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Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
(q, t)-Laguerre polynomials
Ole Warnaar
School of Mathematics and Physics
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Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
The Laguerre polynomials
For > 1 and m a nonnegative integer, Laguerres equation is thelinear second order ODE
x y(x) + ( + 1 x)y(x) + m y(x) = 0
Its polynomial solutions, denoted by L()m (x), are known as the(generalised/associated) Laguerre polynomials.
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Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
The first few Laguerre polynomials are given by
L()0 (x) = 1
L()1 (x) = x + + 1
L()
2 (x) = 12x2 ( + 2)x + 1
2( + 1)( + 2)
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Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
The first few Laguerre polynomials are given by
L()0 (x) = 1
L()1 (x) = x + + 1
L()
2 (x) = 12x2 ( + 2)x + 1
2( + 1)( + 2)
and more generally one has
L()m (x) =mi=0
m + m i
(x)
i
i!
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Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
It is easily verified that the Laguerre polynomials satisfy a three-termrecurrence relation:
(m + 1)L()m+1(x) = (2m + 1 + x)L
()m (x) (m + )L
()m1(x)
According to Favards theorem the Laguerre polynomials thus form a
family of orthogonal poynomials.
(q, t)-Laguerre polynomials
L l i l L l i l ( ) L l i l
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Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
It is easily verified that the Laguerre polynomials satisfy a three-termrecurrence relation:
(m + 1)L()m+1(x) = (2m + 1 + x)L
()m (x) (m + )L
()m1(x)
According to Favards theorem the Laguerre polynomials thus form a
family of orthogonal poynomials.The orthogonalising measure for the Laguerre polynomials is given by
d(x) = xexdx
supported on the positive half-line:0
L()m (x)L()n (x) d(x) = mn
(m + + 1)
m!
(q, t)-Laguerre polynomials
L l i l L l i l ( t) L l i l
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Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
The orthogonality with respect to the Laguerre measure may be provedas follows:
Laguerres equation is equivalent to the statement that L()m (x) isthe eigenfunction with eigenvalue m of the second order differentialoperator
L = xd2
dx2+ (x 1)
d
dx
(q, t)-Laguerre polynomials
Laguerre polynomials q Laguerre polynomials (q t) Laguerre polynomials
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Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
The orthogonality with respect to the Laguerre measure may be provedas follows:
Laguerres equation is equivalent to the statement that L()m (x) isthe eigenfunction with eigenvalue m of the second order differentialoperator
L = xd2
dx2+ (x 1)
d
dx
The operator L is self-adjoint with respect to the inner product
f, g
=
0
f(x)g(x)d(x)
i.e., Lf, g
=
f,Lg
(q, t)-Laguerre polynomials
Laguerre polynomials q Laguerre polynomials (q t) Laguerre polynomials
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Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
The q-Laguerre polynomials
The q-Laguerre polynomials are a discrete or quantum variant of theclassical Laguerre polynomials. They were first studied by Hahn aroundthe 1950s.
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
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Laguerre polynomials q Laguerre polynomials (q, t) Laguerre polynomials
The q-Laguerre polynomials
The q-Laguerre polynomials are a discrete or quantum variant of theclassical Laguerre polynomials. They were first studied by Hahn aroundthe 1950s.
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
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gu p y q gu p y (q, ) gu p y
The q-Laguerre polynomials
The q-Laguerre polynomials are a discrete or quantum variant of theclassical Laguerre polynomials. They were first studied by Hahn aroundthe 1950s.
Differential equation q-difference equationIntegration Jackson-integration
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
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g p y q g p y (q, ) g p y
Let Dq be the q-derivative operator
(Dqf)(x) =f(xq) f(x)
(q 1)x
If f is differentiable at x then
limq1
(Dqf)(x) = f(x)
For example,Dqx
m = [m] xm1
with [a] the q-number [a] = (1 qa)/(1 q).
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
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The q-Laguerre polynomial L(m (x) = L
(m (x; q) is the eigenfunction, with
eigenvalue [m], of the q-difference operator
L = (x + 1)Dq q1Dq1
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
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The q-Laguerre polynomial L(m (x) = L
(m (x; q) is the eigenfunction, with
eigenvalue [m], of the q-difference operator
L = (x + 1)Dq q1Dq1
The first few q-Laguerre polynomials are given by
L()0 (x) = 1
L()1 (x) = q
+1 X + [ + 1]
L()2 (x) =
1[2]
X2 q+1[ + 2] X + 1[2]
[ + 1][ + 2]
where X = x/(1 q).
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
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Again there is a three-term recurrence:
[m + 1]L()m+1(x)
= [m + 1] + q[m + ] q2m++1(1 q)xL
()m (x)
q[m + ]L()m1(x)
Hence there exists an orthogonalising measure such that theq-Laguerre polynomials form an orthogonal family.
To describe this measure we need the Jackson integral.
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
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The Jackson integral over the positive reals is defined as
0
f(x)dqx = (1 q)
n=
f(qi)qi
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
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Because of its discreteness something is lost, and there is no q-analogueof
c
0
f(cx)dx =
0
f(x)dx c > 0
Hence we also definec0
f(x)dqx = (1 q)
n=
f(cqi)cqi
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
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The q-Laguerre polynomials satisfy the discrete orthogonality
c0
L()m (x)L()n (x)d(x)
= mn[c(1 q)]+1qm
q(m + + 1)
q(m + 1)
(acq)
(c)
whered(x) = xeq(x)dqx
and
eq(x) =
n=0
Xn
[n]!
=1
(1 x)(1 xq)(1 xq2
) is the q-exponential function.
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
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The previous results can again be restated as follows:
The q-Laguerre polynomial L
()
m (x) is the eigenfunction witheigenvalue [m] of
L = (x + 1)Dq q1
Dq1
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
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The previous results can again be restated as follows:
The q-Laguerre polynomial L
()
m (x) is the eigenfunction witheigenvalue [m] of
L = (x + 1)Dq q1
Dq1
The operator L is self-adjoint with respect to the inner product
f, g
=
c0
f(x)g(x)d(x)
whered(x) = xeq(x)dqx
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
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The (q, t)-Laguerre polynomials
Motivation. The evaluation of multidimensional integrals related to Lie
algebras.
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
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The (q, t)-Laguerre polynomials
Motivation. The evaluation of multidimensional integrals related to Lie
algebras.Trivial example. The classical Laguerre polynomials
1, 1
=
0
d(x) =
0
xexdx = ( + 1)
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
( )
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The (q, t)-Laguerre polynomials
Motivation. The evaluation of multidimensional integrals related to Lie
algebras.Trivial example. The classical Laguerre polynomials
1, 1
=
0
d(x) =
0
xexdx = ( + 1)
Non-trivial example. The Lie algebra pair (An1, A1)
[0,)n
n
i=1
xi exi
1i
-
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For x = (x1, . . . , xn) let q,i be the partial difference operator
(q,if)(x) =
f(x1, . . . , xi1, qxi, xi+1, . . . , xn) f(x)
(q 1)xi
Define the operators Er(q, t) acting on n = Q(q, t)[x1, . . . , xn]Sn by
Er(q, t) =
ni=1
x
r
i n
j=1j=i
txi xj
xi xj
q,i
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
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For x = (x1, . . . , xn) let q,i be the partial difference operator
(q,if)(x) =
f(x1, . . . , xi1, qxi, xi+1, . . . , xn) f(x)
(q 1)xi
Define the operators Er(q, t) acting on n = Q(q, t)[x1, . . . , xn]Sn by
Er(q, t) =
ni=1
x
r
i n
j=1j=i
txi xj
xi xj
q,i
Note that for n = 1
E0(q, t) = Dq and E1(q, t) = xDq
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
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For x = (x1, . . . , xn) let q,i be the partial difference operator
(q,if)(x) =
f(x1, . . . , xi1, qxi, xi+1, . . . , xn) f(x)
(q 1)xi
Define the operators Er(q, t) acting on n = Q(q, t)[x1, . . . , xn]Sn by
Er(q, t) =
ni=1
x
r
i n
j=1j=i
txi xj
xi xj
q,i
Note that for n = 1
E0(q, t) = Dq and E1(q, t) = xDq
Lemma. For r 0, Er : n n
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
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Let = (1, . . . , n) be a partition (a weakly decreasing sequence of
nonnegative integers) and L : n n the operator
L = E1(q, t) + E0(q, t) q1E0(q
1, t1)
The eigenfunction ofL with eigenvalue
tn1[1] + + t[n1] + [n]
defines the (q, t)-Laguerre polynomial L() (x) = L
() (x; q, t).
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
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Let = (1, . . . , n) be a partition (a weakly decreasing sequence of
nonnegative integers) and L : n n the operator
L = E1(q, t) + E0(q, t) q1E0(q
1, t1)
The eigenfunction ofL with eigenvalue
tn1[1] + + t[n1] + [n]
defines the (q, t)-Laguerre polynomial L() (x) = L
() (x; q, t).
Lemma. The L() (x) form a basis of n.
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
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For (q, t) = (q, q) with q 1 the (q, t)-Laguerre polynomials were
studied by
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
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Let t = q and define an inner product on n by
f, g
=
cx1=0
tx1x2=0
txn1xn=0
f(x)g(x)d(x)
where
d(x) =
1i
-
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Theorem. The operator L is self-adjoint with respectto the inner product on n.
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials
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Theorem. The operator L is self-adjoint with respectto the inner product on n.
Corollary. L() , L() = c,
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q,
t)-Laguerre polynomials
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Theorem. The operator L is self-adjoint with respectto the inner product on n.
Corollary. L() , L() = c,
Theorem c can explicitly be computed in terms ofnice special functions, such as theta functions and
q-Gamma functions.
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q,
t)-Laguerre polynomials
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Ingredients in proofs are:
Macdonald polynomialsKnopOkounkovSahi interpolation polynomials
The double affine Hecke algebra
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q,
t)-Laguerre polynomials
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Ingredients in proofs are:
Macdonald polynomialsKnopOkounkovSahi interpolation polynomials
The double affine Hecke algebra
(q, t)-Laguerre polynomials
Laguerre polynomials q-Laguerre polynomials (q,
t)-Laguerre polynomials
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The End
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