matrix inversions and elliptic hypergeometric series on

Matrix inversions andelliptic hypergeometric serieson root systems

Hjalmar Rosengren

Chalmers University of Technologyand University of Gothenburg

joint work in progress (since 2002)with Michael Schlosser, University of Vienna

Hagenberg, 25 July 2019

Introduction

Matrix inversions give a powerful tool to derive hypergeometricidentities, such as quadratic and cubic summations andtransformations.

Perhaps more importantly, it allows us to look systematically fornew identities and understand how they are related.

We apply matrix inversions to elliptic hypergeometric series onroot systems. Until now, almost nothing is known aboutquadratic and cubic identities for such series.

We will only discuss one matrix inversion and two applications,but more can be done.

Hjalmar Rosengren 2/25

Outline

1 An Ar matrix inversion

2 Application 1: A new Ar Jackson summation

3 Application 2: An Ar quadratic transformation

Hjalmar Rosengren 3/25

Table of Contents




An Ar matrix inversion Hjalmar Rosengren 4/25

Multidimensional matrix inversions

Consider matrices F = (fkl)k,l∈Zr .

Lower-triangular: fkl = 0 unless k ≥ l, that is, ki ≥ li for1 ≤ i ≤ r.

Two such matrices F and G = (gkl)k,l∈Zr are inverse if∑m≤l≤k

fklglm = δkm

or equivalently ∑m≤l≤k

gklflm = δkm.


Case r = 1: Warnaar’s matrix inversionThroughout, |p| < 1 is fixed and

θ(x) =

∞∏j=0

(1− pjx)(1− pj+1/x).

In the trigonometric case p = 0, we have θ(x) = 1− x.

Warnaar (2002) obtained the pair of inverse matrices

fnk =

∏n−1j=k θ(ajck)θ(aj/ck)∏nj=k+1 θ(cjck)θ(cj/ck)

,

gkl =clθ(alcl)θ(al/cl)

ckθ(akck)θ(ak/ck)

∏kj=l+1 θ(ajck)θ(aj/ck)∏k−1j=l θ(cjck)θ(cj/ck)

.

Trigonometric case is due to Krattenthaler (1996).Generalizes Gold & Hsu (1973), Carlitz (1973),Bressoud (1983) etc.


An elliptic Ar matrix inversion, page 1

We give an r-dimensional extension of Warnaar’s inversion.

Let (ak)k∈Z and (cj(k))k∈Z, 1 ≤ j ≤ r, be generic scalars. Thenthe following pair of lower-triangular matrices are mutuallyinverse:

fnk =

∏|n|−1t=|k|

{θ (atc1(k1) · · · cr(kr))

∏rj=1 θ (at/cj(kj))

}∏r

i=1

∏nit=ki+1

{θ(ci(t)c1(k1) · · · cr(kr))

∏rj=1 θ(ci(t)/cj(kj))

}and . . .


An elliptic Ar matrix inversion, page 2

gkl =θ(a|l|c1(l1) · · · cr(lr))θ(a|k|c1(k1) · · · cr(kr))

∏1≤i<j≤r

θ(ci(li)/cj(lj))

θ(ci(ki)/cj(kj))

×r∏

j=1

cj(lj)jθ(a|l|/cj(lj))

cj(kj)jθ(a|k|/cj(kj))

×

∏|k|t=|l|+1

{θ(atc1(k1) · · · cr(kr))

∏rj=1 θ(at/cj(kj))

}∏r

i=1

∏ki−1t=li

{θ(ci(t)c1(k1) · · · cr(kr))

∏rj=1 θ(ci(t)/cj(kj))

} .Trigonometric case due to Schlosser (1997); generalizes Milne& Lilly (1992), Lilly & Milne (1993), Bhatnagar & Milne (1997),Milne (1997).


Why Ar?

The Ar inversion has applications to series of the type∑k1,...,kr

∏1≤i<j≤r

θ(xiqki/xjq

kj ) · · · (elliptic),

∑k1,...,kr

∏1≤i<j≤r

(xjqkj − xiqki) · · · (trigonometric),

∑k1,...,kr

∏1≤i<j≤r

(xj + kj − xi − ki) · · · (rational).

Originate in work of Holman, Biedenharn & Louck (1976) on6j-symbols of SU(r).

The double products are essentially Weyl denominator of rootsystems A(1)

r (elliptic) or Ar (trigonometric/rational).


Proof of elliptic Ar matrix inversion, page 1

We sketch the proof of the Ar matrix inversion.The identity FG = Id is equivalent to

n1,...,nr∑k1,...,kr=0

( ∏1≤i<j≤r

1

θ(ci(ki)/cj(kj))

r∏j=1

1

cj(kj)j

×

∏|n|−1t=1

{θ(atc1(k1) · · · cr(kr))

∏rj=1 θ(at/cj(kj))

}r∏

i=1

ni∏t=0, t 6=ki

{θ(ci(t)c1(k1) · · · cr(kr))

r∏j=1

θ(ci(t)/cj(kj))})

= 0

unless all nj = 0.


Proof of elliptic Ar matrix inversion, page 2

We prove this identity by induction on |n|.

Consider the sum as a function of a1. At a1 = cl(t), 1 ≤ l ≤ r,0 ≤ t ≤ nl, it reduces to a similar sum with nl replaced bynl − 1. That sum vanishes by the induction hypothesis.

By elementary facts on theta functions this is enough toconclude that the sum vanishes.

This proof follows Rains’ (2010) proof of a related identity.It is very simple compared to how various special cases havebeen proved.


Table of Contents




Application 1: A new Ar Jackson summation Hjalmar Rosengren 12/25

Elliptic hypergeometric notation

We use elliptic shifted factorials

(a; q)k = θ(a)θ(qa) · · · θ(qk−1a),

(a1, . . . , am; q)k = (a1; q)k · · · (am; q)k.

In the trigonometric case p = 0 we have

(a; q)k = (1− x)(1− qa) · · · (1− qk−1a).

Elliptic Ar series contain the factor

∆(xqk)

∆(x)=

∏1≤i<j≤r

qkjθ(xiqki/xjq

kj )

θ(xi/xj).


One-variable case

The one-variable elliptic Jackson summation is

n∑k=0

θ(aq2k)

θ(a)

(a, q−n, b, c, d, e; q)k(q, aqn+1, aq/b, aq/c, aq/d, aq/e; q)k

qk

=(aq, aq/bc, aq/bd, aq/cd; q)n(aq/b, aq/c, aq/d, aq/bcd; q)n

where bcde = a2qn+1.Proved by Frenkel and Turaev (1997) based onDate et al. (1988).Trigonometric case is Jackson’s 8W7-summation (1921).


Standard Ar Jackson summationThe “standard” Ar Jackson summation is

n1,...,nr∑k1,...,kr=0

(∆(xqk)

∆(x)

r∏i=1

θ(axiq|k|+ki)

θ(axi)

×r∏

i=1

(axi; q)|k|

(axiq|n|+1; q)|k|·

(b, c; q)|k|

(aq/d, aq/e; q)|k|q|k|

×r∏

i,j=1

(q−njxi/xj ; q)ki(qxi/xj ; q)ki

r∏i=1

(dxi, exi; q)ki(axiq/b, axiq/c; q)ki

=

(aq/bd, aq/cd; q)|n|

(aq/d, aq/bcd; q)|n|

r∏i=1

(axiq, axiq/bc; q)ni

(axiq/b, axiq/c; q)ni

,

where bcde = a2q|n|+1. Due to Milne (1997) in trigonometriccase and R. (2004) in general.


Proof of new Ar Jackson summation

We sketch the proof of our new Ar Jackson summation.

Write the standard Ar Jackson summation in the form∑k

fnkak = bn.

Here, fnk is obtained from our Ar matrix inversion byspecializing all parameters to geometric progressions.

The inverse identity ∑l

gklbl = ak

is our new Ar Jackson summation.


New Ar Jackson summationWe have the new Ar Jackson summation

∑0≤ki≤ni

i=1,...,r

(∆(xqk)

∆(x)

θ(aq2|k|)

θ(a)q|k|

×(a, b, c; q)|k|

(aq|n|+1, aq/b, aq/c; q)|k|

n∏i=1

(bcd/axi; q)|k|−ki(d/xi; q)|k|

(d/xi; q)|k|−ki(bcdq−ni/axi; q)|k|

×r∏

i,j=1

(q−njxi/xj ; q)ki(qxi/xj ; q)ki

r∏i=1

(a2xiq|n|+1/bcd; q)ki

(axiq/d; q)ki

=

(aq, aq/bc; q)|n|

(aq/b, aq/c; q)|n|

r∏i=1

(axiq/bd, axiq/cd; q)ni

(axiq/d, axiq/bcd; q)ni

.

The trigonometric case is due to Schlosser (2008).Application 1: A new Ar Jackson summation Hjalmar Rosengren 17/25

Applications

Our “new” summation has already been announced andapplied in several publications.

In R. (2011) we found that it appears from Felder’s SU(2) ellipticquantum group.

It has also been combined with other Jackson summations toget generalizations of Bailey’s 10W9-transformation:

With a Gustafson–Rakha-type Ar Jackson summation(R. 2017).With the standard Ar Jackson summation(Bhatnagar and Schlosser, 2018).With itself (R. and Schlosser, in preparation).


Table of Contents




Application 2: An Ar quadratic transformation Hjalmar Rosengren 19/25

One-variable case

Warnaar (2002) proved that if cde = qa2, then

n∑k=0

θ(aq3k)

θ(a)

(a, b, q/b; q)k(aq/c, aq/d, aq/e; q)k

(c, d, e; q2)k(q2, aq2/b, aqb; q2)k

qk

=(aq, c/a; q)n

(aq/d, aq/e; q)n

(q2d, q2e; q2)n(q2, q2de; q2)n

×n∑

k=0

θ(deq4k)

θ(de)

(de, q−2n, d, e, deb/a, deq/ab; q2)k(qn+1a; q)2k(q2, q2n+2de, q2e, q2d, q2a/b, qab; q2)k(q−nde/a; q)2k

q2k.

Note that terms on LHS are independent of n.

Trigonometric case is due to Gasper and Rahman (1990);implies quadratic summations of Gessel & Stanton (1983),Gasper (1989), Rahman (1993), Chu (1995) etc.


An Ar Karlsson–Minton identity

We need the case m = 2 of the following Karlsson–Minton-typeidentity. If b1 · · · brx1 · · ·xr = q−mN , then

∑k1+···+kr=N,k1,...,kr≥0

∆(xqmk)

∆(x)

n∏i,j=1

(xibj ; qm)ki

(qmxi/xj ; qm)ki

n∏i=1

(xicqN ; q)mki

(xic; q)mki

= (−1)Nq−mN(N+1)

2

n∏i=1

(c/bi; q)N(cxi; q)N

.

The one-variable case r = 2 is due to Warnaar (2002).

We deduce this from a transformation formula due to Kajihara &Noumi (2003) and R. (2006).


Proof of Ar quadratic transformation

We sketch how to obtain our Ar quadratic transformation.

We first invert the standard Ar Jackson summation (in anotherway than before) to get a quadratic Ar summation (due toSchlosser 1997 in trigonometric case).

We then write down a 2r-dimensional sum, which can bereduced to an r-dimensional sum in two ways: by the Ar

Karlsson–Minton summation or by the quadratic Ar summation.

This gives a transformation between two r-dimensional sums.


An Ar quadratic transformation: Left-hand-side

We have the following Ar quadratic transformation.

If cd1 · · · dr+1x1 · · ·xr = qa2, then

∑k1,...,kr≥0,|k|≤N

(∆(xq2k)

∆(x)

r∏i=1

θ(axiq|k|+2ki)

θ(axi)

×(b, q/b; q)|k|

∏ri=1(axi; q)|k|

(aq/c; q)|k|∏r+1

i=1 (aq/di; q)|k|q|k|

×r∏

i=1

(cxi; q2)ki

∏r+1j=1(djxi; q

2)ki(q2axi/b, qaxib; q2)ki

∏rj=1(q

2xi/xj ; q2)ki

)= · · ·


An Ar quadratic transformation: Right-hand-side

· · · =(c/a; q)N

∏ri=1(aqxi; q)N∏r+1

i=1 (aq/dj ; q)N

∏r+1i=1 (λq2/di; q

2)N(q2; q2)N

∏ri=1(λq

2xi, q2)N

×∑

k1,...,kr≥0,|k|≤N

(∆(xq2k)

∆(x)

r∏i=1

θ(λxiq2|k|+2ki)

θ(λxi)

×(q−2N , λb/a, λq/ab; q2)|k|

∏ri=1(λxi; q

2)|k|

(q−Nλ/a; q)2|k|∏r+1

i=1 (q2λ/di; q2)|k|q2|k|

×r∏

i=1

(qN+1axi; q)2ki∏r+1

j=1(djxi; q2)ki

(q2N+2λxi, q2axi/b, qaxib; q2)ki∏r

j=1(q2xi/xj ; q2)ki

),

where λ = qa2/c = d1 · · · dr+1x1 · · ·xr.

This is new even in the trigonometric (and rational) case.Application 2: An Ar quadratic transformation Hjalmar Rosengren 24/25

Outlook

We expect more applications of our Ar matrix inversion, suchas cubic summations and transformations.

We also have two other matrix inversions, related to BCr or Cr

root systems. Applications of these remain to be investigated.

For the future, it may be interesting to look for integralanalogues of our finite sum identities.May have physical relevance, since elliptic hypergeometricintegrals on root systems appear as indices of four-dimensionalsupersymmetric quantum field theories (Dolan & Osborn 2009,Spiridonov & Vartanov 2011 etc.).

Thanks for your attention!


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