(q; t)-laguerre polynomials_ole warnaar

8/8/2019 (q; t)-Laguerre polynomials_Ole Warnaar

1/36

Laguerre polynomials q-Laguerre polynomials (q, t)-Laguerre polynomials

(q, t)-Laguerre polynomials

Ole Warnaar

School of Mathematics and Physics

http://find/http://goback/


2/36


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.



3/36


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)



4/36


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!



5/36


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.


L l i l L l i l ( ) L l i l


6/36


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!


L l i l L l i l ( t) L l i l


7/36


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


Laguerre polynomials q Laguerre polynomials (q t) Laguerre polynomials


8/36


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


Laguerre polynomials q Laguerre polynomials (q t) Laguerre polynomials


9/36


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.




10/36

Laguerre polynomials q Laguerre polynomials (q, t) Laguerre polynomials






11/36

gu p y q gu p y (q, ) gu p y



Differential equation q-difference equationIntegration Jackson-integration




12/36

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).




13/36

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




14/36

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).




15/36

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.




16/36

The Jackson integral over the positive reals is defined as

0

f(x)dqx = (1 q)

n=

f(qi)qi




17/36

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




18/36

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.




19/36

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




20/36

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




21/36

The (q, t)-Laguerre polynomials

Motivation. The evaluation of multidimensional integrals related to Lie

algebras.




22/36



algebras.Trivial example. The classical Laguerre polynomials

1, 1

=

0

d(x) =

0

xexdx = ( + 1)



( )


23/36



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


24/36

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




25/36


(q,if)(x) =

f(x1, . . . , xi1, qxi, xi+1, . . . , xn) f(x)

(q 1)xi


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




26/36


(q,if)(x) =

f(x1, . . . , xi1, qxi, xi+1, . . . , xn) f(x)

(q 1)xi


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




27/36

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).




28/36

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.




29/36

For (q, t) = (q, q) with q 1 the (q, t)-Laguerre polynomials were

studied by




30/36

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


31/36

Theorem. The operator L is self-adjoint with respectto the inner product on n.




32/36


Corollary. L() , L() = c,


Laguerre polynomials q-Laguerre polynomials (q,

t)-Laguerre polynomials


33/36


Corollary. L() , L() = c,

Theorem c can explicitly be computed in terms ofnice special functions, such as theta functions and

q-Gamma functions.





34/36

Ingredients in proofs are:

Macdonald polynomialsKnopOkounkovSahi interpolation polynomials

The double affine Hecke algebra





35/36

Ingredients in proofs are:

Macdonald polynomialsKnopOkounkovSahi interpolation polynomials

The double affine Hecke algebra





36/36

The End


(q; t)-laguerre polynomials_ole warnaar

Documents