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Hypergroups made concrete for special
orthogonal polynomials, a survey
Tom Koornwinder
Korteweg-de Vries Institute, University of Amsterdam
[email protected]
lecture at Colloque Orthogonal Polynomials and Hypergroups,
Toulouse, France, 18-19 June 2014
last modified: 23 June 2014
Tom Koornwinder
Hypergroups made concrete
Toulouse and Thomas Stieltjes
Thomas J. Stieltjes, 1856–1894
until 1885 in the Netherlands;
assistant at Leiden observatory;
many letters exchanged with Hermite;
1885–1886 in Paris;
from 1886 in Toulouse:
1886–1889 maître de conférences;
1889–1894 professeur
His tomb is at the cemetery of Terre Cabade in Toulouse.
Stieltjes prize: an annual prize for the best PhD thesis in
mathematics defended at a Dutch university, see
http://testweb.science.uu.nl/WONDER/prizes.html
Tom Koornwinder
Hypergroups made concrete
Askey’s sabbatical, Amsterdam, 1969–1970
Mathematisch Centrum, 2e Boerhaavestraat 49, Amsterdam
(earlier situation; in 1969 the top floor looked different)
http://beeldbank.amsterdam.nl/afbeelding/010003018854
Tom Koornwinder
Hypergroups made concrete
Askey’s sabbatical, Amsterdam, 1969–1970 (cntd.)
Dick Askey reading a math book in his Dutch home,
Amstelveen, 1970
Tom Koornwinder
Hypergroups made concrete
Askey’s sabbatical, Amsterdam, 1969–1970 (cntd.)
In his lectures in Amsterdam, 1969–1970, Askey emphasized
positivity properties of (special) orthogonal polynomials related
to generalized translation and convolution. Thus he presented
special hypergroups avant la lettre.
These results were presented with a much wider scope in his
1975 SIAM Lecture Notes.
Tom Koornwinder
Hypergroups made concrete
Askey’s 1975 SIAM Lecture Notes
Tom Koornwinder
Hypergroups made concrete
Four canonical problems
{pn } and {qn } systems of orthogonal polynomials;
Z
pm (x)pn (x) dµ(x) = hn δm,n .
Often pn (x0 ) = 1 = qn (x0 ) for some special x0 .
product formula:
Z
pn (x)pn (y ) =
transmutation:
pn (z)K (x, y , z) dµ(z).
Z
qn (x) = pn (y )A(x, y ) dµ(y ).
linearization of products:
pm (x)pn (x) =
m+n
X
cm,n,k pk (x)/hk .
k =|m−n|
connection formula:
qn (x) =
n
X
an,k pk (x)/hk .
k =0
Find the integral and summation kernels explicitly and/or see
when these kernels are nonnegative.
Tom Koornwinder
Hypergroups made concrete
Four canonical problems (cntd.)
A product formula with nonnegative kernel is a necessary
aspect of a hypergroup associated with orthogonal polynomials.
Similarly, nonnegative linearization coefficients are necessary
for a dual hypergroup associated with orthogonal polynomials.
Transmutations and connection formulas with positive kernel
have some hypergroup flavour, in particular when they connect
with cosine functions or Chebyshev polynomials of the first kind.
Tom Koornwinder
Hypergroups made concrete
Product formulas
Z
pn (x) pn (y ) =
K (x, y , z) =
pn (z) K (x, y , z) dµ(z)
∞
X
pn (x) pn (y ) pn (z)
n=0
Tom Koornwinder
hn
Hypergroups made concrete
Gegenbauer product formula
Jacobi polynomials (α, β > −1) :
(α,β)
(α,β)
Rn
1
∫
−1
(x) :=
Pn
(α,β)
Pn
(x)
(1)
,
(1 − x)α (1 + x)β dx
,
dµα,β (x) := R 1
α (1 + x)β dx
(1
−
x)
−1
(α,β)
(α,β)
Rm (x)Rn (x) dµα,β (x)
(α,β)
= hn
δm,n .
(α,α)
(α,α)
(x)Rn
Gegenbauer product formula (α > − 12 ) : Rn
1
1
1 (α,α)
= ∫ Rn
xy + (1 − x 2 ) 2 (1 − y 2 ) 2 t dµα− 1 ,α− 1 (t).
2
−1
− 12 :
(y )
2
1
2
For α =
cos nφ cos nψ =
cos n(φ + ψ) + cos n(φ − ψ) .
Generalized translation (Levitan, Bochner, Hirschman):
1
1
1 Ty [f ](x) := ∫ f xy + (1 − x 2 ) 2 (1 − y 2 ) 2 t dµα− 1 ,α− 1 (t).
2
−1
2
Positivity of generalized translation:
f (x) =
∞ b
(α,α)
X
f (n)Rn
(x)
n=0
hn
≥ 0 ⇔ Ty [f ](x) =
∞ b
(α,α)
(α,α)
X
f (n)Rn
(x)Rn
(y )
n=0
Tom Koornwinder
Hypergroups made concrete
hn
≥0
Gegenbauer product formula (cntd.)
Product formula and generalized translation in kernel form:
Z 1
(α,α)
(α,α)
(α,α)
Rn
(x)Rn
(y ) =
Rn
(z) Kα,α (x, y , z) dµα,α (z),
−1
Z 1
Ty [f ](x) =
f (z) Kα,α (x, y , z) dµα,α (z), where
−1 ∞
X
Kα,α (x, y , z) =
(α,α)
Rn
(α,α)
(x)Rn
(α,α)
(y )Rn
(α,α)
(z)/hn
n=0
1)2
α− 1
(1 − x 2 − y 2 − z 2 + 2xyz)+ 2
Γ(α +
α ≥ 0.
=
Γ(α + 12 )Γ(α + 23 ) (1 − x 2 )(1 − y 2 )(1 − z 2 )
Convolution:
Z
1
(f ∗ g)(x) :=
Z
1
Z
1
Ty [f ](x) g(y ) dµα,α (y ) =
−1
f (z) g(y )
−1
× Kα,α (x, y , z) dµα,α (y ) dµα,α (z) =
∞ b
X
b(n)Rn(α,α) (x)
f (n)g
(α,α)
n=0
Tom Koornwinder
−1
hn
Hypergroups made concrete
.
Convolution algebra
Z
Put
1
kf k1 :=
|f (x)| dµα,α (x).
Then we conclude:
−1
kf ∗gk1 ≤ kf k1 kgk1 ,
kf ∗gk∞ ≤ kf k∞ kgk1 ,
f , g ≥ 0 ⇒ f ∗g ≥ 0.
The same machinery would work for other orthogonal systems,
provided we have a product formula with positive kernel.
Gegenbauer case α = 21 (d − 3) by group theory:
G = O(d), K = O(d − 1), Ω = G/K = S d−1 (Gelfand pair).
Z
(f ∗ g)(x) = (g ∗ f )(x) =
f (y ) g(y −1 x) dy (f , g K -biinvariant),
G
Z
(F ∗ G)(hx, y i) =
F (hx, zi) G(hz, y i) dω(z) (x, z ∈ Ω),
Ω
Z
(α,α)
φ(x) φ(y ) =
φ(xky ) dk (x, y ∈ G, φ(x) = Rn
(hxe1 , e1 i) ).
K
φ is spherical function (Gelfand); immediate positivity results;
works also for certain other Jacobi parameters (Gangolli).
Tom Koornwinder
Hypergroups made concrete
Jacobi product formula
In the analogous Jacobi problem for α > β > − 12 Gasper
(1971) showed that
(α,β)
Rn
(α,β)
(x)Rn
Z
1
(y ) =
−1
(α,β)
Rn
(z) Kα,β (x, y , z) dµα,β (z)
with Kα,β (x, y , z) ≥ 0 as a definite integral of an explicit
nonnegative elementary function.
In fact he found this by combining two formulas in Watson’s
Treatise on the theory of Bessel functions, see there pages 411
and 413.
Tom Koornwinder
Hypergroups made concrete
Jacobi product formula (cntd.)
Watson essentially has the same nonnegative kernel in the
following two product formulas (α > β > − 12 ):
Z ∞
Jβ (z) e
1
Jα (x) Jβ (y )
= α
Kα,β (x, y , z) z 2β+1 dz,
α
β
x
2 Γ(α + 1) 0
y
zβ
Z π/2
(α,β)
(α,β)
(α,β)
Rn (cos 2θ1 ) Rn (cos 2θ2 ) =
Rn (cos 2θ3 )
0
eα,β (sin θ1 sin θ2 , cos θ1 cos θ2 , cos θ3 ) (cos θ3 )2β+1 sin θ3 dθ3 .
×K
Askey’s question
Rewrite the Gasper-Watson Jacobi product formula as
something similar to the Gegenbauer product formula
(α,α)
Rn
(α,α)
(x)Rn
1
(α,α)
= ∫ Rn
−1
(y )
1
1 xy + (1 − x 2 ) 2 (1 − y 2 ) 2 t dµα− 1 ,α− 1 (t).
2
2
For this purpose work with addition formulas and group theory.
Tom Koornwinder
Hypergroups made concrete
Gegenbauer addition formula
The Gegenbauer product formula gives the constant term in the
Gegenbauer addition formula:
n
X
(−1)k (−n)k (n + 2α + 1)k
(α,α)
2 12
2 12
Rn
xy +(1−x ) (1−y ) t =
(α− 12 ,α− 12 )
k =0 22k ((α + 1)k )2 hk
(α+k ,α+k )
×(1−x 2 )k /2 Rn−k
(α+k ,α+k )
(x) (1−y 2 )k /2 Rn−k
(α− 21 ,α− 12 )
(y ) Rk
(t).
For α = 12 (d − 3) by group theory: G = O(d) ⊃ K = O(d − 1) ⊃
M = O(d − 2), A = SO(2) ⊂ G commuting with M;
φ spherical function for (G, K ) and ψδ for (K , M) (Gelfand pairs):
X
φ(a1 k a2 ) =
φba1 ,a2 (δ) dδ ψδ (k ) (k ∈ K , a1 , a2 ∈ A).
δ∈(K /M)b
Or as reproducing kernel for spherical harmonics of degree n:
(α,α)
Rn
(hx, y i)
dn
1 X
=
Yn,k (x) Yn,k (y )
dn
(x, y ∈ S d−1 = O(d)/O(d−1)).
k =1
Tom Koornwinder
Hypergroups made concrete
disk polynomials
(d−2,0)
Gangolli: Jacobi polynomials Rn
are spherical functions on
complex projective space P d−1 (C) = U(d)/(U(1) × U(d − 1))
(a compact Riemannian symmetric space of rank one).
But this is the space of U(1)-orbits on S 2d−1 = U(d)/U(d − 1)
(unit sphere in Cd ). Functions on P d−1 (C) are U(1)-invariant
functions on S 2d−1 .
Moreover (U(d), U(d − 1)) is Gelfand pair with Zernike’s disk
α (z) (α = d − 2) as spherical functions.
polynomials Rm,n
(α,|m−n|)
α
Rm,n
(reiφ ) := Rmin(m,n) (2r 2 − 1) r |m−n| ei(m−n)φ ,
Z
α
Rm,n
(x + iy ) Rkα,l (x + iy ) (1 − x 2 − y 2 )α dx dy = 0
D
((m, n) 6= (k , l); D unit disk).
Work with complex spherical harmonics on Cd : refinement of
ordinary spherical harmonics on R2d .
Tom Koornwinder
Hypergroups made concrete
disk polynomials (cntd.)
The disk polynomials, introduced by the Dutch Nobel prize
winner Zernike, find important applications at the Dutch world
leading chip machine maker ASML.
Tom Koornwinder
Hypergroups made concrete
Addition formula for disk polynomials
R. L. Šapiro (1968), K (1972):
1
1
α
Rm,n
z1 z2 + (1 − |z1 |2 ) 2 (1 − |z2 |2 ) 2 w
=
m X
n
X
1
α
2 2 (k +l) α+k +l
cm,n,k
Rm−k ,n−l (z1 )
,l (1 − |z1 | )
k =0 l=0
1
α+k +l
α−1
× (1 − |z2 |2 ) 2 (k +l) Rm−k
,n−l (z2 ) Rk ,l (w).
(α,0)
This yields an addition formula for Jacobi polynomials Rn
and next, by differentiation and by analytic continuation in the
(α,β)
parameters, an addition formula for Jacobi polynomials Rn
1
(α > β > − 2 ). It involves an expansion in terms of orthogonal
polynomials in two variables on a parabolic biangle.
Tom Koornwinder
Hypergroups made concrete
Orthogonal polynomials on the parabolic biangle
(α,β+n−k + 21 )
(α,β)
Rn,k (x, y ) := Rk
Z
1
y =0
1
(β,β)
1
(2y − 1)y 2 (n−k ) Rn−k (y − 2 x).
1
Z
y2
(α,β)
x=−y
1
2
(α,β)
Rn,k (x, y )Rm,l (x, y )
× (1 − y )α (y − x 2 )β dx dy = 0 ((n.k ) 6= (m, l)).
Parametrize this region by
(x, y ) = (r cos φ, r 2 )
(0 ≤ r ≤ 1, 0 ≤ φ ≤ π).
r 2β+2 (1 − r 2 )α (sin φ)2β+1 dr dφ
Put dνα,β (r , φ) := R 1 R π
.
2β+2 (1 − r 2 )α (sin φ)2β+1 dr dφ
r
r =0 φ=0
Tom Koornwinder
Hypergroups made concrete
Addition formula for Jacobi polynomials
Λ(x, y , r , φ) := 21 (1 + x)(1 + y ) + 12 (1 − x)(1 − y ) r 2
1
1
+ (1 − x 2 ) 2 (1 − y 2 ) 2 r cos φ − 1.
(α,β)
Rn
×
(Λ(x, y , r , φ)) =
n X
k
X
1
(α,β)
1
cn,k ,l (1 − x) 2 (k +l) (1 + x) 2 (k −l)
k =0 l=0
1
(α+k +l,β+k −l)
Rn−k
(x) (1 − y ) 2 (k +l) (1
1
(α+k +l,β+k −l)
+ y ) 2 (k −l) Rn−k
(α−β− 21 ,β− 12 )
× Rk ,l
(y )
(r cos φ, r 2 ).
Constant term in the expansion is the Jacobi product formula
(α,β)
(α,β)
Rn (x)Rn (y )
Z
1
Z
π
(α,β)
Rn
=
r =0
φ=0
(Λ(x, y , r , φ)) dνα−β− 1 ,β− 1 (r , φ).
2
2
Conversely, the product formula implies the addition formula by
integration by parts and Rodrigues type formulas.
Tom Koornwinder
Hypergroups made concrete
Laplace type integral representation
(Askey, 1974; K, 1974)
The Gegenbauer Laplace type integral representation is a
degenerate case of the Gegenbauer product formula:
(α,α)
Rn
(x)
Z
1
=
−1
1
(x + i(1 − x 2 ) 2 t)n dµα− 1 ,α− 1 (t).
2
2
Combine this with a fractional integral or degenerate the Jacobi
product formula for obtaining
(α,β)
Z
1
r =0
Jacobi Laplace type integral representation: Rn (x) =
Z π
n
2
2 21
1
1
(1
+
x)
−
(1
−
x)r
+
i(1
−
x
)
r
cos
φ
dνα−β− 1 ,β− 1 (r , φ).
2
2
2
φ=0
2
One can go back and forth between this integral representation
and the Jacobi product formula by Bateman’s bilinear sum and
its inverse.
Tom Koornwinder
Hypergroups made concrete
Bateman’s bilinear sum and its inverse
(x + y )
where
n
(α,β)
Rn
1 + xy
x +y
(x + 1)n =
(α,β)
(α,β)
Rn (x)Rn (y )
where
(α,β)
Rn
n
X
=
(α,β)
an,k Rk
(α,β)
(x)Rk
(y ),
k =0
(α,β)
an,k Rk
k =0
n
X
=
(x) =
n
X
(x);
k
bn,k (x + y )
(α,β)
Rk
k =0
n
X
1 + xy
x +y
,
bn,k (x + 1)k .
k =0
These connect
Z 1 Z π
n (α,β) 1 + xy
(x + y ) Rn
=
(Λ(x, y , r , φ) + 1)n dνα−β− 1 ,β− 1 (r , φ)
2
2
x +y
r =0 φ=0
Z 1 Z π
(α,β)
(α,β)
(α,β)
Rn (Λ(x, y , r , φ)) dνα−β− 1 ,β− 1 (r , φ).
and Rn (x)Rn (y ) =
r =0
2
φ=0
Tom Koornwinder
Hypergroups made concrete
2
Hypergroups
The term hypergroup seems to have been used first in the
paper
H. S. Wall, Hypergroups, Amer. J. Math. 59 (1937), 77–98.
It generalizes the definition of a group in the sense that the
product of two elements is a sum of finitely many elements.
Commutativity is not required and each element has an inverse.
Dunkl (TAMS, 1973) first defines a hypergroup as a locally
compact space on which the space of finite regular Borel
measures has a convolution structure preserving the probability
measures. He requires commutativity and there is no inverse.
Similar but differently phrased definitions were given by Jewett
(Adv. Math., 1975) and Spector (TAMS, 1978). An imported
added axiom involves the so-called Michael topology of the
collection of compact subsets of a locally compact space.
Hypergroups according to Dunkl, Jewett and Spector are
sometimes called DJS-hypergroups.
Tom Koornwinder
Hypergroups made concrete
Further developments
Gasper (1972): Extension of Jacobi generalized translation
to absolutely bounded (not necessarily positive) case.
Dunkl: Addition formulas for Krawtchouk, Hahn and
q-Hahn polynomials from interpretation on finite groups.
D. Stanton: Similarly for q-Krawtchouk polynomials.
K: Addition formula for little q-Legendre polynomials from
quantum group interpretation.
Floris: Addition formula for q-disk polynomials in noncommuting variables from quantum group interpretation.
Koelink: addition formulas in many q-cases, both from
quantum groups and analytic.
Rahman: analytic proofs of q-addition formulas.
K & A. Schwartz: positive convolution for orthogonal
polynomials on triangle and simplex.
Carlen, Geronimo & Loss (2011) proved Gssper’s positivity
of Jacobi generalized translation by probabilistic means.
Tom Koornwinder
Hypergroups made concrete
Heckman-Opdam Jacobi polynomials
The big open problem: Show the positivity of convolution for
Heckman-Opdam Jacobi polynomials.
Partial results by Rösler and by Remling & Rösler in BCn case
Partial, yet unpublished results in A2 case by Dominique Bakry
and co-workers.
Tom Koornwinder
Hypergroups made concrete
Transmutation
Z
qn (x) =
pn (y ) A(x, y ) dµ(y )
A(x, y ) =
∞
X
qn (x) pn (y )
n=0
Tom Koornwinder
hn
Hypergroups made concrete
Fractional integrals
Riemann-Liouville:
1
Γ(µ)
Z
1
(Wµ f )(x) :=
Γ(µ)
Z
(Rµ f )(x) :=
x
f (y ) (x − y )µ−1 dy
(Re µ > 0).
0
Weyl:
∞
f (y ) (y − x)µ−1 dy
(Re µ > 0).
x
Askey & Fitch (1969) emphasized Bateman’s integral:
Z x c−1
x c+µ−1
a, b
1
y
a, b
;x =
; y (x−y )µ−1 dy ,
2 F1
2 F1
Γ(c + µ)
c+µ
Γ(µ) 0 Γ(c)
c
(Re µ, Re c > 0). Hence, for Re µ > 0:
Z 1
(1 − x)α+µ (α+µ,β−µ)
1
(1 − y )α (α,β)
Rn
(x) =
R
(y ) (y −x)µ−1 dy .
Γ(α + µ + 1)
Γ(µ) x Γ(α + 1) n
Tom Koornwinder
Hypergroups made concrete
Transmutation property
Bateman’s integral in kernel form:
Z
(α+µ,β−µ)
(α,β)
Rn
(x) = Rn (y ) A(x, y ) dµα,β (y ),
where
A(x, y ) =
(y − x)µ−1
2α+β+1 Γ(α + µ + 1) Γ(β + 1)
+
.
Γ(α + β + 2) Γ(µ)
(1 − x)α+µ (1 + y )β
Transmutation Theorem. Let {pn } and {qn } be complete
orthogonal systems with respect to measures dµ and dν,
respectively. Let D and E be operators having the pn
repectively the qn as eigenfunctions with
R the same
eigenvalue λn . Suppose that qn (x) = pnR(y ) A(x, y ) dµ(y ).
Then the operator A given by (Af )(y ) := f (x) A(x, y ) dν(x)
satisfies the transmutation property A ◦ E = D ◦ A.
Hence in case of Bateman’s integral : D = Dα,β , E = Dα+µ,β−µ ,
(α,β)
(α,β)
where Dα,β Rn
= −n(n + α + β + 1) Rn .
Tom Koornwinder
Hypergroups made concrete
Feldheim-Vilenkin integral
Feldheim-Vilenkin integral
(not of the desired transmutation form):
(x − 1)α+µ 1 n (α+µ,α+µ) − 1
(x 2 )
x 2 Rn
Γ(α + µ + 1)
Z x
(y − 1)α 1 n (α,α) − 1
1
y 2 Rn
(y 2 ) (x − y )µ−1 dy
=
Γ(µ) 1 Γ(α + 1)
(µ > 0).
Remark. Both the Bateman and Feldheim-Vilenkin integral
can be obtained from spherical harmonics. For Bateman also
use that
( 1 p−1, 12 q−1)
2
(x12 +· · ·+xq+p
)n Rn 2
2
2 )
(x12 + · · · + xq2 ) − (xq+1
+ · · · + xq+p
2
x12 + · · · + xq+p
is an O(q) × O(p)-invariant homogeneous harmonic polynomial
of degree 2n on Rq+p .
Tom Koornwinder
Hypergroups made concrete
!
Transmutation in the non-compact case
Jacobi functions (surveyed by K, 1984). These form a
continuous orthogonal system of Gauss hypergeometric
functions. They are noncompact analogues of Jacobi
polynomials. They have richer transmutation properties.
!
1
1
(ρ
+
iλ),
(ρ
−
iλ)
(α,β)
2
; − sinh2 t , ρ := α + β + 1;
φλ (t) := 2 F1 2
α+1
Z ∞
Z ∞
bf (λ) |cα,β (λ)|−2 dλ.
bf (λ) =
f (t) ∆α,β (t) dt,
f (t) =
0
(α,β)
Dα,β φλ
= −λ
Transmutation:
2
0
(− 1 ,− 1 )
φλ 2 2 (t)
(α,β)
φλ ;
(α+µ,β±µ)
φλ
(t)
Z
=
0
t
(α,β)
φλ
= cos(λt).
(s) A(s, t) ∆α,β (s) ds
with A(s, t) positive and elementary if µ > 0. Relationship with
Abel transform on noncompact semisimple Lie groups.
Generalization to Chébli-Trimèche hypergroups.
Tom Koornwinder
Hypergroups made concrete
Linearization of products
pm (x) pn (x) =
m+n
X
cm,n,k pk (x)/hk
k =|m−n|
Z
cm,n,k =
pm (x) pn (x) pk (x) dµ(x)
Tom Koornwinder
Hypergroups made concrete
Linearization of products (cntd.)
Jacobi polynomials:
(α,β)
Rm
(α,β)
(x) Rn
(x) =
Theorem (Gasper, 1970)
(α,β)
k =|m−n| cm,n,k
Pm+n
(α,β)
Rk
(α,β)
(x)/hk
(a) ⇐⇒ (b) ⇐= (c)
(α,β)
cm,n,k
(a)
≥ 0 for all m, n, k .
(b) some quartic polynomial in α, β is nonnegative.
(c) α ≥ β > −1 and α + β > −1.
General monic orthogonal polynomials pn :
p1 (x) pn (x) = pn+1 (x) + an pn (x) + bn pn−1 (x),
P
pm (x)pn (x) = m+n
k =|m−n| cm,n,k pk (x)/hk .
Theorem (Askey, 1970)
∀n an , bn , an+1 − an , bn+1 − bn ≥ 0 =⇒ ∀m, n, k
(α,β)
cm,n,k ≥ 0.
This covers: If α ≥ β and α + β ≥ 1 then cm,n,k ≥ 0.
Tom Koornwinder
Hypergroups made concrete
Linearization of products (cntd.)
Remark 1. A function f on a group G is called positive definite
if for all x1 , . . . , xk ∈ G and all c1 , . . . , ck ∈ C
k
X
f (xi xj−1 )ci cj ≥ 0.
i,j=1
If (G, K ) is a Gelfand pair P
with G, K compact and with spherical
functions φλ then φλ φµ = ν cλ,µ,ν φν with cλ,µ,ν ≥ 0.
Indeed, spherical functions are elementary positive definite
functions, a product of positive definite functions is again
positive definite, and a K -biinvariant positive definite function is
a nonnegative linear combination of spherical functions.
Thus for special parameter values the theorems of Gasper and
Askey also follow from group theory.
Tom Koornwinder
Hypergroups made concrete
Linearization of products (cntd.)
Remark 2. K (1978):
An addition formula obtained for a spherical function on a
Gelfand pair carries the essential information making it positive
definite and leading to nonnegative linearization coefficients.
This last information is preserved in an addition formula for
other parameter values which do not come from group theory.
The addition formula needs to have certain properties. In
particular, the expansion coefficients in the addition formula
should be nonnegative. Then it implies the nonnegativity of the
linearization coefficients.
This works in the Jacobi case for α ≥ β ≥ − 21 .
Tom Koornwinder
Hypergroups made concrete
An application to Laguere polynomials
This works also for disk polynomials.
If these are rewritten in terms of Jacobi polynomials and next
the limit to the Laguerre case is taken then:
Z ∞
Lαk (x) Lαm (λx) Lαn ((1−λ)x) x α e−x dx ≥ 0 (α ≥ 0, λ ∈ [0, 1]).
0
By iteration:
Z ∞
Lαn1 (x) Lαn2 (x) Lαn3 (x) Lαn4 (x) x α e−2x dx > 0 (α > 0).
0
This leads to the four boxes paper by Askey, Ismail & K (1978).
Tom Koornwinder
Hypergroups made concrete
Connection formula
qn (x) =
n
X
an,k pk (x)/hk
k =0
Z
an,k =
qn (x) pk (x) dµ(x)
Tom Koornwinder
Hypergroups made concrete
Connection formula (cntd.)
(γ,δ)
Rn
(x) =
n
X
(α,β)
an,k Rk
(α,β)
(x)/hk
=⇒ an,k = stuff × 3 F2 (1).
k =0
In particular, an,k is elementary and nonnegative in the cases
(γ,γ)
Rn (x)
(γ,β)
Rn
=
(x) =
n
X
k =0
n
X
(α,α)
(x)/hk
(α,β)
(x)/hk
an,k Rk
an,k Rk
(α,α)
(γ > α > −1),
(α,β)
(γ > α > −1).
k =0
Askey & Gasper (1971) give sufficient conditions for
(γ,δ),(α,β)
nonnegativity of an,k = an,k
. For given (α, β) this includes
an infinite region in the (γ, δ) plane bounded by three lines with
(γ, δ) = (2α + 1, 2β + 1) as one of the vertices.
Askey (1968): Certain of these positivity cases from isometric
embeddings of projective spaces.
Tom Koornwinder
Hypergroups made concrete
Connection formula (cntd.)
Nevai (1979): Connection coefficients for pn in terms of
Chebyshev polynomials Tk are limits of linearization
coefficients for pn .
Lasser (1994): Under certain assumptions positivity of
linearization coefficients implies positivity of connection
coefficients with Chebyshev.
Further work by Szwarc.
It seems that certain conditions on the coefficients in the
three-term recurrence relation can identify a class of orthogonal
polynomials giving rise to the dual case of the Chébli-Trimèche
hypergroups.
Tom Koornwinder
Hypergroups made concrete

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