλ−µ(xt^µ;z⁰)Fλ(x;w), (B.12)

where the coefficient Eλ−µ(xt^µ;z⁰) in the last line is regarded as 0 if λ−µ6∈N^s. On the other hand, from (B.7) withz=x^n−|µ|µ we have

Ψ(x^n−|µ|_µ ;w) = X

λ∈N^s

|λ|≤n

E_λ x;x^n−|µ|_µ

F_λ(x;w). (B.13)

Since Fλ(x;w) is a basis of H^w_n,s, equating coefficients ofFλ(x;w) on (B.12) and (B.13), we have E_λ x;x^n−|µ|_µ

=Eλ−µ xt^µ;z₁, . . . , z_n−|λ|

, (B.14)

where the right-hand side of (B.14) vanishes ifλ−µ6∈N^s.

Now we prove the vanishing properties (1), (2) in Lemma B.7. (1) We first consider the case |λ| = |µ| and λ 6= µ. Then we have λ−µ 6∈ N^s. From (B.14) we therefore obtain E_λ x;x^n−|µ|µ

= 0. Next we suppose the case |λ| = |µ| and λ = µ. Because of Lemma B.6, from (B.14) we obtain Eλ x;x^n−|λ|_λ

=E0 xt^λ;z⁰

=Qn−|λ|

i=1

j=1 zi−xjt^λ^j

. (2) Suppose the case |λ|<|µ|. Then we have λ−µ6∈N^s. From (B.14) we obtain E_λ x;x^n−|µ|µ

= 0.

B.2 Explicit expression for E_λ(x;z)

Lemma B.8. For z∈(C^∗)ⁿ, x∈(C^∗)^s, the functions Eλ(x;z) for λ∈N^s, where |λ| ≤n, may be expressed as

E_λ(x;z) = X

(i1,...,in)∈{0,1,...,s}ⁿ i1+···+_in=λ

E_i

1(x;z₁)E_i

2 xtⁱ¹;z₂ E_i

3 xtⁱ¹⁺ⁱ²;z₃

· · ·

×E_ir xtⁱ¹^+···+ⁱⁿ⁻¹;z_n

, (B.15)

where the symbol ₀ denotes ₀ =0 ∈N^s and E_ik(x;z_k), k = 1, . . . , n, are given by (B.2). In other words, if |λ|=r, r= 0,1, . . . , n, then Eλ(x;z) are rewritten as

Eλ(x;z) = X

1≤i1<···<ir≤n

(j1,...,jr)∈{1,...,s}^r j1+···+_jr=λ

E0(x;z1)· · ·E0(x;zi1−1)Ej1(x;zi1)

×E₀ xt^j¹;z_i₁₊₁

· · ·E₀ xt^j¹;z_i₂−1 E_j

2 xt^j¹;z_i₂

· · ·

×E₀ xt^j¹^+···+^jr−1;z_i_r−1₊₁

· · ·E₀ xt^j¹^+···+^jr−1;z_i_r−1

E_jr xt^j¹^+···+^jr−1;z_i_r

×E0 xt^j¹^+···+^jr;zir+1

· · ·E0 xt^j¹^+···+^jr;zn

In particular, the leading term of E_λ(x;z) with|λ|=r as a symmetric polynomial in z is equal to m_((s−1)^r_s^n−r₎(z) up to a constant.

Before proving LemmaB.8, we present several special cases below.

Example B.9. If |λ| = 0, i.e., λ = 0, then E0(x;z) = Qn

i=1E0(x;zi) = Qn i=1

j=1(zi −xj), which we already saw in (B.10).

Example B.10. If|λ|=n, then E_λ(x;z) = X

(i1,...,in)∈{1,...,s}ⁿ i1+···+_in=λ

Ei1(x;z1)Ei2(xtⁱ¹;z2)Ei3(xtⁱ¹⁺ⁱ²;z3)· · ·E_ir(xtⁱ¹^+···+ⁱⁿ⁻¹;zn)

= X

K1t···tKs

={1,2,...,n}

i=1

k∈Ki

1≤j≤s j6=i

z_k−x_jt^λ

(k−1) j

x_it^λ^(k−1)ⁱ −x_jt^λ

(k−1) j

whereλ^(k)_i =|K_i∩ {1,2, . . . , k}|, and the summation is taken over all partitionsK1t · · · tKs= {1,2, . . . , n} such that|K_i|=λ_i,i= 1,2, . . . , s.

Example B.11 (shifted symmetric polynomials). Lets= 1. Then we have En−r(x1;z) = X

1≤i1<···<ir≤n r

k=1

zi_k −x1tⁱ^k^−k

for r = 0,1, . . . , n,

which coincide with the Knop–Sahi shifted symmetric polynomials attached to partitions of a single column, see [21, p. 476, Proposition 3.1]. See also [13, Appendix] for an application of these polynomials to theq-Selberg integral.

Proof of Lemma B.8. Since we have Qs

j=1(z1 −wj) = Ps

i=0Ei(x;z1)Fi(x;w) from (B.8), the function Ψ(z;w) may be expanded as

Ψ(z;w) =Ψ(z1;w)Ψ(z2;w)· · ·Ψ(zn;w)

i1=0

Ei1(x;z1)Fi1(x;w)

i2=0

Ei2 xtⁱ¹;z2

Fi2 xtⁱ¹;w

· · ·

in=0

E_in xtⁱ¹^+···+ⁱⁿ⁻¹;zn

F_in xtⁱ¹^+···+ⁱⁿ⁻¹;w

. (B.16)

From (B.5), ifλ=i1+· · ·+in, we have F_i

1(x;w)F_i

2 xtⁱ¹;w

· · ·F_in xtⁱ¹^+···+ⁱⁿ⁻¹;w

=F_λ(x;w).

Then, comparing the coefficient ofF_λ(x;w) in (B.16) with those in (B.7), we obtain (B.15).

Remark. Similar to (B.16), we have the recursion formula E_λ(x;z1, . . . , zn) =

k=1

Eλ−_k(x;z1, . . . , zn−1)E_k xt^λ−^k;z1

(B.17)

from the identity (B.5) of Fλ(x;w).

Lemma B.12. Forz∈(C^∗)ⁿ, x∈(C^∗)^s, the functions Eλ(x;z) for λ∈N^s, where |λ| ≤n, can be expanded as

Eλ(x;z) =Cλ(x)m_(s^n−|λ|_(s−1)^|λ|₎(z) +· · · , (B.18)

where the coefficient C_λ(x) of the leading term is given by C_λ(x) = (t;t)_|λ|

Q_s

i=1(t;t)_λ_i

i=1

1≤j≤s j6=i

λi

k=1

xit^k−1−xjt^λ^j. (B.19)

Proof . Since we have (B.18) immediately from Lemma B.8, we prove (B.19) by induction on r =|λ|. We denote byC_λ^(r)(x) the coefficientC_λ(x) in (B.18). Whenr = 1, we immediately find

C⁽¹⁾_i (x) = Y

1≤j≤s j6=i

x_i−x_j for i= 1, . . . , s

from the explicit expression (B.2) ofEi(x;z). From the recursion formula (B.17) we have the relation

C_λ^(r)(x) =

k=1

C_λ−^(r−1)

k (x)C⁽¹⁾

k xt^λ−^k

. (B.20)

Since (B.19) holds for r = 1, it suffices to show that the right-hand side of (B.19) satisfies the same recurrence formula as (B.20). One can directly verify that the corresponding formula for the right-hand side of (B.19) reduces to the identity

1 =

k=1

1−t^λ^k 1−t^r

1≤i≤s i6=k

λi

l=1

x_it^l−1−x_kt^λ^k

x_it^l−1−x_kt^λ^k⁻¹, (B.21)

which is equivalent to the identity [24, p. 46, Lemma 1.51] with (xi, yi)→ t^λⁱ, xit^λⁱ

Remark. The identity (B.21) is very well known and has played a very important role in the theory of multiple basic hypergeometric series. In the paper [28], Rosengren extended this identity to its elliptic form and applied it to his multiple elliptic hypergeometric series. It is really remarkable that, according to his paper, this identity in elliptic form already appeared in Tannery and Molk’s book [31, p. 34] (which was published in 1898) and also in Whittaker and Watson [35, p. 451]. See his very detailed discussion about this identity in [28].

Lemma B.13. For z= (z₁, . . . , z_n) andx= (x₁, . . . , x_s) we have E_(0,µ₂_,...,µ_s₎(x;z) = E0(x1;z)E_(µ₂_,...,µ_s₎(x_b₁;z)

i=2 µi

k=1

xit^k−1−x1

, (B.22)

where x

b1= (x₂, . . . , x_s)∈(C^∗)^s−1.

Proof . From (B.7) of Lemma B.4with substitution ws=x1, we have Ψ(z, w)

ws=x1

= X

µ∈N^s

|µ|≤n

Eµ(x;z)Fµ(x;w) ws=x1

Here, from the definition (B.3) ofFµ(x;w), we have

F_µ(x;w) _w

s=x1







0 ifµ1 >0,

F_(µ₂_,...,µ_s₎(x

b1;w

bs)

i=2 µi

k=1

(x_it^k−1−x₁) ifµ₁ = 0, where w

bs= (w1, . . . , ws−1)∈(C^∗)^s−1. Thus we obtain Ψ(z, w)

ws=x1

= X

(µ2,...,µs)∈N^s−1 µ2+···+µs≤n

E_(0,µ₂_,...,µ_s₎(x;z)

i=2 µi

k=1

xit^k−1−x1

×F_(µ₂_,...,µ_s₎(x_b₁;w_b_s). (B.23)

On the other hand, from (B.6), (B.7) and (B.10), we also obtain Ψ(z, w)

ws=x1

=Ψ(z, w_b_s)

i=1

(zi−x1) =E0(x1;z) X

µ∈Ns−1

|µ|≤n

Eµ(x_b₁;z)Fµ(x_b₁;w_b_s)

= X

(µ2,...,µs)∈N^s−1 µ2+···+µs≤n

E0(x1;z)E_(µ₂_,...,µ_s₎(x

b1;z)

F_(µ₂_,...,µ_s₎(x

b1;w

bs). (B.24)

Comparing (B.23) with (B.24), we therefore obtain (B.22).

Acknowledgements

The authors would like to express their gratitude to the referees for providing them many useful suggestions. This work is supported by JSPS Kakenhi Grants (C)25400118, (B)15H03626 and (C)18K03339.

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In document Connection Formula for the Jackson Integral of Type A (Stránka 34-42)