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# B Proofs for the Lagrange interpolation polynomials of type A

z=ξλλ0+1,(n)

= (−1)n+s−j qαb1b2· · ·bstn−j −a−11 a−12 · · ·a−1s 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)

zs+n−11 zs+n−22 · · ·zλs+n−λ0−1

0+1

z=ξλλ0+1,(n)

= (n−1)!

λ0+1

X

j=1

(−1)j lim

δ→∞

Fjε(z)−Gεj(z) z1z2· · ·zj−1zjs+n−j

× Eλλ0,(n−1)(z

bj)∆(n−1)(z

bj)

z1s+n−2z2s+n−3· · ·zj−1s+n−jzs+n−j−1j+1 · · ·zs+n−λλ 0−1

0+1

z=ξλλ0+1,(n)

= (n−1)!(−1)n+s

λ0+1

X

j=1

qαb1b2· · ·bstn−j−a−11 a−12 · · ·a−1s t−(n−j)

×∆(n−λ0−1) ξλ0,(n−λ0−1)

= (n−1)!(−1)n+s−1 1−qαa1a2· · ·asb1b2· · ·bst2n−2−λ0

1−tλ0+1 a1a2· · ·astn−1(1−t)

×∆(n−λ0−1) ξλ0,(n−λ0−1)

. (A.21)

Comparing (A.10) with (A.21), we therefore obtain the explicit expression ofcε0 =cε

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 Hzs,n be the C-linear space specified by (4.1). For x = (x1, . . . , xs) ∈ (C)s and µ = (µ1, . . . , µs)∈Ns with

|µ|=µ1+· · ·+µs≤n, we set

xn−|µ|µ = z1, z2, . . . , zn−|µ|, x1, x1t, . . . , x1tµ1−1

| {z }

µ1

, . . . , xs, xst, . . . , xstµs−1

| {z }

µs

∈(C)n, (B.1)

leaving the first n− |µ|variables unspecialized.

Theorem B.1. For genericx= (x1, . . . , xs)∈(C)sthere exists a uniqueC-basis{Eµ(x;z)|µ∈ Ns,|µ| ≤n} of the C-linear spaceHzs,n satisfying

(1) |λ|=|µ| =⇒ Eλ x;xn−|µ|µ

λµQn−|λ|

i=1

Qs

j=1 zi−xjtλj ,

(2) |λ|<|µ| =⇒ Eλ x;xn−|µ|µ

= 0.

Remark. The functions Eµ(x, z) for µ = (µ1, . . . , µs) ∈ Ns, where |µ| ≤ n, in the above theorem are the same as Eµµ01,...,µs(x, z), (µ0, µ1, . . . , µs) ∈Zs+1,n, in Theorem 4.1, because the map (µ0, µ1, . . . , µs)7→(µ1, . . . , µs) defines a bijection Zs+1,n→ {µ∈Ns| |µ| ≤n}. Throughout this section, for simplicity we use the notationEµ(x, z) (µ∈Ns,|µ| ≤n) instead ofEµµ01,...,µs(x, z), µ∈Zs+1,n.

We provide a proof of this theorem in this subsection.

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

x10=z∈C and x0i =xi∈C for i= 1, . . . , s, where0= (0, . . . ,0)∈Nsand i = (0, . . . ,0,

i

^

1,0, . . . ,0)∈Ns. Then it immediately follows that the functions

E0(x;z) =

s

Y

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 Hwn,s=

f(w)∈C[w1, . . . , ws]Ss|degwif(w)≤nfori= 1, . . . , s , which satisfies dimCHwn,s = n+ss

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

Fµ(x;w) =

s

Y

i=1 s

Y

j=1 µi

Y

k=1

xitk−1−wj

, (B.3)

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

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

s

Y

j=1

(xi−wj). (B.4)

By definition we have the relation Fµ(x;w)Fν xtµ;w

=Fµ+ν(x;w), (B.5)

where xtµ is given by (2.4). This relation is the fundamental tool for studying Eλ(x;z) in the succeeding arguments.

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

This lemma implies that the matrix F = Fµ(x;xtν)

µ,ν∈Ns

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

Proof . By definition, for µ, ν ∈ Ns we have Fµ(x;xtν) = Qs i=1

Qs j=1

Qµi

k=1 xitk−1 −xjtνj . Thus if there exists i∈ {1, . . . , s} such thatνi < µi, then Fµ(x;xtν) = 0. If ν ≺µ, then νi< µi for some i∈ {1,2, . . . , s} by definition, and hence we obtain Fµ(x;xtν) = 0 ifν ≺µ. Moreover, from (B.3), we obtain

Fν(x;xtν) =

s

Y

i=1

xνiit(νi2)(t;t)νi Y

1≤j≤s j6=i

νi

Y

k=1

xitk−1−xjtνj 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) =

n

Y

i=1 s

Y

j=1

(zi−wj). (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∈Ns

|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) ∈ Hzs,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

s

Y

j=1

(z−wj) =

s

Y

i=1

(z−xi) +

s

X

i=1

s

Y

j=1

(xi−wj)

 Y

1≤j≤s j6=i

z−xj xi−xj

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

s

X

i=1

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

Proof . From LemmaB.3the set{Fµ(x;w)|µ∈Ns,|µ| ≤n} ⊂Hwn,sis linearly independent and

|{Fµ(x;w)|µ∈Ns,|µ| ≤n}|= n+sn

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

Next we proveEλ(x;z)∈Hzs,n. From (B.7) we have Ψ(z;xtν) = X

µ∈Ns

|µ|≤n

Eµ(x;z)Fµ x;xtν

. (B.9)

We denote by G = Gµν(x)

µ,ν∈Ns

|µ|,|ν|≤n

the inverse matrix of F = Fµ(x;xtν)

µ,ν∈Ns

|µ|,|ν|≤n

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

Eλ(x;z) = X

ν∈Ns

|ν|≤n

Ψ(z;xtν)Gνλ(x), where |λ| ≤n.

Since Ψ(z;xtν)∈Hzs,n, we obtainEλ(x;z)∈Hzs,n.

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

n

Y

i=1 s

Y

j=1

(zi−xj). (B.10)

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

µ∈Ns

|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;xn−|µ|µ

λµQn−|λ|

i=1

Qs

j=1 zi−xjtλj , (2) |λ|<|µ| =⇒ Eλ x;xn−|µ|µ

= 0.

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

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

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

n−|µ|

Y

i=1 s

Y

j=1

(zi−wj) =Fµ(x;w) X

ν∈Ns

|ν|≤n−|µ|

Eν(xtµ;z0)Fν(xtµ;w), (B.11)

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

Ψ(xn−|µ|µ ;w) = X

ν∈Ns

|ν+µ|≤n

Eν(xtµ;z0)Fν+µ(x;w) = X

λ−µ∈Ns

|λ|≤n

Eλ−µ(xtµ;z0)Fλ(x;w)

= X

λ∈Ns

|λ|≤n

Eλ−µ(xtµ;z0)Fλ(x;w), (B.12)

where the coefficient Eλ−µ(xtµ;z0) in the last line is regarded as 0 if λ−µ6∈Ns. On the other hand, from (B.7) withz=xn−|µ|µ we have

Ψ(xn−|µ|µ ;w) = X

λ∈Ns

|λ|≤n

Eλ x;xn−|µ|µ

Fλ(x;w). (B.13)

Since Fλ(x;w) is a basis of Hwn,s, equating coefficients ofFλ(x;w) on (B.12) and (B.13), we have Eλ x;xn−|µ|µ

=Eλ−µ xtµ;z1, . . . , zn−|λ|

, (B.14)

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

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∈ Ns. From (B.14) we therefore obtain Eλ x;xn−|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;xn−|λ|λ

=E0 xtλ;z0

=Qn−|λ|

i=1

Qs

j=1 zi−xjtλj

. (2) Suppose the case |λ|<|µ|. Then we have λ−µ6∈Ns. From (B.14) we obtain Eλ x;xn−|µ|µ

= 0.

B.2 Explicit expression for Eλ(x;z)

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

Eλ(x;z) = X

(i1,...,in)∈{0,1,...,s}n i1+···+in

Ei

1(x;z1)Ei

2 xti1;z2 Ei

3 xti1+i2;z3

· · ·

×Eir xti1+···+in−1;zn

, (B.15)

where the symbol 0 denotes 0 =0 ∈Ns and Eik(x;zk), 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

X

(j1,...,jr)∈{1,...,s}r j1+···+jr

E0(x;z1)· · ·E0(x;zi1−1)Ej1(x;zi1)

×E0 xtj1;zi1+1

· · ·E0 xtj1;zi2−1 Ej

2 xtj1;zi2

· · ·

×E0 xtj1+···+jr−1;zir−1+1

· · ·E0 xtj1+···+jr−1;zir−1

Ejr xtj1+···+jr−1;zir

×E0 xtj1+···+jr;zir+1

· · ·E0 xtj1+···+jr;zn

.

In particular, the leading term of Eλ(x;z) with|λ|=r as a symmetric polynomial in z is equal to m((s−1)rsn−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

Qs

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

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

(i1,...,in)∈{1,...,s}n i1+···+in

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

= X

K1t···tKs

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

s

Y

i=1

Y

k∈Ki

Y

1≤j≤s j6=i

zk−xjtλ

(k−1) j

xitλ(k−1)i −xjtλ

(k−1) j

,

whereλ(k)i =|Ki∩ {1,2, . . . , k}|, and the summation is taken over all partitionsK1t · · · tKs= {1,2, . . . , n} such that|Ki|=λi,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

Y

k=1

zik −x1tik−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)

=

s

X

i1=0

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

s

X

i2=0

Ei2 xti1;z2

Fi2 xti1;w

· · ·

×

s

X

in=0

Ein xti1+···+in−1;zn

Fin xti1+···+in−1;w

. (B.16)

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

1(x;w)Fi

2 xti1;w

· · ·Fin xti1+···+in−1;w

=Fλ(x;w).

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

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

s

X

k=1

Eλ−k(x;z1, . . . , zn−1)Ek xtλ−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∈Ns, where |λ| ≤n, can be expanded as

Eλ(x;z) =Cλ(x)m(sn−|λ|(s−1)|λ|)(z) +· · · , (B.18)

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

Qs

i=1(t;t)λi

s

Y

i=1

Y

1≤j≤s j6=i

λi

Y

k=1

1

xitk−1−xjtλj. (B.19)

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

C(1)i (x) = Y

1≤j≤s j6=i

1

xi−xj for i= 1, . . . , s

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

Cλ(r)(x) =

s

X

k=1

Cλ−(r−1)

k (x)C(1)

k xtλ−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 =

s

X

k=1

1−tλk 1−tr

Y

1≤i≤s i6=k

λi

Y

l=1

xitl−1−xktλk

xitl−1−xktλk−1, (B.21)

which is equivalent to the identity [24, p. 46, Lemma 1.51] with (xi, yi)→ tλi, xitλi

.

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 , 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 .

Lemma B.13. For z= (z1, . . . , zn) andx= (x1, . . . , xs) we have E(0,µ2,...,µs)(x;z) = E0(x1;z)E2,...,µs)(xb1;z)

s

Q

i=2 µi

Q

k=1

xitk−1−x1

, (B.22)

where x

b1= (x2, . . . , xs)∈(C)s−1.

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

ws=x1

= X

µ∈Ns

|µ|≤n

Eµ(x;z)Fµ(x;w) ws=x1

.

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

Fµ(x;w) w

s=x1

=





0 ifµ1 >0,

F2,...,µs)(x

b1;w

bs)

s

Q

i=2 µi

Q

k=1

(xitk−1−x1) ifµ1 = 0, where w

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

ws=x1

= X

2,...,µs)∈Ns−1 µ2+···+µs≤n

"

E(0,µ2,...,µs)(x;z)

s

Y

i=2 µi

Y

k=1

xitk−1−x1

#

×F2,...,µs)(xb1;wbs). (B.23)

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

ws=x1

=Ψ(z, wbs)

n

Y

i=1

(zi−x1) =E0(x1;z) X

µ∈Ns−1

|µ|≤n

Eµ(xb1;z)Fµ(xb1;wbs)

= X

2,...,µs)∈Ns−1 µ2+···+µs≤n

E0(x1;z)E2,...,µs)(x

b1;z)

F2,...,µs)(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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