Polynomials, simple and complex
Michael Anshelevich
Texas A&M University
November 16, 2011
Michael Anshelevich
Polynomials, simple and complex
Part I.
Michael Anshelevich
Polynomials, simple and complex
Meixner polynomials.
Parameters: a, b, c ∈ R and t ≥ 0.
Polynomials {Pn : n ≥ 0}: defined by P0 = 1, P1 = x − at, and
the recursion
Pn+1 (x) = xPn (x) − (at + (b + c)n)Pn (x)
− n(t + bc(n − 1))Pn−1 (x).
Linear functional ϕ : C[x] → C: defined by
ϕ[1] = 1,
ϕ[Pn (x)] = 0 for n > 0.
Michael Anshelevich
Polynomials, simple and complex
Moments.
Facts.
1
ϕ arises from a (probability) measure µ:
Z
ϕ[Q(x)] =
Q(x) dµ(x).
R
Measures µa,b,c,t look quite different.
2
We have not only ϕ[Pn ] = 0 but
ϕ[Pn Pk ] = 0 for n 6= k
(polynomials are orthogonal).
3
The n’th moment of ϕ
ϕ[xn ] = Polynomial in a, b, c, t with positive integer coefficients.
Michael Anshelevich
Polynomials, simple and complex
Permutation statistics.
π ∈ S(n) = permutation. Write it in the cycle notation. Let
1 ≤ i ≤ n.


fixed point of π
i = ascent of π


descent of π
if π(i) = i,
if π(i) > i,
if π(i) < i.
Theorem. (Kim, Zeng 2001)
For the functional ϕ from before,
X
ϕ[xn ] =
a#fixed points b# ascents−#cycles c# descents−#cycles t#cycles .
π∈S(n)
Michael Anshelevich
Polynomials, simple and complex
Example: Gamma.
Set a = b = c = 1. Recursion
Pn+1 (x) = xPn (x) − (t + 2n)Pn (x) − n(t + (n − 1))Pn−1 (x).
Laguerre polynomials. Orthogonality measure:
Gamma distribution
dµ(x) =
1 t−1 −x
x e 1[0,∞) dx.
Γ(t)
Thus
X
ϕ[xn ] =
t#cycles .
π∈S(n)
In particular, for t = 1,
R∞
0
xn e−x dx = n!.
Michael Anshelevich
Polynomials, simple and complex
Example: derangements.
Set instead a = 0, b = c = 1, t = 1. Recursion
Pn+1 (x) = xPn (x) − 2n Pn (x) − n2 Pn−1 (x).
Shifted Laguerre polynomials. Orthogonality measure: shifted
Gamma distribution
dµ(x) = e−x−1 1[−1,∞) dx.
π ∈ S(n) with no fixed points
↔
Derangements π ∈ D(n).
Thus
Z
∞
−1
xn e−x−1 dx =
Z
∞
(x − 1)n e−x dx = |D(n)|.
0
Note: no formula for this number.
Michael Anshelevich
Polynomials, simple and complex
Example: Poisson.
Set a = b = 1, c = 0. Recursion
Pn+1 (x) = xPn (x) − (t + n)Pn (x) − nt Pn−1 (x).
Charlier polynomials. Orthogonality measure:
Poisson distribution
dµ(x) = e−t
∞ k
X
t
k=0
k!
δk (x).
π ∈ S(n) with (# descents = #cycles) ↔ Partitions π ∈ P(n).
Thus
ϕ[xn ] =
X
t#subsets .
π∈P(n)
Michael Anshelevich
Polynomials, simple and complex
Example: Gaussian.
Set a = b = c = 0. Recursion
Pn+1 (x) = xPn (x) − nt Pn−1 (x).
Hermite polynomials. Orthogonality measure:
Gaussian (or normal) distribution
dµ(x) = √
1 −x2 /2t
e
dx.
2πt
π ∈ S(n) with no fixed points and
↔ Pairings π ∈ P2 (n).
# ascents = # descents = #cycles
Michael Anshelevich
Polynomials, simple and complex
Example: Gaussian.
π ∈ S(n) with no fixed points and
↔ Pairings π ∈ P2 (n).
# ascents = # descents = #cycles
Thus
X
ϕ[xn ] =
t#pairs
π∈P2 (n)
Zero if n odd.
ϕ[x2n ] = tn |P2 (n)| = (2n − 1)(2n − 3) . . . 3 · 1 · tn .
Michael Anshelevich
Polynomials, simple and complex
q-integers.
A new wrinkle: Instead of using n ∈ N, use
[n]q = 1 + q + q 2 + . . . + q n−1 =
1 − qn
.
1−q
q-integers (Ramanujan, quantum groups, etc.)
As before define {Pn : n ≥ 0} by
Pn+1 (x) = xPn (x) − (at + (b + c)[n]q )Pn (x) − [n]q (t + bc[n − 1]q )Pn−1 (x).
Again orthogonal polynomials for a measure µ = µa,b,c,t,q .
Moments of µ? Answer seems to involve crossings.
Michael Anshelevich
Polynomials, simple and complex
Crossings.
q-Gaussian case [Ismail, Stanton, Viennot 1987]:
Pn+1 (x) = xPn (x) − [n]q tPn−1 (x).
(q-Hermite polynomials, Rogers (1894)). Moments:
X
ϕ[x2n ] = tn
q #crossings in the pairing .
π∈P2 (2n)
q-Poisson case [A 2005], [Kim, Stanton, Zeng 2006]:
Pn+1 (x) = xPn (x) − (t + [n]q )Pn (x) − [n]q tPn−1 (x).
Moments:
ϕ[xn ] =
X
t#subsets q #crossings in a partition .
π∈P(n)
Michael Anshelevich
Polynomials, simple and complex
Project.
Full case: unknown. Difficulty: how to count crossings of a
permutation.
Have an approach (linked partitions), want to start work soon.
More general question: Linearization coefficients
ϕ[Pn1 Pn2 . . . Pnk ].
Again expect combinatorial interpretations!
Michael Anshelevich
Polynomials, simple and complex
Why do analysts care?
a, b, c some parameters.
t = convolution parameter = time parameter.
Recall: Gaussian, Poisson, Gamma
dµt (x) = √
1 −x2 /2t
e
dx,
2πt
e−t
∞ k
X
t
k=0
k!
δk (x),
xt−1 e−x 1[0,∞) dx.
In each case, form a convolution semigroup:
µt ∗ µs = µt+s ,
where (roughly)
Z
µ(y − x) dν(y).
d(µ ∗ ν)(x) =
R
Thus Combinatorics comes from Analysis.
Michael Anshelevich
Polynomials, simple and complex
Independence.
Even more: if X, Y are independent random variables, with
distributions µX , µY , the distribution of the sum is the
convolution:
µX+Y = µX ∗ µY .
So a convolution semigroup of measures
{µt : t ≥ 0}
comes from a process with independent increments
{X(t) : t ≥ 0}.
Thus Analysis comes from Probability.
Michael Anshelevich
Polynomials, simple and complex
Stochastic integrals.
Fancier relations:
Hn (X(t))
(Hermite polynomial of a Brownian motion)
Z
=
dX(t1 ) dX(t2 ) . . . dX(tn )
0≤t1 ≤t2 ≤...≤tn ≤t
(multiple stochastic integral)
= W a(t) + a∗ (t), a(t) + a∗ (t), . . . , a(t) + a∗ (t)
(Wick product on a Fock space).
Michael Anshelevich
Polynomials, simple and complex
Probability theories.
Gaussian, Poisson, Gamma come from Probability. q = 1 case.
Well understood.
Another well-understood case: q = 0. t again a time parameter,
but with respect to the operation of free convolution . Arises
from free independence. All in the context of Free Probability
(Voiculescu 1986).
General q? “q-deformed” probability? Gaussian, Poisson case
well understood [Bożejko, Kümmerer, Speicher 1997] [A 2001],
used in proofs above. In general, no. Processes with
q-independent increments?
Michael Anshelevich
Polynomials, simple and complex
Project.
Kim, Zeng (2001): full Meixner case q = 1, linearization
coefficients. Combinatorial proof.
My approach (Analysis / Probability): full Meixner case q = 1,
moments only (unpublished).
General q, linearization coefficients, Gaussian and Poisson only
(2005).
General Meixner case? General q? Linearization coefficients?
Maybe using combinatorics, maybe probability.
Michael Anshelevich
Polynomials, simple and complex
Part II.
Michael Anshelevich
Polynomials, simple and complex
Matrix-valued polynomials.
Polynomials: spanned by monomials Axn , for A ∈ R or C.
What if A ∈ A, for
A = algebra, matrix algebra, C ∗ -algebra, etc.?
So that
(Axn )(Bxk ) = (AB)xn+k 6= (BA)xn+k .
Orthogonal polynomials?
Interesting theory already for A = M2×2 , connected to to
random walks.
Difficulty: Gram-Schmidt does not always work.
P2 (x) = x2 −
ϕ[x2 P1 (x)]
P1 (x) − ϕ[x2 ]
ϕ[P1 (x)2 ]
but may have ϕ[P1 (x)2 ] ∈ M2×2 non-zero but not invertible.
Michael Anshelevich
Polynomials, simple and complex
Operator-valued polynomials.
A non-commutative, but Ax = xA. Free probability: maximally
non-commutative. Thus instead of A[x] want Ahxi spanned by
A0 xA1 xA2 . . . An−1 xAn .
To even prove existence of orthogonal polynomials, one needs
free probability (Fock space constructions). Gram-Schmidt
does not even make sense. New objects, little known about
them. Both algebraic and analytic difficulties. Joint work with
Belinschi, Popa, etc.
Michael Anshelevich
Polynomials, simple and complex