Askey-Wilson algebra and a generalization of the
Heun operators
Alexei Zhedanov
Renmin University, Beijing, China

based on joint work with Pascal Baseilhac (Tours, Franca), Alberto Grünbaum (Berkeley), Satoshi
Tsujimoto (Kyoto) and Luc Vinet (CRM, Montreal)

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

1 / 34

Bispectral pairs

Bispectral pair is a generalization of Leonard pair.
Formal denition:
The pair of operators X, Y which satisfy AW (3) algebra relations[X, Y ] = Z,
[Y, Z] = −2νY XY + A2 Y 2 + A1 {X, Y } + BY + C1 X + E1 ,[Z, X] = −2νXY X + A1 X 2 + A2 {X, Y } + BX + C2 Y + E2 ,A1,2 , C1,2 , E1,2 , B, ν are

Lutzenko, AZ (1992)).
The Jacobi identity

structure constants of the AW-algebra (Granovskii,[[X, Y ], Z] + [[Y, Z], X] + [Z, X], Y ] = 0

holds for AW-algebra.
Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

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In nite-dimensional case

Casimir operator

bispectral pair ⇐⇒ Leonard pair
Q = Z 2 −ν XY 2 X + Y X 2 Y +(2+ν) (A1 XY X + A2 Y XY )+ quadratic terms

commutes with generators

Ane property:[Q, X] = [Q, Y ] = [Q, Z] = 0changeX → α1 X + α2 , Y → β1 Y + β2

leads to the same algebra with "shifted" parameters

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

3 / 34

q-algebra and q-polynomials
When q 6= 0, ±1 the parameter ν 6= 0. New parameter q instead of ν2ν = q + q −1 − 2Then AW-algebra can be presented in terms of two relations (Terwilliger,
2000)
X 2 Y + Y X 2 − (q + q −1 )XY X = A1 Y + BX + C1 ,Y 2 X + XY 2 − (q + q −1 )Y XY = A2 X + BY + C2

Another form in terms of "linear q-algebra" (Granovskii, AZ 1993):[X, Y ]q = Z, [Y, Z]q = A1 Y + BX + C1 , [Z, X]q = A2 Y + BX + C2

where [X, Y ]q = XY − qY X is "q-mutator"

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

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Z3 form (Wiegmann, Zabrodin, 1995, Terwilliger, 2004)[X, Y ]q = a3 Z + ω3 , [Y, Z]q = a1 X + ω1 , [Z, X]q = a2 Y + ω2

All q-polynomials from Askey scheme are described by representations ofabove algebra.
In case if a1 a2 a3 6= 0 it is possible to put
a1 = a2 = a3 = 1

Then there are 4 independent parameters: ω1 , ω2 , ω3 and the value Q of the
Casimir operator. They correspond to 4 parameters of Askey-Wilson
polynomials.

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

5 / 34

Degenerations
When some parameters ai are zero then the Aksey-Wilson polynomials
degenerate to other polynomials from the Askey scheme.

big q-Jacobi polynomials (or q-Hahn

For example when a1 = 0 one has
polynomials in nite-dimensional case).
The case a1 = ω1 = 0 corresponds tolittle q-Jacobi polynomials etc.

The maximal degeneration a1 = a2 = a3 = 0 leads to the algebra[X, Y ]q = 1, [Y, Z]q = 1, [Z, X]q = 1

q-oscillator

Each relation is the: XY − qY X = 1. This algebra is called
equitable slq (2) algebra (Terwilliger, 2005)

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

6 / 34

Bannai-Ito algebra

Put q = −1:{X, Y } = a3 Z + ω3 , {Y, Z} = a1 X + ω1 , {Z, X} = a2 Y + ω2

Describes Bannai-Ito polynomials (=limit q → −1 of Askey-Wilson
polynomials)

Degenerations: a = 0 corresponds to big -1 Jacobi polynomials (L.Vinet,
AZ, 2010);

1

a1 = ω1 = 0 corresponds tolittle -1 Jacobi polynomials.

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

7 / 34

Algebraic Heun pencil

Generic bilinear operator constructed from X and Y :M = τ1 XY + τ2 Y X + τ3 X + τ4 Y + τ0 I

We call M thealgebraic Heun operator

Properties(i) M is generic bilinear operator with respect to X and Y .
(ii) M possesses a bispectrality property.

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

8 / 34

Why Heun operator?

Example:Jacobi algebra
X = x,

Y = x(1 − x)∂x2 + (ν1 x + ν2 )∂x

The Jacobi polynomials are eigenfunctions of the operator Y
Y Pn(ω1 ,ω2 ) (x) = λn Pn(ω1 ,ω2 ) (x)The operator X is 3-diagonal:
XPn (x) = Pn+1 (x) + bn Pn (x) + un Pn−1 (x)The operator M is 2nd-order dierential operator which is tridiagonal withrespect to Jacobi polynomials
M Pn (x) = ξn+1 Pn+1 (x) + ηn Pn (x) + ζn Pn−1 (x)Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

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Main statements (A.Grünbaum, L.Vinet, AZ, 2016)

Proposition 1. The most general 2nd-order dierential operator which is
3-diagonal on Jacobi polynomials coincides with M .

Proposition 2. The operator M coincides with generic Heun operator.

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

10 / 34

Assume that X and Y are nite-dimensional. Then
Xen = λn en ,Y dn = µn dn , n = 0, 1, 2, . . . , N

and
Xdn = an+1 dn+1 + bn dn + an dn−1 ,Y en = ξn+1 en+1 + ηn en + ξn en−1

It is clear that M is 3-diagonal with respect to both bases en and dn :M en = {en−1 , en , en+1 },M dn = {dn−1 , dn , dn+1 }Inverse statement: the most general operator with this (bispectrality)

property is bilinear in X and Y and hence = algebraic Heun operator(Nomura and Terwilliger, 2007)

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

11 / 34

Time and band limiting

Time and band limiting in Fourier analysis

Fourier transform
Inverse transform1
F (ω) = √
2π

Z ∞

1
f (t) = √2πZ ∞e−iωt f (t)dt−∞eiωt F (ω)dω−∞In practice we always have restrictions −T < t < T and −Ω < ω < Ω. For
example Ω = 20KHz for sound.

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

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Corresponding projection operators D(T ) and B(Ω). They dened as
D(T )f (t) =
B(Ω)F (ω) =f (t) if |t| ≤ T
0 if |t| > TF (ω) if |ω| ≤ Ω
0 if |ω| > Ω

What is action of B(Ω) on functions f (t)? The
B(Ω)f (t) =D(T ) and B(Ω) are1πintegral operator:Z ∞sin Ω(t − s)f (s)dst−s−∞time- and band-limiting operators.

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

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Is it possible to achieve simultaneous restriction of Fourier analysis to theintervals [−T, T ] and [−Ω, Ω]? The answer is NOT.
Uncertainty principle
σf σF ≥ 1/2

where
σf2 =

Z ∞

t2 |f (t)|2 dt

−∞

σF2 =

Z ∞

ω 2 |F (ω)|2 dt−∞

I.e. if the signal is restricted on small interval [T, T ] then the Fouriertransform F (ω) should be unrestricted.

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

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However, one can consider theintervalconcentration of the signal on the prescribedRT|f (t)|2 dt
2αf = R −T∞|f (t)|2 dt−∞

if the frequency function is already restricted F (ω) = 0, |ω| > Ω.

The problem: nd the band-limited functions f (t) which provide maximal
value of the concentration αf2 .

Answer: we should solve the eigenvalue problem
VT,Ω f (x) = λf (x)where VT,Ω is theintegral operator
VT,Ω f (x) =

1
π

Z T

sin Ω(x − s)f (s)dsx−s−TAlexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

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Equivalent presentation
VT,Ω = BΩ DT BΩ

The operator VT,Ω is compact and self-adjoint and has only discrete spectrum
0 < λ < 1.
Themaximal eigenvalue λ yields the solution:0

max{αf2 } = λ0

Because VT,Ω is integral operator, the eigenvalue problem is complicated tosolve. There are no eective methods to nd solutions.

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

16 / 34

Main idea of BTL
Find 2nd-order dierential operator L
L = A(x)∂x2 + B(x)∂x + C(x)which commutes with VT,Ω
LVT,Ω = VT,Ω L

Then all eigenfunctions of VT,Ω are also eigenfunctions of L and can be found(at least approximately) by standard methods.
Solution was found by Slepian, Pollak and Landau:
L = ∂x (1 − x2 )∂x − Ω2 x2 ,where T = 1 (standard normalization of t).

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

17 / 34

Equation Lf (x) = µf (x) has
Why?

prolate spheroidal functions as solutions.

Explanation: L is a special case of algebraic Heun operator.
Indeed, choose

X = x2 ,Y = ∂x2

Then X, Y is a bispectral pair because[X, [X, Y ]] = 8X,[Y, [Y, Z] = 8Y

Restriction of the operator X corresponds to time-limiting. Restriction of Y
corresponds to band-limiting.

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

18 / 34

Try to construct genericsymmetric bilinear operator from X and Y
L = α{X, Y } + βX + γY

such that [L, DT ] = [L, BΩ ] = 0. Then L commutes with VT,Ω = BΩ DT BΩ .
Easily veried that1
L = − {X, Y } − Ω2 X + Y
2Prolate spheroidal equation = special case of conuent Heun equation.

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

19 / 34

Finite dimensional BTL

Consider nite-dimensional case: n = 0, 1, . . . , N . Two eigenbases en and dn
and two projectors π1 and π2

en , if n ≤ J1
0, if n > J1

dn , if n ≤ J2
0, if n > J2

π1 e n =π2 d n =They satisfy

π12 = π1 ,

π22 = π2 .

If J1 = N then π1 = I . Similarly, if J2 = N then π2 = I .

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

20 / 34

Combination of projectors yields two restriction operators:
V1 = π1 π2 π1 = K1 K2 ,V 2 = π2 π1 π2 = K 2 K 1 ,where
K1 = π1 π2 ,K 2 = π2 π1

Operators V1 and V2 are symmetric and hence are diagonalizable and have thesame eigenvalues (may be degenerate).
When J1 = J2 = N both V1 and V2 are identity operators V1 = V2 = I . When
only J2 = N and J1 arbitrary then V1 = V2 = J1 and the operator V1 has
J1 + 1 eigenvalues equal to 1 and N − J1 zero eigenvalues.
But what are eigenvectors and eigenvalues of operators V1 , V2 for arbitrary

J1 , J2 ?

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

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The operators V1 , V2 are highly nonlocal. For example:
(1)

Vik =

J2
X

w̃s χi (µs )χk (µs )s=0Hence the problem of funding eigenvectors and eigenvalues iscomplicated.

very

Main idea - to nd a 3-diagonal matrix M which commutes withboth V1 and V2[M, V1 ] = [M, V2 ] = 0.

Because eigenvalue problem for 3-diagonal matrices is much easier to solve.

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

22 / 34

Perline's bispectral BTL operator

Assume that X and Y is a Leonard pair
Try thesymmetric bilinear operator (idea of Perline, 1987)
M = τ1 {X, Y } + τ3 X + τ4 Y + τ0such that M commutes with both projectors[M, π1 ] = [M, π2 ] = 0

Then M will also commute with V1 , V2 .

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

23 / 34


∗
∗



M en = 





∗
∗ ∗
∗ ∗ ∗
∗ ∗
0



0
∗
∗







∗ 

∗ ∗
∗ ∗


∗
∗



M dn = 





∗
∗ ∗
∗ ∗
0


0
∗
∗






∗


∗ ∗

∗ ∗ ∗
∗ ∗Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

24 / 34

This leads to restrictions on coecients τ3 , τ4
τ1 (λJ1 + λJ1 +1 ) + τ4 = 0,

τ1 (µJ2 + µJ2 +1 ) + τ3 = 0

always possible to nd coecients τ , τ .
The only exception - Bannai-Ito spectrum:
It is

3

4

λn = (−1)n (nα + β)

In this case degeneration happens

∗
∗




M en = 







∗
∗
∗
∗

∗
∗
∗
∗

∗
∗

..

.






,






∗






M dn = 







∗
∗

∗
∗
∗
∗

∗
∗
∗
∗

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

∗
∗

..

.







.






25 / 34BTL for anti-Krawtchouk polynomials

For BI polynomials the above method does not work - there are no nontrivial
3-diagonal commuting operator M commuting with V1 , V2 .
Instead, one can try 5-diagonal operator M .
Consider special example: anti-Krawtchouk polynomials (V.Genest, L.Vinet,
AZ, 2014).

Anti-spin algebra{L1 , L2 } = L3 ,{L2 , L3 } = L1 ,{L3 , L1 } = L2 ,Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

26 / 34

Casimir operator

Q = L21 + L22 + L23

takes the value
Q = (N + 1/2)(N + 3/2),

Search commuting operator M in terms of

N = 1, 2, . . .

quartic and cubic expressions

M = {L21 , L22 } + α1 {L21 , L2 } + α2 {L22 , L1 }+M ispentadiagonal

α3 L21 + α4 L22 + α5 L1 + α6 L2

M en = Gn en−2 + Fn en−1 + Hn en + Fn+1 en+1 + Gn+2 en+2

Projector


πN1 en =en , n ≤ N1.
0, n > N1

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

27 / 34

Commutativity [M, πN1 ] = 0 holds i
GN1 +1 = GN1 +2 = FN1 +1 = 0

Similalrly, "dual" projector
πN2 dn =dn , n ≤ N2.
0, n > N2

Commutativity [M, πN2 ] = 0 conditions
G̃N2 +1 = G̃N2 +2 = F̃N2 +1 = 0

6 equations for 6 unknowns. Solution:
α1 = (−1)N1 , α2 = (−1)N2 , α3 = −1 − κ1 ,α4 = −1 − κ2 , α5 = (−1)N2 κ1 , α6 = (−1)N1 κ2 ,where

κ1 = 2N1 2 + 4N1 + 5/2,

κ2 = 2N2 2 + 4N2 + 5/2.

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

28 / 34

Classication of algebraic Heun operators
Askey scheme ⇒ Algebraic Heun operatorsu(1, 1) and Heisenberg-Weyl algebras ⇒ Hermite andLaguerre polynomials ⇒ conuent Heun operators(F.A.Grünbaum, L.Vinet, AZ, 2016)

Jacobi algebra ⇒ Jacobi polynomials ⇒ ordinary Heun
operator (F.A.Grünbaum, L.Vinet, AZ, 2016)
Hahn algebra ⇒ Hahn polynomials ⇒ Heun-Hahn andHeun-Krawtchouk operators (L.Vinet, AZ, 2018)
q-Hahn algebra ⇒ big (little) q-Jacobi, q-Hahn
polynomials ⇒ big (little) q-Heun operators (P.Bseilhac,

L.Vinet, AZ, 2018)

AW algebra ⇒ AW polynomials ⇒ Heun-AW
operators (P.Bseilhac, S.Tsujimoto, L.Vinet, AZ, to appear)
Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

29 / 34

Evolution of algebraic approach to Heun operators
1. A.Erdelyi (40-th) - tridiagonal property of solutions of Heun equation withrespect to Jacobi polynomials.
2. R.Perline (1987) - rst appearance of special nite-dimensional AHO.
3. A.Turbiner (end of 80-th - beginning of 90-th) - description of the ordinary
Heun operator in terms of quadratic combination of sl2 generators ⇒equivalent to Krawtchouk or Meixner AHO.
4. K.Nomura, P.Terwilliger (2007) - nite-dimensional analogue of AHO andgeneric bispectrality property.
5. M.Ismail, E.Koelink (2010-2016) - Method of 3-diagonalization andconstruction of special cases of AHO.
6. A.Grümbaum, L.Vinet, AZ (2016) - ordinary Heun operator as bilinear
pencil on Jacobi algebra.
7. A.Grümbaum, L.Vinet, AZ (2017) - generic AHO and BTL problem
Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

30 / 34

Perspectives and open problems
Elementary reductions of algebraic Heun operators.
When equation M ψ = 0 is reduced to (generalized) hypergeometricequation?

Elliptic Heun operator. Krichever and Zabrodin (1995)
constructed elliptic Lamé equation on the base of Sklyanin algebra.
How to construct corresponding Heun pencil?
Multivariate analogs of AHO. Ruijsenaars and Van Diejen
considered multivariate exactly solvable models related with generalized
Heun operators. What is their algebraic description?

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

31 / 34

Heun pencil and biorthogonal functions. The AHO is alinear pencil
M = τ1 XY + τ2 Y X + τ3 X + τ4 Y + τ0

Consider the equationMψ = 0If eigenvalue λ = −τ0 then we have the ordinary eigenvalue problem.
Other choices lead to generalized eigenvalue problem
M1 ψ = λ M 2 ψ

where M1 , M2 - 3-diagonal operators. This leads to theory ofbiorthogonal rational functions (AZ, 1999). What are the mostgeneral BRF arising in this problem?

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

32 / 34

Applications to integrable systems. It is known that after
separation of variables for some integrable systems (i.e. Kepler problem)in elliptic coordinates the Heun operators appear as integrals. How tounderstand this phenomenon from algebraic point of view?
BTL and random matrix models. Integral equations (andcommuting dierential operators) for some RMM areas forspecial examples of BTL (Mehta, Tracy, Widom). What is more deep
relation between two problems?

the same

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

33 / 34

Thank you for your attention!

Alexei Zhedanov (Renmin University, Beijing,
Askey-Wilson
China) algebra and a generalization of the Heun operators

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