Moments of Askey-Wilson polynomials Jang Soo Kim Joint work with - - PowerPoint PPT Presentation

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Moments of Askey-Wilson polynomials Jang Soo Kim

Joint work with Dennis Stanton Korea Institute for Advanced Study (KIAS) June 25, 2013

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Definition of orthogonal polynomials

Definition

{Pn(x)}n≥0 are orthogonal polynomials w.r.t. weight function w(x) if

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Definition of orthogonal polynomials

Definition

{Pn(x)}n≥0 are orthogonal polynomials w.r.t. weight function w(x) if deg Pn(x) = n

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Definition of orthogonal polynomials

Definition

{Pn(x)}n≥0 are orthogonal polynomials w.r.t. weight function w(x) if deg Pn(x) = n

• Pn(x)Pm(x)w(x)dx =

if n = m nonzero if n = m

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Definition of orthogonal polynomials

Definition

{Pn(x)}n≥0 are orthogonal polynomials w.r.t. weight function w(x) if deg Pn(x) = n

• Pn(x)Pm(x)w(x)dx =

if n = m nonzero if n = m

Example

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Definition of orthogonal polynomials

Definition

{Pn(x)}n≥0 are orthogonal polynomials w.r.t. weight function w(x) if deg Pn(x) = n

• Pn(x)Pm(x)w(x)dx =

if n = m nonzero if n = m

Example

Recall π cos nθ cos mθdθ = if n = m nonzero if n = m

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Definition of orthogonal polynomials

Definition

{Pn(x)}n≥0 are orthogonal polynomials w.r.t. weight function w(x) if deg Pn(x) = n

• Pn(x)Pm(x)w(x)dx =

if n = m nonzero if n = m

Example

Recall π cos nθ cos mθdθ = if n = m nonzero if n = m cos nθ is a polynomial in cos θ.

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Definition of orthogonal polynomials

Definition

{Pn(x)}n≥0 are orthogonal polynomials w.r.t. weight function w(x) if deg Pn(x) = n

• Pn(x)Pm(x)w(x)dx =

if n = m nonzero if n = m

Example

Recall π cos nθ cos mθdθ = if n = m nonzero if n = m cos nθ is a polynomial in cos θ. Let Tn(cos θ) = cos nθ. (Tchebyshev polynomial of 1st kind)

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Definition of orthogonal polynomials

Definition

{Pn(x)}n≥0 are orthogonal polynomials w.r.t. weight function w(x) if deg Pn(x) = n

• Pn(x)Pm(x)w(x)dx =

if n = m nonzero if n = m

Example

Recall π cos nθ cos mθdθ = if n = m nonzero if n = m cos nθ is a polynomial in cos θ. Let Tn(cos θ) = cos nθ. (Tchebyshev polynomial of 1st kind) Tn(x) are orthogonal w.r.t. w(x) = (1 − x2)−1/2. 1

−1

Tn(x)Tm(x)(1 − x2)−1/2dx = if n = m nonzero if n = m

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Moments of orthogonal polynomials

Let L(p(x)) :=

• p(x)w(x)dx.

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Moments of orthogonal polynomials

Let L(p(x)) :=

• p(x)w(x)dx.

In order to determine L(p(x)) it is enough to know L(xn) for n = 0, 1, 2, . . . .

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Moments of orthogonal polynomials

Let L(p(x)) :=

• p(x)w(x)dx.

In order to determine L(p(x)) it is enough to know L(xn) for n = 0, 1, 2, . . . . The nth moment is defined by µn := L(xn) =

• xnw(x)dx.

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Moments of orthogonal polynomials

Let L(p(x)) :=

• p(x)w(x)dx.

In order to determine L(p(x)) it is enough to know L(xn) for n = 0, 1, 2, . . . . The nth moment is defined by µn := L(xn) =

• xnw(x)dx.

Example

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Moments of orthogonal polynomials

Let L(p(x)) :=

• p(x)w(x)dx.

In order to determine L(p(x)) it is enough to know L(xn) for n = 0, 1, 2, . . . . The nth moment is defined by µn := L(xn) =

• xnw(x)dx.

Example

The normalized (µ0 = 1) nth moment of {Tn(x)} is µn = 1 π 1

−1

xn(1 − x2)−1/2dx

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Moments of orthogonal polynomials

Let L(p(x)) :=

• p(x)w(x)dx.

In order to determine L(p(x)) it is enough to know L(xn) for n = 0, 1, 2, . . . . The nth moment is defined by µn := L(xn) =

• xnw(x)dx.

Example

The normalized (µ0 = 1) nth moment of {Tn(x)} is µn = 1 π 1

−1

xn(1 − x2)−1/2dx µ2n+1 = 0

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Moments of orthogonal polynomials

Let L(p(x)) :=

• p(x)w(x)dx.

In order to determine L(p(x)) it is enough to know L(xn) for n = 0, 1, 2, . . . . The nth moment is defined by µn := L(xn) =

• xnw(x)dx.

Example

The normalized (µ0 = 1) nth moment of {Tn(x)} is µn = 1 π 1

−1

xn(1 − x2)−1/2dx µ2n+1 = 0 µ2n = 1 22n

• 2n

n

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Three-term recurrence and Viennot’s theorem

Theorem (Favard, 1935)

Monic orthogonal polynomials Pn(x) satisfy a three-term recurrence Pn+1(x) = (x − bn)Pn(x) − λnPn−1(x), n ≥ 0 for b0, b1, . . . and λ1, λ2, . . . .

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Three-term recurrence and Viennot’s theorem

Theorem (Favard, 1935)

Monic orthogonal polynomials Pn(x) satisfy a three-term recurrence Pn+1(x) = (x − bn)Pn(x) − λnPn−1(x), n ≥ 0 for b0, b1, . . . and λ1, λ2, . . . .

Theorem (Viennot, 1983)

If Pn+1(x) = (x − bn)Pn(x) − λnPn−1(x), the nth moment is µn =

• P∈Motn

wt(P). i − 1 i 1 i − 1 i λi i bi

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Three-term recurrence and Viennot’s theorem

Theorem (Favard, 1935)

Monic orthogonal polynomials Pn(x) satisfy a three-term recurrence Pn+1(x) = (x − bn)Pn(x) − λnPn−1(x), n ≥ 0 for b0, b1, . . . and λ1, λ2, . . . .

Theorem (Viennot, 1983)

If Pn+1(x) = (x − bn)Pn(x) − λnPn−1(x), the nth moment is µn =

• P∈Motn

wt(P). i − 1 i 1 i − 1 i λi i bi

Example

1 1 1 λ3 b2 1 b3 λ3 λ2 b1 λ1

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Hermite polynomials

The Hermite polynomials Hn(x) satisfy Hn+1(x) = xHn(x) − nHn−1(x), bn = 0, λn = n.

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Hermite polynomials

The Hermite polynomials Hn(x) satisfy Hn+1(x) = xHn(x) − nHn−1(x), bn = 0, λn = n. A Hermite history is a Dyck path in which a down step of height i has label in {0, 1, . . . , i − 1}.

b b b b b b b b b b b

2 1 1

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Hermite polynomials

The Hermite polynomials Hn(x) satisfy Hn+1(x) = xHn(x) − nHn−1(x), bn = 0, λn = n. A Hermite history is a Dyck path in which a down step of height i has label in {0, 1, . . . , i − 1}.

b b b b b b b b b b b

2 1 1 µn is the number of Hermite histories of length n

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Hermite polynomials

The Hermite polynomials Hn(x) satisfy Hn+1(x) = xHn(x) − nHn−1(x), bn = 0, λn = n. A Hermite history is a Dyck path in which a down step of height i has label in {0, 1, . . . , i − 1}.

b b b b b b b b b b b

2 1 1 µn is the number of Hermite histories of length n # Hermite histories of length n = # perfect matchings of [n] := {1, 2, . . . , n}

b b b b b b b b b b b

2 1 1 ⇔

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Hermite polynomials

The Hermite polynomials Hn(x) satisfy Hn+1(x) = xHn(x) − nHn−1(x), bn = 0, λn = n. A Hermite history is a Dyck path in which a down step of height i has label in {0, 1, . . . , i − 1}.

b b b b b b b b b b b

2 1 1 µn is the number of Hermite histories of length n # Hermite histories of length n = # perfect matchings of [n] := {1, 2, . . . , n}

b b b b b b b b b b b

2 1 1 ⇔ µ2n+1 = 0 and µ2n = (2n − 1)!! = 1 · 3 · · · (2n − 1).

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Combinatorial interpretation for Hermite polynomials

µn = L(xn) =# perfect matchings on [n]

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Combinatorial interpretation for Hermite polynomials

µn = L(xn) =# perfect matchings on [n] Hn(x) also has a combinatorial meaning Hn(x) =

• π∈M(n)

(−1)edge(π)xfix(π)

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Combinatorial interpretation for Hermite polynomials

µn = L(xn) =# perfect matchings on [n] Hn(x) also has a combinatorial meaning Hn(x) =

• π∈M(n)

(−1)edge(π)xfix(π) H3(x) = x3 − 3x x x x x −1 x −1 x −1

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Combinatorial interpretation for Hermite polynomials

µn = L(xn) =# perfect matchings on [n] Hn(x) also has a combinatorial meaning Hn(x) =

• π∈M(n)

(−1)edge(π)xfix(π) H3(x) = x3 − 3x x x x x −1 x −1 x −1 Combinatorial proof of L(Hn(x)Hm(x)) = 0 if n = m?

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Combinatorial interpretation for Hermite polynomials

µn = L(xn) =# perfect matchings on [n] Hn(x) also has a combinatorial meaning Hn(x) =

• π∈M(n)

(−1)edge(π)xfix(π) H3(x) = x3 − 3x x x x x −1 x −1 x −1 Combinatorial proof of L(Hn(x)Hm(x)) = 0 if n = m? Yes!

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Product of Hermite polynomials

Theorem (Azor, Gillis, and Victor, 1982)

L(Hn1(x) · · · Hnk(x)) = # perfect matchings on k sections [n1] ⊎ · · · ⊎ [nk] without homogeneous edges.

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Product of Hermite polynomials

Theorem (Azor, Gillis, and Victor, 1982)

L(Hn1(x) · · · Hnk(x)) = # perfect matchings on k sections [n1] ⊎ · · · ⊎ [nk] without homogeneous edges.

Example

L(Hn1(x)Hn2(x)Hn3(x)Hn4(x)) is # perfect matchings such as n1 n2 n3 n4

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Product of Hermite polynomials

Theorem (Azor, Gillis, and Victor, 1982)

L(Hn1(x) · · · Hnk(x)) = # perfect matchings on k sections [n1] ⊎ · · · ⊎ [nk] without homogeneous edges.

Example

L(Hn1(x)Hn2(x)Hn3(x)Hn4(x)) is # perfect matchings such as n1 n2 n3 n4 L(Hn(x)Hm(x)) = 0 if n = m

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Product of Hermite polynomials

Theorem (Azor, Gillis, and Victor, 1982)

L(Hn1(x) · · · Hnk(x)) = # perfect matchings on k sections [n1] ⊎ · · · ⊎ [nk] without homogeneous edges.

Example

L(Hn1(x)Hn2(x)Hn3(x)Hn4(x)) is # perfect matchings such as n1 n2 n3 n4 L(Hn(x)Hm(x)) = 0 if n = m L(Hn(x)Hn(x)) = n!

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Askey scheme

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Askey scheme

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Notations

[n]q = 1 + q + q2 + · · · + qn−1

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Notations

[n]q = 1 + q + q2 + · · · + qn−1 [n]q! = [1]q[2]q · · · [n]q

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Notations

[n]q = 1 + q + q2 + · · · + qn−1 [n]q! = [1]q[2]q · · · [n]q q-binomial coefficient

• n

k

• q

= [n]q! [k]q![n − k]q!

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Notations

[n]q = 1 + q + q2 + · · · + qn−1 [n]q! = [1]q[2]q · · · [n]q q-binomial coefficient

• n

k

• q

= [n]q! [k]q![n − k]q! q-multinomial coefficient

• a1 + · · · + ak

a1, . . . , ak

• q

= [a1 + · · · + ak]q! [a1]q! · · · [ak]q!

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Notations

[n]q = 1 + q + q2 + · · · + qn−1 [n]q! = [1]q[2]q · · · [n]q q-binomial coefficient

• n

k

• q

= [n]q! [k]q![n − k]q! q-multinomial coefficient

• a1 + · · · + ak

a1, . . . , ak

• q

= [a1 + · · · + ak]q! [a1]q! · · · [ak]q! q-shifted factorial (q-Pochhammer symbol) (a)n = (a; q)n = (1 − a)(1 − aq) · · · (1 − aqn−1)

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Notations

[n]q = 1 + q + q2 + · · · + qn−1 [n]q! = [1]q[2]q · · · [n]q q-binomial coefficient

• n

k

• q

= [n]q! [k]q![n − k]q! q-multinomial coefficient

• a1 + · · · + ak

a1, . . . , ak

• q

= [a1 + · · · + ak]q! [a1]q! · · · [ak]q! q-shifted factorial (q-Pochhammer symbol) (a)n = (a; q)n = (1 − a)(1 − aq) · · · (1 − aqn−1) (a1, . . . , ak)n = (a1)n · · · (ak)n

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Askey-Wilson polynomials

Askey-Wilson polynomials Pn(x) = Pn(x; a, b, c, d; q) with x = cos θ Pn(x; a, b, c, d|q) = (ab, ac, ad)n an

4φ3

• q−n, abcdqn−1, aeiθ, ae−iθ

ab, ac, ad

• q; q
• .

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Askey-Wilson polynomials

Askey-Wilson polynomials Pn(x) = Pn(x; a, b, c, d; q) with x = cos θ Pn(x; a, b, c, d|q) = (ab, ac, ad)n an

4φ3

• q−n, abcdqn−1, aeiθ, ae−iθ

ab, ac, ad

• q; q
• .

Orthogonality 1 2π 1

−1

Pm(x)Pn(x)w(x) dx √ 1 − x2 = hnδnm, where w(x) = w(x; a, b, c, d; q) is w(cos θ, a, b, c, d; q) = (e2iθ, e−2iθ)∞ (aeiθ, ae−iθ, beiθ, be−iθ, ceiθ, ce−iθ, deiθ, de−iθ)∞ .

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Askey-Wilson polynomials

Askey-Wilson polynomials Pn(x) = Pn(x; a, b, c, d; q) with x = cos θ Pn(x; a, b, c, d|q) = (ab, ac, ad)n an

4φ3

• q−n, abcdqn−1, aeiθ, ae−iθ

ab, ac, ad

• q; q
• .

Orthogonality 1 2π 1

−1

Pm(x)Pn(x)w(x) dx √ 1 − x2 = hnδnm, where w(x) = w(x; a, b, c, d; q) is w(cos θ, a, b, c, d; q) = (e2iθ, e−2iθ)∞ (aeiθ, ae−iθ, beiθ, be−iθ, ceiθ, ce−iθ, deiθ, de−iθ)∞ . The normalized nth moment µn(a, b, c, d; q) is (µ0 = 1) µn(a, b, c, d; q) = C 1

−1

xnw(x) dx √ 1 − x2 .

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Askey-Wilson polynomials

Askey-Wilson polynomials Pn(x) = Pn(x; a, b, c, d; q) with x = cos θ Pn(x; a, b, c, d|q) = (ab, ac, ad)n an

4φ3

• q−n, abcdqn−1, aeiθ, ae−iθ

ab, ac, ad

• q; q
• .

Orthogonality 1 2π 1

−1

Pm(x)Pn(x)w(x) dx √ 1 − x2 = hnδnm, where w(x) = w(x; a, b, c, d; q) is w(cos θ, a, b, c, d; q) = (e2iθ, e−2iθ)∞ (aeiθ, ae−iθ, beiθ, be−iθ, ceiθ, ce−iθ, deiθ, de−iθ)∞ . The normalized nth moment µn(a, b, c, d; q) is (µ0 = 1) µn(a, b, c, d; q) = C 1

−1

xnw(x) dx √ 1 − x2 . µn(a, b, c, d; q) is symmetrical in a, b, c, d.

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Known formulas for Askey-Wilson moments

Theorem (Corteel, Stanley, Stanton, Williams, 2010)

µn(a, b, c, d; q) = 1 2n

n

• m=0

(ab, ac, ad)m (abcd)m qm

m

• j=0

q−j2a−2j(aqj + q−j/a)n (q, q1−2j/a2)j(q, q2j+1a2)m−j .

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Known formulas for Askey-Wilson moments

Theorem (Corteel, Stanley, Stanton, Williams, 2010)

µn(a, b, c, d; q) = 1 2n

n

• m=0

(ab, ac, ad)m (abcd)m qm

m

• j=0

q−j2a−2j(aqj + q−j/a)n (q, q1−2j/a2)j(q, q2j+1a2)m−j .

Theorem (Ismail and Rahman, 2011)

µn(a, b, c, d; q) = (ab, qac, qad)n (2a)n(q, qa2, abcd)n

n

• k=0

1 − a2q2k 1 − a2 · (a2, q−n)k (q, a2qn+1)k (1 + a2q2k)n × qk(n+1) (1 − ac)(1 − ad) (1 − acqk)(1 − adqk)

4φ3

• qk−n, q, cd, aqk+1/b

acqk+1, adqk+1, q1−n/ab

• q, q

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Known formulas for Askey-Wilson moments

Theorem (Corteel, Stanley, Stanton, Williams, 2010)

µn(a, b, c, d; q) = 1 2n

n

• m=0

(ab, ac, ad)m (abcd)m qm

m

• j=0

q−j2a−2j(aqj + q−j/a)n (q, q1−2j/a2)j(q, q2j+1a2)m−j .

Theorem (Ismail and Rahman, 2011)

µn(a, b, c, d; q) = (ab, qac, qad)n (2a)n(q, qa2, abcd)n

n

• k=0

1 − a2q2k 1 − a2 · (a2, q−n)k (q, a2qn+1)k (1 + a2q2k)n × qk(n+1) (1 − ac)(1 − ad) (1 − acqk)(1 − adqk)

4φ3

• qk−n, q, cd, aqk+1/b

acqk+1, adqk+1, q1−n/ab

• q, q
• Proposition (K., Stanton, 2012)

2n(abcd)nµn(a, b, c, d; q) is a polynomial in a, b, c, d, q with integer coefficients.

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Main purpose

Give 3 combinatorial methods for computing µn.

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Main purpose

Give 3 combinatorial methods for computing µn. Motzkin paths

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Main purpose

Give 3 combinatorial methods for computing µn. Motzkin paths staircase tableaux

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Main purpose

Give 3 combinatorial methods for computing µn. Motzkin paths staircase tableaux q-Hermite polynomials and matchings

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Motzkin paths

Image stolen from Wikipedia

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Motzkin paths

The Askey-Wilson polynomials Pn = Pn(x; a, b, c, d; q) satisfy Pn+1 = (x − bn)Pn − λnPn−1, bn = 1 2(a + a−1 − (An + Cn)), λn = 1 4An−1Cn, where An = (1 − abqn)(1 − acqn)(1 − adqn)(1 − abcdqn−1) a(1 − abcdq2n−1)(1 − abcdq2n) , Cn = a(1 − qn)(1 − bcqn−1)(1 − bdqn−1)(1 − cdqn−1) (1 − abcdq2n−2)(1 − abcdq2n−1) .

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Motzkin paths

The Askey-Wilson polynomials Pn = Pn(x; a, b, c, d; q) satisfy Pn+1 = (x − bn)Pn − λnPn−1, bn = 1 2(a + a−1 − (An + Cn)), λn = 1 4An−1Cn, where An = (1 − abqn)(1 − acqn)(1 − adqn)(1 − abcdqn−1) a(1 − abcdq2n−1)(1 − abcdq2n) , Cn = a(1 − qn)(1 − bcqn−1)(1 − bdqn−1)(1 − cdqn−1) (1 − abcdq2n−2)(1 − abcdq2n−1) . If c = d = 0, then bi = aqi + bqi and λi = (1 − abqi−1)(1 − qi).

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Motzkin paths

The Askey-Wilson polynomials Pn = Pn(x; a, b, c, d; q) satisfy Pn+1 = (x − bn)Pn − λnPn−1, bn = 1 2(a + a−1 − (An + Cn)), λn = 1 4An−1Cn, where An = (1 − abqn)(1 − acqn)(1 − adqn)(1 − abcdqn−1) a(1 − abcdq2n−1)(1 − abcdq2n) , Cn = a(1 − qn)(1 − bcqn−1)(1 − bdqn−1)(1 − cdqn−1) (1 − abcdq2n−2)(1 − abcdq2n−1) . If c = d = 0, then bi = aqi + bqi and λi = (1 − abqi−1)(1 − qi). Doubly striped skew shapes: generalization of Dongsu Kim’s striped skew shapes. ⇔

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The c = d = 0 case: Al-Salam-Chihara polynomials

Theorem (K., Stanton, 2012)

2nµn(a, b, 0, 0; q) =

n

• k=0
• n

n−k 2

• −
• n

n−k 2

− 1

• u+v+2t=k

aubv(−1)tq(t+1

2 )

• u + v + t

u, v, t

• q

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The c = d = 0 case: Al-Salam-Chihara polynomials

Theorem (K., Stanton, 2012)

2nµn(a, b, 0, 0; q) =

n

• k=0
• n

n−k 2

• −
• n

n−k 2

− 1

• u+v+2t=k

aubv(−1)tq(t+1

2 )

• u + v + t

u, v, t

• q

This is equivalent to a formula of Josuat-Vergès.

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The c = d = 0 case: Al-Salam-Chihara polynomials

Theorem (K., Stanton, 2012)

2nµn(a, b, 0, 0; q) =

n

• k=0
• n

n−k 2

• −
• n

n−k 2

− 1

• u+v+2t=k

aubv(−1)tq(t+1

2 )

• u + v + t

u, v, t

• q

This is equivalent to a formula of Josuat-Vergès.

Theorem (Corteel, Josuat-Vergès, Rubey, Prellberg, 2009)

The nth moment of q-Laguerre polynomials is equal to

• π∈Sn

ywex(π)qcr(π) =

• T∈PT n

yrow(π)qso(π) = 1 (1 − q)n

n

• k=0

n−k

• j=0

yj

• n

j

• n

j + k

• −
• n

j − 1

• n

j + k + 1

• k
• i=0

(−1)kyiqi(k+1−i).

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The c = d = 0 case: Al-Salam-Chihara polynomials

Theorem (K., Stanton, 2012)

2nµn(a, b, 0, 0; q) =

n

• k=0
• n

n−k 2

• −
• n

n−k 2

− 1

• u+v+2t=k

aubv(−1)tq(t+1

2 )

• u + v + t

u, v, t

• q

This is equivalent to a formula of Josuat-Vergès.

Theorem (Corteel, Josuat-Vergès, Rubey, Prellberg, 2009)

The nth moment of q-Laguerre polynomials is equal to

• π∈Sn

ywex(π)qcr(π) =

• T∈PT n

yrow(π)qso(π) = 1 (1 − q)n

n

• k=0

n−k

• j=0

yj

• n

j

• n

j + k

• −
• n

j − 1

• n

j + k + 1

• k
• i=0

(−1)kyiqi(k+1−i). Our proof is the first combinatorial proof of CJRP .

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Open problem

If d = 0, bn = (a + b + c)qn − abcq2n − abcq2n−1 λn = (1 − qn)(1 − abqn−1)(1 − bcqn−1)(1 − caqn−1). i − 1 i 1 i − 1 i λi i bi

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Open problem

If d = 0, bn = (a + b + c)qn − abcq2n − abcq2n−1 λn = (1 − qn)(1 − abqn−1)(1 − bcqn−1)(1 − caqn−1). i − 1 i 1 i − 1 i λi i bi

Problem

Find a combinatorial proof using Motzkin paths of the following identity:

• P∈Motn

wt(P) =

n

• k=0
• n

n−k 2

• −
• n

n−k 2

− 1

• ×
• u+v+w+2t=k

aubvcw(−1)tq(t+1

2 )

• u + v + t

v

• q
• v + w + t

w

• q
• w + u + t

u

• q

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Staircase tableaux

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Staircase tableaux

A staircase tableau of size n is a filling of the Young diagram of the staircase partition (n, n − 1, . . . , 1) with α, β, γ, δ satisfying certain conditions. β γ γ α α δ δ γ β δ β

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Staircase tableaux

A staircase tableau of size n is a filling of the Young diagram of the staircase partition (n, n − 1, . . . , 1) with α, β, γ, δ satisfying certain conditions. β γ γ α α δ δ γ β δ β Introduced by Corteel and Williams (2010).

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Staircase tableaux

A staircase tableau of size n is a filling of the Young diagram of the staircase partition (n, n − 1, . . . , 1) with α, β, γ, δ satisfying certain conditions. β γ γ α α δ δ γ β δ β Introduced by Corteel and Williams (2010). Have connection with asymmetric exclusion process (ASEP) and moments of Askey-Wilson.

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Staircase tableaux

A staircase tableau of size n is a filling of the Young diagram of the staircase partition (n, n − 1, . . . , 1) with α, β, γ, δ satisfying certain conditions. β γ γ α α δ δ γ β δ β Introduced by Corteel and Williams (2010). Have connection with asymmetric exclusion process (ASEP) and moments of Askey-Wilson. If there are no γ and δ, we get permutation tableaux.

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Staircase tableaux

Theorem (Corteel, Stanley, Stanton, Williams, 2010)

2n(abcd)nµn(a, b, c, d; q) = i−n

T∈T (n)

(−1)b(T)(1 − q)A(T)+B(T)+C(T)+D(T)−nqE(T) × (ac)C(T)(bd)D(T) (1 + ai)(1 + ci) n−A(T)−C(T) (1 − bi)(1 − di) n−B(T)−D(T).

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Staircase tableaux

Theorem (Corteel, Stanley, Stanton, Williams, 2010)

2n(abcd)nµn(a, b, c, d; q) = i−n

T∈T (n)

(−1)b(T)(1 − q)A(T)+B(T)+C(T)+D(T)−nqE(T) × (ac)C(T)(bd)D(T) (1 + ai)(1 + ci) n−A(T)−C(T) (1 − bi)(1 − di) n−B(T)−D(T).

Theorem (K., Stanton, 2012)

We have

• anbncndnq(n

2)

2n(abcd)nµn(a, b, c, d; q) = Cat n 2

• ,
• an−1bncndnq(n

2)

2n(abcd)nµn(a, b, c, d; q) = − Cat n + 1 2

• ,
• an−1bn−1cndnq(n

2)

2n(abcd)nµn(a, b, c, d; q) = Cat n + 2 2

• − Cat

n 2

• ,

where Cat(n) =

1 n+1

2n

n

• if n is a nonnegative integer, and Cat(n) = 0
• therwise.

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q-Hermite polynomials

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q-Hermite polynomials

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q-Hermite polynomials

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Back to the definition of µn(a, b, c, d; q)

Recall µn(a, b, c, d; q) = C 1

−1

xnw(x) dx √ 1 − x2 = C π (cos θ)nw(cos θ)dθ, where w(cos θ, a, b, c, d; q) = (e2iθ, e−2iθ)∞ (aeiθ, ae−iθ, beiθ, be−iθ, ceiθ, ce−iθ, deiθ, de−iθ)∞ .

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Back to the definition of µn(a, b, c, d; q)

Recall µn(a, b, c, d; q) = C 1

−1

xnw(x) dx √ 1 − x2 = C π (cos θ)nw(cos θ)dθ, where w(cos θ, a, b, c, d; q) = (e2iθ, e−2iθ)∞ (aeiθ, ae−iθ, beiθ, be−iθ, ceiθ, ce−iθ, deiθ, de−iθ)∞ . Let In = (q)∞ 2π 1

−1

xnw(x) dx √ 1 − x2 = (q)∞ 2π π (cos θ)nw(cos θ)dθ

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Back to the definition of µn(a, b, c, d; q)

Recall µn(a, b, c, d; q) = C 1

−1

xnw(x) dx √ 1 − x2 = C π (cos θ)nw(cos θ)dθ, where w(cos θ, a, b, c, d; q) = (e2iθ, e−2iθ)∞ (aeiθ, ae−iθ, beiθ, be−iθ, ceiθ, ce−iθ, deiθ, de−iθ)∞ . Let In = (q)∞ 2π 1

−1

xnw(x) dx √ 1 − x2 = (q)∞ 2π π (cos θ)nw(cos θ)dθ Then the normalized nth moment is µn = In I0 .

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Back to the definition of µn(a, b, c, d; q)

Recall µn(a, b, c, d; q) = C 1

−1

xnw(x) dx √ 1 − x2 = C π (cos θ)nw(cos θ)dθ, where w(cos θ, a, b, c, d; q) = (e2iθ, e−2iθ)∞ (aeiθ, ae−iθ, beiθ, be−iθ, ceiθ, ce−iθ, deiθ, de−iθ)∞ . Let In = (q)∞ 2π 1

−1

xnw(x) dx √ 1 − x2 = (q)∞ 2π π (cos θ)nw(cos θ)dθ Then the normalized nth moment is µn = In I0 . I0 is the Askey-Wilson integral I0 = (q)∞ 2π 1

−1

w(x) dx √ 1 − x2 = (abcd)∞ (ab, ac, ad, bc, bd, cd)∞ .

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q-Hermite polynomials

Ismail, Stanton, and Viennot computed I0 using q-Hermite polynomials.

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q-Hermite polynomials

Ismail, Stanton, and Viennot computed I0 using q-Hermite polynomials. The q-Hermite polynomials Hn(x|q) are defined by

• n≥0

Hn(cos θ|q) zn (q)n = 1 (zeiθ, ze−iθ)∞

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q-Hermite polynomials

Ismail, Stanton, and Viennot computed I0 using q-Hermite polynomials. The q-Hermite polynomials Hn(x|q) are defined by

• n≥0

Hn(cos θ|q) zn (q)n = 1 (zeiθ, ze−iθ)∞ L(HnHm) = (q)∞ 2π π Hn(cos θ|q)Hm(cos θ|q)(e2iθ, e−2iθ)∞dθ = 0, n = m

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q-Hermite polynomials

Ismail, Stanton, and Viennot computed I0 using q-Hermite polynomials. The q-Hermite polynomials Hn(x|q) are defined by

• n≥0

Hn(cos θ|q) zn (q)n = 1 (zeiθ, ze−iθ)∞ L(HnHm) = (q)∞ 2π π Hn(cos θ|q)Hm(cos θ|q)(e2iθ, e−2iθ)∞dθ = 0, n = m Thus w(cos θ) = (e2iθ, e−2iθ)∞ (aeiθ, ae−iθ, beiθ, be−iθ, ceiθ, ce−iθ, deiθ, de−iθ)∞ =

• n1,n2,n3,n4≥0

an1bn2cn3dn4 (q)n1(q)n2(q)n3(q)n4 Hn1Hn2Hn3Hn4(e2iθ, e−2iθ)∞

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q-Hermite polynomials

Ismail, Stanton, and Viennot computed I0 using q-Hermite polynomials. The q-Hermite polynomials Hn(x|q) are defined by

• n≥0

Hn(cos θ|q) zn (q)n = 1 (zeiθ, ze−iθ)∞ L(HnHm) = (q)∞ 2π π Hn(cos θ|q)Hm(cos θ|q)(e2iθ, e−2iθ)∞dθ = 0, n = m Thus w(cos θ) = (e2iθ, e−2iθ)∞ (aeiθ, ae−iθ, beiθ, be−iθ, ceiθ, ce−iθ, deiθ, de−iθ)∞ =

• n1,n2,n3,n4≥0

an1bn2cn3dn4 (q)n1(q)n2(q)n3(q)n4 Hn1Hn2Hn3Hn4(e2iθ, e−2iθ)∞ We can write I0 = (q)∞ 2π π w(cos θ)dθ =

• n1,n2,n3,n4≥0

an1bn2cn3dn4 (q)n1(q)n2(q)n3(q)n4 L(Hn1Hn2Hn3Hn4)

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Combinatorial description for I0

Theorem (Ismail, Stanton, and Viennot (1985))

I0 =

• n1,n2,n3,n4≥0
• an1

bn2 cn3 dn4 [n1]q![n2]q![n3]q![n4]q!

• σ∈PM(n1,n2,n3,n4)

qcr(σ) where a = a/√1 − q, b = b/√1 − q, c = c/√1 − q, d = d/√1 − q and PM(n1, n2, n3, n4) is the set of perfect matchings on [n1] ⊎ [n2] ⊎ [n3] ⊎ [n4] without homogeneous edges. n1 n2 n3 n4

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Combinatorial description for I0

Theorem (Ismail, Stanton, and Viennot (1985))

I0 =

• n1,n2,n3,n4≥0
• an1

bn2 cn3 dn4 [n1]q![n2]q![n3]q![n4]q!

• σ∈PM(n1,n2,n3,n4)

qcr(σ) where a = a/√1 − q, b = b/√1 − q, c = c/√1 − q, d = d/√1 − q and PM(n1, n2, n3, n4) is the set of perfect matchings on [n1] ⊎ [n2] ⊎ [n3] ⊎ [n4] without homogeneous edges. n1 n2 n3 n4

Question

How about In?

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Combinatorial description for In

I0 is the generating function for perfect matchings with 4 sections: I0 = (q)∞ 2π π w(cos θ)dθ =

• n1,n2,n3,n4≥0

an1bn2cn3dn4 (q)n1(q)n2(q)n3(q)n4 L(Hn1Hn2Hn3Hn4)

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Combinatorial description for In

I0 is the generating function for perfect matchings with 4 sections: I0 = (q)∞ 2π π w(cos θ)dθ =

• n1,n2,n3,n4≥0

an1bn2cn3dn4 (q)n1(q)n2(q)n3(q)n4 L(Hn1Hn2Hn3Hn4) In is the generating function for perfect matchings with 5 sections: In = (q)∞ 2π π (cos θ)nw(cos θ)dθ =

• n1,n2,n3,n4≥0

an1bn2cn3dn4 (q)n1(q)n2(q)n3(q)n4 L(xnHn1Hn2Hn3Hn4)

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Combinatorial description for In

I0 is the generating function for perfect matchings with 4 sections: I0 = (q)∞ 2π π w(cos θ)dθ =

• n1,n2,n3,n4≥0

an1bn2cn3dn4 (q)n1(q)n2(q)n3(q)n4 L(Hn1Hn2Hn3Hn4) In is the generating function for perfect matchings with 5 sections: In = (q)∞ 2π π (cos θ)nw(cos θ)dθ =

• n1,n2,n3,n4≥0

an1bn2cn3dn4 (q)n1(q)n2(q)n3(q)n4 L(xnHn1Hn2Hn3Hn4)

Theorem (K., Stanton, 2012)

In = √1 − q 2 n

• n1,n2,n3,n4≥0
• an1

bn2 cn3 dn4 [n1]q![n2]q![n3]q![n4]q!

• σ∈PMn(n1,n2,n3,n4)

qcr(σ) where PMn(n1, n2, n3, n4) is the set of perfect matchings on [n] ⊎ [n1] ⊎ [n2] ⊎ [n3] ⊎ [n4] with homogeneous edges only in the first section. n n1 n2 n3 n4

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Combinatorial intepretation for µn(a, b, c, d; q)

Theorem (K., Stanton, 2012)

2nµn(a, b, c, d; q) = (1 − q)n/2In/I0 where In is the generating function for n n1 n2 n3 n4

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Combinatorial intepretation for µn(a, b, c, d; q)

Theorem (K., Stanton, 2012)

2nµn(a, b, c, d; q) = (1 − q)n/2In/I0 where In is the generating function for n n1 n2 n3 n4

Theorem (K., Stanton, 2012)

2nµn(a, b, c, d; q) =

n

• k=0
• n

n−k 2

• −
• n

n−k 2

− 1

• α+β+γ+δ+2t=k

aαbβcγdδ (ac)β(bd)γ (abcd)β+γ × (−1)tq(t+1

2 )

• α + β + γ + t

α

• q
• β + γ + δ + t

β, γ, δ + t

• q
• δ + α + t

δ

• q

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Corollaries

Corollary (K., Stanton, 2012)

2nµn(a, b, c, 0; q) =

n

• k=0
• n

n−k 2

• −
• n

n−k 2

− 1

• ×
• u+v+w+2t=k

aubvcw(−1)tq(t+1

2 )

• u + v + t

v

• q
• v + w + t

w

• q
• w + u + t

u

• q

.

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Corollaries

Corollary (K., Stanton, 2012)

2nµn(a, b, c, 0; q) =

n

• k=0
• n

n−k 2

• −
• n

n−k 2

− 1

• ×
• u+v+w+2t=k

aubvcw(−1)tq(t+1

2 )

• u + v + t

v

• q
• v + w + t

w

• q
• w + u + t

u

• q

.

Corollary (K., Stanton, 2012)

2nµn(a, b, q/a, q/b; q) =

n

• k=0
• n

n−k 2

• −
• n

n−k 2

− 1

• 1

[k + 1]q

• |A|+|B|≤k

A+B≡k mod 2

aAbBq

k−A−B 2

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Corollaries

Corollary (K., Stanton, 2012)

2nµn(a, b, c, 0; q) =

n

• k=0
• n

n−k 2

• −
• n

n−k 2

− 1

• ×
• u+v+w+2t=k

aubvcw(−1)tq(t+1

2 )

• u + v + t

v

• q
• v + w + t

w

• q
• w + u + t

u

• q

.

Corollary (K., Stanton, 2012)

2nµn(a, b, q/a, q/b; q) =

n

• k=0
• n

n−k 2

• −
• n

n−k 2

− 1

• 1

[k + 1]q

• |A|+|B|≤k

A+B≡k mod 2

aAbBq

k−A−B 2

Corollary (K., Stanton, 2012)

[n + 1]q!2nµn(a, b, q/a, q/b; q) is a Laurent polynomial in a and b whose coefficients are positive polynomials in q.

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Open problems

Problem

Find a combinatorial proof of 2nµn(a, b, q/a, q/b; q) =

n

• k=0
• n

n−k 2

• −
• n

n−k 2

− 1

• 1

[k + 1]q

• |A|+|B|≤k

A+B≡k mod 2

aAbBq

k−A−B 2

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Open problems

Problem

Find a combinatorial proof of 2nµn(a, b, q/a, q/b; q) =

n

• k=0
• n

n−k 2

• −
• n

n−k 2

− 1

• 1

[k + 1]q

• |A|+|B|≤k

A+B≡k mod 2

aAbBq

k−A−B 2

If ac = q and bd = q,

27 / 28

Open problems

Problem

Find a combinatorial proof of 2nµn(a, b, q/a, q/b; q) =

n

• k=0
• n

n−k 2

• −
• n

n−k 2

− 1

• 1

[k + 1]q

• |A|+|B|≤k

A+B≡k mod 2

aAbBq

k−A−B 2

If ac = q and bd = q,

Corollary (K., Stanton, 2012)

[n + 1]q!2nµn(a, b, q/a, q/b; q) is a Laurent polynomial in a and b whose coefficients are positive polynomials in q.

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Open problems

Problem

Find a combinatorial proof of 2nµn(a, b, q/a, q/b; q) =

n

• k=0
• n

n−k 2

• −
• n

n−k 2

− 1

• 1

[k + 1]q

• |A|+|B|≤k

A+B≡k mod 2

aAbBq

k−A−B 2

If ac = q and bd = q,

Corollary (K., Stanton, 2012)

[n + 1]q!2nµn(a, b, q/a, q/b; q) is a Laurent polynomial in a and b whose coefficients are positive polynomials in q. If ac = qi and bd = qj,

27 / 28

Open problems

Problem

Find a combinatorial proof of 2nµn(a, b, q/a, q/b; q) =

n

• k=0
• n

n−k 2

• −
• n

n−k 2

− 1

• 1

[k + 1]q

• |A|+|B|≤k

A+B≡k mod 2

aAbBq

k−A−B 2

If ac = q and bd = q,

Corollary (K., Stanton, 2012)

[n + 1]q!2nµn(a, b, q/a, q/b; q) is a Laurent polynomial in a and b whose coefficients are positive polynomials in q. If ac = qi and bd = qj,

Conjecture

For positive integers i and j, 2n[n + i + j − 1]q!µn(a, b, qi/a, qj/b; q) is a Laurent polynomial in a, b whose coefficients are positive polynomials in q.

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Math Genealogy

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Math Genealogy

SCHMIDT

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Math Genealogy

SCHMIDT BOCHNER

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Math Genealogy

SCHMIDT BOCHNER ASKEY

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Math Genealogy

SCHMIDT BOCHNER ASKEY STANTON

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Math Genealogy

SCHMIDT BOCHNER ASKEY STANTON DONGSU KIM

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Math Genealogy

SCHMIDT BOCHNER ASKEY STANTON DONGSU KIM JANG SOO KIM

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Math Genealogy

SCHMIDT BOCHNER ASKEY STANTON DONGSU KIM JANG SOO KIM