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

7 / 28 Product of Hermite polynomials Theorem (Azor, Gillis, and Victor, 1982) L ( H n 1 ( x ) · · · H n k ( x )) = # perfect matchings on k sections [ n 1 ] ⊎ · · · ⊎ [ n k ] without homogeneous edges. Example L ( H n 1 ( x ) H n 2 ( x ) H n 3 ( x ) H n 4 ( x )) is # perfect matchings such as n 1 n 2 n 3 n 4 L ( H n ( x ) H m ( x )) = 0 if n � = m

7 / 28 Product of Hermite polynomials Theorem (Azor, Gillis, and Victor, 1982) L ( H n 1 ( x ) · · · H n k ( x )) = # perfect matchings on k sections [ n 1 ] ⊎ · · · ⊎ [ n k ] without homogeneous edges. Example L ( H n 1 ( x ) H n 2 ( x ) H n 3 ( x ) H n 4 ( x )) is # perfect matchings such as n 1 n 2 n 3 n 4 L ( H n ( x ) H m ( x )) = 0 if n � = m L ( H n ( x ) H n ( x )) = n !

8 / 28 Askey scheme

8 / 28 Askey scheme

9 / 28 Notations [ n ] q = 1 + q + q 2 + · · · + q n − 1

9 / 28 Notations [ n ] q = 1 + q + q 2 + · · · + q n − 1 [ n ] q ! = [ 1 ] q [ 2 ] q · · · [ n ] q

9 / 28 Notations [ n ] q = 1 + q + q 2 + · · · + q n − 1 [ n ] q ! = [ 1 ] q [ 2 ] q · · · [ n ] q q -binomial coefficient � � [ n ] q ! n = k [ k ] q ![ n − k ] q ! q

9 / 28 Notations [ n ] q = 1 + q + q 2 + · · · + q n − 1 [ n ] q ! = [ 1 ] q [ 2 ] q · · · [ n ] q q -binomial coefficient � � [ n ] q ! n = k [ k ] q ![ n − k ] q ! q q -multinomial coefficient � � a 1 + · · · + a k = [ a 1 + · · · + a k ] q ! a 1 , . . . , a k [ a 1 ] q ! · · · [ a k ] q ! q

9 / 28 Notations [ n ] q = 1 + q + q 2 + · · · + q n − 1 [ n ] q ! = [ 1 ] q [ 2 ] q · · · [ n ] q q -binomial coefficient � � [ n ] q ! n = k [ k ] q ![ n − k ] q ! q q -multinomial coefficient � � a 1 + · · · + a k = [ a 1 + · · · + a k ] q ! a 1 , . . . , a k [ a 1 ] q ! · · · [ a k ] q ! q q -shifted factorial ( q -Pochhammer symbol ) ( a ) n = ( a ; q ) n = ( 1 − a )( 1 − aq ) · · · ( 1 − aq n − 1 )

9 / 28 Notations [ n ] q = 1 + q + q 2 + · · · + q n − 1 [ n ] q ! = [ 1 ] q [ 2 ] q · · · [ n ] q q -binomial coefficient � � [ n ] q ! n = k [ k ] q ![ n − k ] q ! q q -multinomial coefficient � � a 1 + · · · + a k = [ a 1 + · · · + a k ] q ! a 1 , . . . , a k [ a 1 ] q ! · · · [ a k ] q ! q q -shifted factorial ( q -Pochhammer symbol ) ( a ) n = ( a ; q ) n = ( 1 − a )( 1 − aq ) · · · ( 1 − aq n − 1 ) ( a 1 , . . . , a k ) n = ( a 1 ) n · · · ( a k ) n

10 / 28 Askey-Wilson polynomials Askey-Wilson polynomials P n ( x ) = P n ( x ; a , b , c , d ; q ) with x = cos θ � � � � q − n , abcdq n − 1 , ae i θ , ae − i θ P n ( x ; a , b , c , d | q ) = ( ab , ac , ad ) n � 4 φ 3 � q ; q . ab , ac , ad a n

10 / 28 Askey-Wilson polynomials Askey-Wilson polynomials P n ( x ) = P n ( x ; a , b , c , d ; q ) with x = cos θ � � � � q − n , abcdq n − 1 , ae i θ , ae − i θ P n ( x ; a , b , c , d | q ) = ( ab , ac , ad ) n � 4 φ 3 � q ; q . ab , ac , ad a n � 1 1 dx Orthogonality P m ( x ) P n ( x ) w ( x ) √ 1 − x 2 = h n δ nm , where 2 π − 1 w ( x ) = w ( x ; a , b , c , d ; q ) is ( e 2 i θ , e − 2 i θ ) ∞ w ( cos θ, a , b , c , d ; q ) = ( ae i θ , ae − i θ , be i θ , be − i θ , ce i θ , ce − i θ , de i θ , de − i θ ) ∞ .

10 / 28 Askey-Wilson polynomials Askey-Wilson polynomials P n ( x ) = P n ( x ; a , b , c , d ; q ) with x = cos θ � � � � q − n , abcdq n − 1 , ae i θ , ae − i θ P n ( x ; a , b , c , d | q ) = ( ab , ac , ad ) n � 4 φ 3 � q ; q . ab , ac , ad a n � 1 1 dx Orthogonality P m ( x ) P n ( x ) w ( x ) √ 1 − x 2 = h n δ nm , where 2 π − 1 w ( x ) = w ( x ; a , b , c , d ; q ) is ( e 2 i θ , e − 2 i θ ) ∞ w ( cos θ, a , b , c , d ; q ) = ( ae i θ , ae − i θ , be i θ , be − i θ , ce i θ , ce − i θ , de i θ , de − i θ ) ∞ . The normalized n th moment µ n ( a , b , c , d ; q ) is ( µ 0 = 1 ) � 1 dx x n w ( x ) µ n ( a , b , c , d ; q ) = C √ 1 − x 2 . − 1

10 / 28 Askey-Wilson polynomials Askey-Wilson polynomials P n ( x ) = P n ( x ; a , b , c , d ; q ) with x = cos θ � � � � q − n , abcdq n − 1 , ae i θ , ae − i θ P n ( x ; a , b , c , d | q ) = ( ab , ac , ad ) n � 4 φ 3 � q ; q . ab , ac , ad a n � 1 1 dx Orthogonality P m ( x ) P n ( x ) w ( x ) √ 1 − x 2 = h n δ nm , where 2 π − 1 w ( x ) = w ( x ; a , b , c , d ; q ) is ( e 2 i θ , e − 2 i θ ) ∞ w ( cos θ, a , b , c , d ; q ) = ( ae i θ , ae − i θ , be i θ , be − i θ , ce i θ , ce − i θ , de i θ , de − i θ ) ∞ . The normalized n th moment µ n ( a , b , c , d ; q ) is ( µ 0 = 1 ) � 1 dx x n w ( x ) µ n ( a , b , c , d ; q ) = C √ 1 − x 2 . − 1 µ n ( a , b , c , d ; q ) is symmetrical in a , b , c , d .

11 / 28 Known formulas for Askey-Wilson moments Theorem (Corteel, Stanley, Stanton, Williams, 2010) q − j 2 a − 2 j ( aq j + q − j / a ) n � n � m ( ab , ac , ad ) m µ n ( a , b , c , d ; q ) = 1 q m ( q , q 1 − 2 j / a 2 ) j ( q , q 2 j + 1 a 2 ) m − j . 2 n ( abcd ) m m = 0 j = 0

11 / 28 Known formulas for Askey-Wilson moments Theorem (Corteel, Stanley, Stanton, Williams, 2010) q − j 2 a − 2 j ( aq j + q − j / a ) n � n � m ( ab , ac , ad ) m µ n ( a , b , c , d ; q ) = 1 q m ( q , q 1 − 2 j / a 2 ) j ( q , q 2 j + 1 a 2 ) m − j . 2 n ( abcd ) m m = 0 j = 0 Theorem (Ismail and Rahman, 2011) n � 1 − a 2 q 2 k · ( a 2 , q − n ) k ( ab , qac , qad ) n ( q , a 2 q n + 1 ) k ( 1 + a 2 q 2 k ) n µ n ( a , b , c , d ; q ) = ( 2 a ) n ( q , qa 2 , abcd ) n 1 − a 2 k = 0 � � � q k − n , q , cd , aq k + 1 / b � ( 1 − ac )( 1 − ad ) × q k ( n + 1 ) � 4 φ 3 � q , q acq k + 1 , adq k + 1 , q 1 − n / ab ( 1 − acq k )( 1 − adq k )

11 / 28 Known formulas for Askey-Wilson moments Theorem (Corteel, Stanley, Stanton, Williams, 2010) q − j 2 a − 2 j ( aq j + q − j / a ) n � n � m ( ab , ac , ad ) m µ n ( a , b , c , d ; q ) = 1 q m ( q , q 1 − 2 j / a 2 ) j ( q , q 2 j + 1 a 2 ) m − j . 2 n ( abcd ) m m = 0 j = 0 Theorem (Ismail and Rahman, 2011) n � 1 − a 2 q 2 k · ( a 2 , q − n ) k ( ab , qac , qad ) n ( q , a 2 q n + 1 ) k ( 1 + a 2 q 2 k ) n µ n ( a , b , c , d ; q ) = ( 2 a ) n ( q , qa 2 , abcd ) n 1 − a 2 k = 0 � � � q k − n , q , cd , aq k + 1 / b � ( 1 − ac )( 1 − ad ) × q k ( n + 1 ) � 4 φ 3 � q , q acq k + 1 , adq k + 1 , q 1 − n / ab ( 1 − acq k )( 1 − adq k ) Proposition (K., Stanton, 2012) 2 n ( abcd ) n µ n ( a , b , c , d ; q ) is a polynomial in a , b , c , d , q with integer coefficients.

12 / 28 Main purpose Give 3 combinatorial methods for computing µ n .

12 / 28 Main purpose Give 3 combinatorial methods for computing µ n . Motzkin paths

12 / 28 Main purpose Give 3 combinatorial methods for computing µ n . Motzkin paths staircase tableaux

12 / 28 Main purpose Give 3 combinatorial methods for computing µ n . Motzkin paths staircase tableaux q -Hermite polynomials and matchings

13 / 28 Motzkin paths Image stolen from Wikipedia

14 / 28 Motzkin paths The Askey-Wilson polynomials P n = P n ( x ; a , b , c , d ; q ) satisfy P n + 1 = ( x − b n ) P n − λ n P n − 1 , b n = 1 λ n = 1 2 ( a + a − 1 − ( A n + C n )) , 4 A n − 1 C n , where A n = ( 1 − abq n )( 1 − acq n )( 1 − adq n )( 1 − abcdq n − 1 ) , a ( 1 − abcdq 2 n − 1 )( 1 − abcdq 2 n ) C n = a ( 1 − q n )( 1 − bcq n − 1 )( 1 − bdq n − 1 )( 1 − cdq n − 1 ) . ( 1 − abcdq 2 n − 2 )( 1 − abcdq 2 n − 1 )

14 / 28 Motzkin paths The Askey-Wilson polynomials P n = P n ( x ; a , b , c , d ; q ) satisfy P n + 1 = ( x − b n ) P n − λ n P n − 1 , b n = 1 λ n = 1 2 ( a + a − 1 − ( A n + C n )) , 4 A n − 1 C n , where A n = ( 1 − abq n )( 1 − acq n )( 1 − adq n )( 1 − abcdq n − 1 ) , a ( 1 − abcdq 2 n − 1 )( 1 − abcdq 2 n ) C n = a ( 1 − q n )( 1 − bcq n − 1 )( 1 − bdq n − 1 )( 1 − cdq n − 1 ) . ( 1 − abcdq 2 n − 2 )( 1 − abcdq 2 n − 1 ) If c = d = 0 , then b i = aq i + bq i and λ i = ( 1 − abq i − 1 )( 1 − q i ) .

14 / 28 Motzkin paths The Askey-Wilson polynomials P n = P n ( x ; a , b , c , d ; q ) satisfy P n + 1 = ( x − b n ) P n − λ n P n − 1 , b n = 1 λ n = 1 2 ( a + a − 1 − ( A n + C n )) , 4 A n − 1 C n , where A n = ( 1 − abq n )( 1 − acq n )( 1 − adq n )( 1 − abcdq n − 1 ) , a ( 1 − abcdq 2 n − 1 )( 1 − abcdq 2 n ) C n = a ( 1 − q n )( 1 − bcq n − 1 )( 1 − bdq n − 1 )( 1 − cdq n − 1 ) . ( 1 − abcdq 2 n − 2 )( 1 − abcdq 2 n − 1 ) If c = d = 0 , then b i = aq i + bq i and λ i = ( 1 − abq i − 1 )( 1 − q i ) . Doubly striped skew shapes : generalization of Dongsu Kim’s striped skew shapes. ⇔

15 / 28 The c = d = 0 case: Al-Salam-Chihara polynomials Theorem (K., Stanton, 2012) �� � � �� � � � n � n n u + v + t a u b v ( − 1 ) t q ( t + 1 2 ) 2 n µ n ( a , b , 0 , 0 ; q ) = − n − k n − k − 1 u , v , t 2 2 k = 0 u + v + 2 t = k q

15 / 28 The c = d = 0 case: Al-Salam-Chihara polynomials Theorem (K., Stanton, 2012) �� � � �� � � � n � n n u + v + t a u b v ( − 1 ) t q ( t + 1 2 ) 2 n µ n ( a , b , 0 , 0 ; q ) = − n − k n − k − 1 u , v , t 2 2 k = 0 u + v + 2 t = k q This is equivalent to a formula of Josuat-Vergès.

15 / 28 The c = d = 0 case: Al-Salam-Chihara polynomials Theorem (K., Stanton, 2012) �� � � �� � � � n � n n u + v + t a u b v ( − 1 ) t q ( t + 1 2 ) 2 n µ n ( a , b , 0 , 0 ; q ) = − n − k n − k − 1 u , v , t 2 2 k = 0 u + v + 2 t = k q This is equivalent to a formula of Josuat-Vergès. Theorem (Corteel, Josuat-Vergès, Rubey, Prellberg, 2009) The n th moment of q -Laguerre polynomials is equal to � � y wex ( π ) q cr ( π ) = y row ( π ) q so ( π ) = π ∈ S n T ∈PT n �� �� � � �� �� n − k � n � � k 1 n n n n y j ( − 1 ) k y i q i ( k + 1 − i ) . − ( 1 − q ) n j + k j − 1 j + k + 1 j k = 0 j = 0 i = 0

15 / 28 The c = d = 0 case: Al-Salam-Chihara polynomials Theorem (K., Stanton, 2012) �� � � �� � � � n � n n u + v + t a u b v ( − 1 ) t q ( t + 1 2 ) 2 n µ n ( a , b , 0 , 0 ; q ) = − n − k n − k − 1 u , v , t 2 2 k = 0 u + v + 2 t = k q This is equivalent to a formula of Josuat-Vergès. Theorem (Corteel, Josuat-Vergès, Rubey, Prellberg, 2009) The n th moment of q -Laguerre polynomials is equal to � � y wex ( π ) q cr ( π ) = y row ( π ) q so ( π ) = π ∈ S n T ∈PT n �� �� � � �� �� n − k � n � � k 1 n n n n y j ( − 1 ) k y i q i ( k + 1 − i ) . − ( 1 − q ) n j + k j − 1 j + k + 1 j k = 0 j = 0 i = 0 Our proof is the first combinatorial proof of CJRP .

16 / 28 Open problem If d = 0 , b n = ( a + b + c ) q n − abcq 2 n − abcq 2 n − 1 λ n = ( 1 − q n )( 1 − abq n − 1 )( 1 − bcq n − 1 )( 1 − caq n − 1 ) . i i i i − 1 i − 1 λ i b i 1

16 / 28 Open problem If d = 0 , b n = ( a + b + c ) q n − abcq 2 n − abcq 2 n − 1 λ n = ( 1 − q n )( 1 − abq n − 1 )( 1 − bcq n − 1 )( 1 − caq n − 1 ) . i i i i − 1 i − 1 λ i b i 1 Problem Find a combinatorial proof using Motzkin paths of the following identity: �� � � �� � � n n n wt ( P ) = − n − k n − k − 1 2 2 P ∈ Mot n k = 0 � � � � � � � u + v + t v + w + t w + u + t a u b v c w ( − 1 ) t q ( t + 1 2 ) × v w u u + v + w + 2 t = k q q q

17 / 28 Staircase tableaux

18 / 28 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. γ β γ α α δ γ δ β δ β

18 / 28 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. γ β γ α α δ γ δ β δ β Introduced by Corteel and Williams (2010).

18 / 28 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. γ β γ α α δ γ δ β δ β Introduced by Corteel and Williams (2010). Have connection with asymmetric exclusion process (ASEP) and moments of Askey-Wilson.

18 / 28 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. γ β γ α α δ γ δ 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 .

19 / 28 Staircase tableaux Theorem (Corteel, Stanley, Stanton, Williams, 2010) 2 n ( abcd ) n µ n ( a , b , c , d ; q ) = i − n � ( − 1 ) b ( T ) ( 1 − q ) A ( T )+ B ( T )+ C ( T )+ D ( T ) − n q E ( T ) T ∈T ( n ) × ( ac ) C ( T ) ( bd ) D ( T ) � � n − A ( T ) − C ( T ) � � n − B ( T ) − D ( T ) . ( 1 + ai )( 1 + ci ) ( 1 − bi )( 1 − di )

19 / 28 Staircase tableaux Theorem (Corteel, Stanley, Stanton, Williams, 2010) 2 n ( abcd ) n µ n ( a , b , c , d ; q ) = i − n � ( − 1 ) b ( T ) ( 1 − q ) A ( T )+ B ( T )+ C ( T )+ D ( T ) − n q E ( T ) T ∈T ( n ) × ( ac ) C ( T ) ( bd ) D ( T ) � � n − A ( T ) − C ( T ) � � n − B ( T ) − D ( T ) . ( 1 + ai )( 1 + ci ) ( 1 − bi )( 1 − di ) Theorem (K., Stanton, 2012) We have � 2 ) � � n � a n b n c n d n q ( n 2 n ( abcd ) n µ n ( a , b , c , d ; q ) = Cat , 2 � n + 1 � � 2 ) � a n − 1 b n c n d n q ( n 2 n ( abcd ) n µ n ( a , b , c , d ; q ) = − Cat , 2 � n + 2 � � 2 ) � � n � a n − 1 b n − 1 c n d n q ( n 2 n ( abcd ) n µ n ( a , b , c , d ; q ) = Cat − Cat , 2 2 � 2 n � 1 where Cat ( n ) = if n is a nonnegative integer, and Cat ( n ) = 0 n + 1 n otherwise.

20 / 28 q -Hermite polynomials

20 / 28 q -Hermite polynomials

20 / 28 q -Hermite polynomials

21 / 28 Back to the definition of µ n ( a , b , c , d ; q ) Recall � 1 � π dx x n w ( x ) ( cos θ ) n w ( cos θ ) d θ, µ n ( a , b , c , d ; q ) = C √ 1 − x 2 = C − 1 0 where ( e 2 i θ , e − 2 i θ ) ∞ w ( cos θ, a , b , c , d ; q ) = ( ae i θ , ae − i θ , be i θ , be − i θ , ce i θ , ce − i θ , de i θ , de − i θ ) ∞ .

21 / 28 Back to the definition of µ n ( a , b , c , d ; q ) Recall � 1 � π dx x n w ( x ) ( cos θ ) n w ( cos θ ) d θ, µ n ( a , b , c , d ; q ) = C √ 1 − x 2 = C − 1 0 where ( e 2 i θ , e − 2 i θ ) ∞ w ( cos θ, a , b , c , d ; q ) = ( ae i θ , ae − i θ , be i θ , be − i θ , ce i θ , ce − i θ , de i θ , de − i θ ) ∞ . Let � 1 � π I n = ( q ) ∞ 1 − x 2 = ( q ) ∞ dx x n w ( x ) ( cos θ ) n w ( cos θ ) d θ √ 2 π 2 π − 1 0

21 / 28 Back to the definition of µ n ( a , b , c , d ; q ) Recall � 1 � π dx x n w ( x ) ( cos θ ) n w ( cos θ ) d θ, µ n ( a , b , c , d ; q ) = C √ 1 − x 2 = C − 1 0 where ( e 2 i θ , e − 2 i θ ) ∞ w ( cos θ, a , b , c , d ; q ) = ( ae i θ , ae − i θ , be i θ , be − i θ , ce i θ , ce − i θ , de i θ , de − i θ ) ∞ . Let � 1 � π I n = ( q ) ∞ 1 − x 2 = ( q ) ∞ dx x n w ( x ) ( cos θ ) n w ( cos θ ) d θ √ 2 π 2 π − 1 0 Then the normalized n th moment is µ n = I n I 0 .

21 / 28 Back to the definition of µ n ( a , b , c , d ; q ) Recall � 1 � π dx x n w ( x ) ( cos θ ) n w ( cos θ ) d θ, µ n ( a , b , c , d ; q ) = C √ 1 − x 2 = C − 1 0 where ( e 2 i θ , e − 2 i θ ) ∞ w ( cos θ, a , b , c , d ; q ) = ( ae i θ , ae − i θ , be i θ , be − i θ , ce i θ , ce − i θ , de i θ , de − i θ ) ∞ . Let � 1 � π I n = ( q ) ∞ 1 − x 2 = ( q ) ∞ dx x n w ( x ) ( cos θ ) n w ( cos θ ) d θ √ 2 π 2 π − 1 0 Then the normalized n th moment is µ n = I n I 0 . I 0 is the Askey-Wilson integral � 1 I 0 = ( q ) ∞ ( abcd ) ∞ dx w ( x ) √ 1 − x 2 = ( ab , ac , ad , bc , bd , cd ) ∞ . 2 π − 1

22 / 28 q -Hermite polynomials Ismail, Stanton, and Viennot computed I 0 using q -Hermite polynomials.

22 / 28 q -Hermite polynomials Ismail, Stanton, and Viennot computed I 0 using q -Hermite polynomials. The q -Hermite polynomials H n ( x | q ) are defined by � H n ( cos θ | q ) z n 1 ( q ) n = ( ze i θ , ze − i θ ) ∞ n ≥ 0

22 / 28 q -Hermite polynomials Ismail, Stanton, and Viennot computed I 0 using q -Hermite polynomials. The q -Hermite polynomials H n ( x | q ) are defined by � H n ( cos θ | q ) z n 1 ( q ) n = ( ze i θ , ze − i θ ) ∞ n ≥ 0 � π L ( H n H m ) = ( q ) ∞ H n ( cos θ | q ) H m ( cos θ | q )( e 2 i θ , e − 2 i θ ) ∞ d θ = 0 , n � = m 2 π 0

22 / 28 q -Hermite polynomials Ismail, Stanton, and Viennot computed I 0 using q -Hermite polynomials. The q -Hermite polynomials H n ( x | q ) are defined by � H n ( cos θ | q ) z n 1 ( q ) n = ( ze i θ , ze − i θ ) ∞ n ≥ 0 � π L ( H n H m ) = ( q ) ∞ H n ( cos θ | q ) H m ( cos θ | q )( e 2 i θ , e − 2 i θ ) ∞ d θ = 0 , n � = m 2 π 0 Thus ( e 2 i θ , e − 2 i θ ) ∞ w ( cos θ ) = ( ae i θ , ae − i θ , be i θ , be − i θ , ce i θ , ce − i θ , de i θ , de − i θ ) ∞ � a n 1 b n 2 c n 3 d n 4 H n 1 H n 2 H n 3 H n 4 ( e 2 i θ , e − 2 i θ ) ∞ = ( q ) n 1 ( q ) n 2 ( q ) n 3 ( q ) n 4 n 1 , n 2 , n 3 , n 4 ≥ 0

22 / 28 q -Hermite polynomials Ismail, Stanton, and Viennot computed I 0 using q -Hermite polynomials. The q -Hermite polynomials H n ( x | q ) are defined by � H n ( cos θ | q ) z n 1 ( q ) n = ( ze i θ , ze − i θ ) ∞ n ≥ 0 � π L ( H n H m ) = ( q ) ∞ H n ( cos θ | q ) H m ( cos θ | q )( e 2 i θ , e − 2 i θ ) ∞ d θ = 0 , n � = m 2 π 0 Thus ( e 2 i θ , e − 2 i θ ) ∞ w ( cos θ ) = ( ae i θ , ae − i θ , be i θ , be − i θ , ce i θ , ce − i θ , de i θ , de − i θ ) ∞ � a n 1 b n 2 c n 3 d n 4 H n 1 H n 2 H n 3 H n 4 ( e 2 i θ , e − 2 i θ ) ∞ = ( q ) n 1 ( q ) n 2 ( q ) n 3 ( q ) n 4 n 1 , n 2 , n 3 , n 4 ≥ 0 We can write � π � a n 1 b n 2 c n 3 d n 4 I 0 = ( q ) ∞ w ( cos θ ) d θ = L ( H n 1 H n 2 H n 3 H n 4 ) 2 π ( q ) n 1 ( q ) n 2 ( q ) n 3 ( q ) n 4 0 n 1 , n 2 , n 3 , n 4 ≥ 0

23 / 28 Combinatorial description for I 0 Theorem (Ismail, Stanton, and Viennot (1985)) � � a n 1 � b n 2 � c n 3 � d n 4 � q cr ( σ ) I 0 = [ n 1 ] q ![ n 2 ] q ![ n 3 ] q ![ n 4 ] q ! n 1 , n 2 , n 3 , n 4 ≥ 0 σ ∈PM ( n 1 , n 2 , n 3 , n 4 ) a = a / √ 1 − q , � b = b / √ 1 − q , � c = c / √ 1 − q , � d = d / √ 1 − q and where � PM ( n 1 , n 2 , n 3 , n 4 ) is the set of perfect matchings on [ n 1 ] ⊎ [ n 2 ] ⊎ [ n 3 ] ⊎ [ n 4 ] without homogeneous edges. n 1 n 2 n 3 n 4

23 / 28 Combinatorial description for I 0 Theorem (Ismail, Stanton, and Viennot (1985)) � � a n 1 � b n 2 � c n 3 � d n 4 � q cr ( σ ) I 0 = [ n 1 ] q ![ n 2 ] q ![ n 3 ] q ![ n 4 ] q ! n 1 , n 2 , n 3 , n 4 ≥ 0 σ ∈PM ( n 1 , n 2 , n 3 , n 4 ) a = a / √ 1 − q , � b = b / √ 1 − q , � c = c / √ 1 − q , � d = d / √ 1 − q and where � PM ( n 1 , n 2 , n 3 , n 4 ) is the set of perfect matchings on [ n 1 ] ⊎ [ n 2 ] ⊎ [ n 3 ] ⊎ [ n 4 ] without homogeneous edges. n 1 n 2 n 3 n 4 Question How about I n ?

24 / 28 Combinatorial description for I n I 0 is the generating function for perfect matchings with 4 sections: � π � a n 1 b n 2 c n 3 d n 4 I 0 = ( q ) ∞ w ( cos θ ) d θ = L ( H n 1 H n 2 H n 3 H n 4 ) 2 π ( q ) n 1 ( q ) n 2 ( q ) n 3 ( q ) n 4 0 n 1 , n 2 , n 3 , n 4 ≥ 0

24 / 28 Combinatorial description for I n I 0 is the generating function for perfect matchings with 4 sections: � π � a n 1 b n 2 c n 3 d n 4 I 0 = ( q ) ∞ w ( cos θ ) d θ = L ( H n 1 H n 2 H n 3 H n 4 ) 2 π ( q ) n 1 ( q ) n 2 ( q ) n 3 ( q ) n 4 0 n 1 , n 2 , n 3 , n 4 ≥ 0 I n is the generating function for perfect matchings with 5 sections: � π � a n 1 b n 2 c n 3 d n 4 I n = ( q ) ∞ ( cos θ ) n w ( cos θ ) d θ = L ( x n H n 1 H n 2 H n 3 H n 4 ) 2 π ( q ) n 1 ( q ) n 2 ( q ) n 3 ( q ) n 4 0 n 1 , n 2 , n 3 , n 4 ≥ 0

24 / 28 Combinatorial description for I n I 0 is the generating function for perfect matchings with 4 sections: � π � a n 1 b n 2 c n 3 d n 4 I 0 = ( q ) ∞ w ( cos θ ) d θ = L ( H n 1 H n 2 H n 3 H n 4 ) 2 π ( q ) n 1 ( q ) n 2 ( q ) n 3 ( q ) n 4 0 n 1 , n 2 , n 3 , n 4 ≥ 0 I n is the generating function for perfect matchings with 5 sections: � π � a n 1 b n 2 c n 3 d n 4 I n = ( q ) ∞ ( cos θ ) n w ( cos θ ) d θ = L ( x n H n 1 H n 2 H n 3 H n 4 ) 2 π ( q ) n 1 ( q ) n 2 ( q ) n 3 ( q ) n 4 0 n 1 , n 2 , n 3 , n 4 ≥ 0 Theorem (K., Stanton, 2012) � √ 1 − q � n � a n 1 � c n 3 � � b n 2 � d n 4 � q cr ( σ ) I n = [ n 1 ] q ![ n 2 ] q ![ n 3 ] q ![ n 4 ] q ! 2 n 1 , n 2 , n 3 , n 4 ≥ 0 σ ∈PM n ( n 1 , n 2 , n 3 , n 4 ) where PM n ( n 1 , n 2 , n 3 , n 4 ) is the set of perfect matchings on [ n ] ⊎ [ n 1 ] ⊎ [ n 2 ] ⊎ [ n 3 ] ⊎ [ n 4 ] with homogeneous edges only in the first section. n 1 n 2 n 3 n 4 n

25 / 28 Combinatorial intepretation for µ n ( a , b , c , d ; q ) Theorem (K., Stanton, 2012) 2 n µ n ( a , b , c , d ; q ) = ( 1 − q ) n / 2 I n / I 0 where I n is the generating function for n n 1 n 2 n 3 n 4

25 / 28 Combinatorial intepretation for µ n ( a , b , c , d ; q ) Theorem (K., Stanton, 2012) 2 n µ n ( a , b , c , d ; q ) = ( 1 − q ) n / 2 I n / I 0 where I n is the generating function for n n 1 n 2 n 3 n 4 Theorem (K., Stanton, 2012) �� � � �� � n � a α b β c γ d δ ( ac ) β ( bd ) γ n n 2 n µ n ( a , b , c , d ; q ) = − n − k n − k − 1 ( abcd ) β + γ 2 2 k = 0 α + β + γ + δ + 2 t = k � � � � � � α + β + γ + t β + γ + δ + t δ + α + t × ( − 1 ) t q ( t + 1 2 ) α β, γ, δ + t δ q q q

26 / 28 Corollaries Corollary (K., Stanton, 2012) �� � � �� � n n n 2 n µ n ( a , b , c , 0 ; q ) = − n − k n − k − 1 2 2 k = 0 � � � � � � � u + v + t v + w + t w + u + t a u b v c w ( − 1 ) t q ( t + 1 2 ) × . v w u u + v + w + 2 t = k q q q

26 / 28 Corollaries Corollary (K., Stanton, 2012) �� � � �� � n n n 2 n µ n ( a , b , c , 0 ; q ) = − n − k n − k − 1 2 2 k = 0 � � � � � � � u + v + t v + w + t w + u + t a u b v c w ( − 1 ) t q ( t + 1 2 ) × . v w u u + v + w + 2 t = k q q q Corollary (K., Stanton, 2012) �� � � �� � n � n n 1 k − A − B 2 n µ n ( a , b , q / a , q / b ; q ) = a A b B q − 2 n − k n − k − 1 [ k + 1 ] q 2 2 k = 0 | A | + | B |≤ k A + B ≡ k mod 2

26 / 28 Corollaries Corollary (K., Stanton, 2012) �� � � �� � n n n 2 n µ n ( a , b , c , 0 ; q ) = − n − k n − k − 1 2 2 k = 0 � � � � � � � u + v + t v + w + t w + u + t a u b v c w ( − 1 ) t q ( t + 1 2 ) × . v w u u + v + w + 2 t = k q q q Corollary (K., Stanton, 2012) �� � � �� � n � n n 1 k − A − B 2 n µ n ( a , b , q / a , q / b ; q ) = a A b B q − 2 n − k n − k − 1 [ k + 1 ] q 2 2 k = 0 | A | + | B |≤ k A + B ≡ k mod 2 Corollary (K., Stanton, 2012) [ n + 1 ] q ! 2 n µ n ( a , b , q / a , q / b ; q ) is a Laurent polynomial in a and b whose coefficients are positive polynomials in q .

27 / 28 Open problems Problem Find a combinatorial proof of �� � � �� � n � n n 1 k − A − B 2 n µ n ( a , b , q / a , q / b ; q ) = a A b B q − 2 n − k n − k − 1 [ k + 1 ] q 2 2 k = 0 | A | + | B |≤ k A + B ≡ k mod 2

27 / 28 Open problems Problem Find a combinatorial proof of �� � � �� � n � n n 1 k − A − B 2 n µ n ( a , b , q / a , q / b ; q ) = a A b B q − 2 n − k n − k − 1 [ k + 1 ] q 2 2 k = 0 | A | + | B |≤ k A + B ≡ k mod 2 If ac = q and bd = q ,

27 / 28 Open problems Problem Find a combinatorial proof of �� � � �� � n � n n 1 k − A − B 2 n µ n ( a , b , q / a , q / b ; q ) = a A b B q − 2 n − k n − k − 1 [ k + 1 ] q 2 2 k = 0 | A | + | B |≤ k A + B ≡ k mod 2 If ac = q and bd = q , Corollary (K., Stanton, 2012) [ n + 1 ] q ! 2 n µ n ( a , b , q / a , q / b ; q ) is a Laurent polynomial in a and b whose coefficients are positive polynomials in q .

27 / 28 Open problems Problem Find a combinatorial proof of �� � � �� � n � n n 1 k − A − B 2 n µ n ( a , b , q / a , q / b ; q ) = a A b B q − 2 n − k n − k − 1 [ k + 1 ] q 2 2 k = 0 | A | + | B |≤ k A + B ≡ k mod 2 If ac = q and bd = q , Corollary (K., Stanton, 2012) [ n + 1 ] q ! 2 n µ n ( a , b , q / a , q / b ; q ) is a Laurent polynomial in a and b whose coefficients are positive polynomials in q . If ac = q i and bd = q j ,

27 / 28 Open problems Problem Find a combinatorial proof of �� � � �� � n � n n 1 k − A − B 2 n µ n ( a , b , q / a , q / b ; q ) = a A b B q − 2 n − k n − k − 1 [ k + 1 ] q 2 2 k = 0 | A | + | B |≤ k A + B ≡ k mod 2 If ac = q and bd = q , Corollary (K., Stanton, 2012) [ n + 1 ] q ! 2 n µ n ( a , b , q / a , q / b ; q ) is a Laurent polynomial in a and b whose coefficients are positive polynomials in q . If ac = q i and bd = q j , Conjecture For positive integers i and j , 2 n [ n + i + j − 1 ] q ! µ n ( a , b , q i / a , q j / b ; q ) is a Laurent polynomial in a , b whose coefficients are positive polynomials in q .

28 / 28 Math Genealogy

28 / 28 Math Genealogy SCHMIDT

28 / 28 Math Genealogy SCHMIDT BOCHNER

28 / 28 Math Genealogy SCHMIDT BOCHNER ASKEY