Negative thinking and polynomial analogs OPSFA-15, Hagenberg, - PowerPoint PPT Presentation

Negative thinking and polynomial analogs OPSFA-15, Hagenberg, Austria Armin Straub July 26, 2019 University of South Alabama includes joint work with: Sam Formichella (University of South Alabama) Negative thinking and polynomial analogs Armin Straub 1 / 40

Basic q -analogs q -binomial coefficients A q -analog reduces to the classical object in the limit q → 1 . IDEA [ n ] q = q n − 1 DEF q − 1 = 1 + q + . . . q n − 1 • q -number: Negative thinking and polynomial analogs Armin Straub 2 / 40

Basic q -analogs q -binomial coefficients A q -analog reduces to the classical object in the limit q → 1 . IDEA [ n ] q = q n − 1 DEF q − 1 = 1 + q + . . . q n − 1 • q -number: ( q ; q ) n • q -factorial: [ n ] q ! = [ n ] q [ n − 1] q · · · [1] q = (1 − q ) n For q -series fans: � n � [ n ] q ! D1 ( q ; q ) n • q -binomial: = [ k ] q ! [ n − k ] q ! = k ( q ; q ) k ( q ; q ) n − k q Negative thinking and polynomial analogs Armin Straub 2 / 40

Basic q -analogs q -binomial coefficients A q -analog reduces to the classical object in the limit q → 1 . IDEA [ n ] q = q n − 1 DEF q − 1 = 1 + q + . . . q n − 1 • q -number: ( q ; q ) n • q -factorial: [ n ] q ! = [ n ] q [ n − 1] q · · · [1] q = (1 − q ) n For q -series fans: � n � [ n ] q ! D1 ( q ; q ) n • q -binomial: = [ k ] q ! [ n − k ] q ! = k ( q ; q ) k ( q ; q ) n − k q � 6 � = 6 · 5 EG = 3 · 5 2 2 = (1 + q + q 2 + q 3 + q 4 + q 5 )(1 + q + q 2 + q 3 + q 4 ) � 6 � 2 1 + q q (1 + q + q 2 + q 3 + q 4 ) = (1 − q + q 2 ) (1 + q + q 2 ) � �� � � �� � =[3] q =[5] q Negative thinking and polynomial analogs Armin Straub 2 / 40

Basic q -analogs q -binomial coefficients A q -analog reduces to the classical object in the limit q → 1 . IDEA [ n ] q = q n − 1 DEF q − 1 = 1 + q + . . . q n − 1 • q -number: ( q ; q ) n • q -factorial: [ n ] q ! = [ n ] q [ n − 1] q · · · [1] q = (1 − q ) n For q -series fans: � n � [ n ] q ! D1 ( q ; q ) n • q -binomial: = [ k ] q ! [ n − k ] q ! = k ( q ; q ) k ( q ; q ) n − k q � 6 � = 6 · 5 EG = 3 · 5 2 2 = (1 + q + q 2 + q 3 + q 4 + q 5 )(1 + q + q 2 + q 3 + q 4 ) � 6 � 2 1 + q q (1 + q + q 2 + q 3 + q 4 ) = (1 − q + q 2 ) (1 + q + q 2 ) Φ 6 (1) = 1 � �� � � �� � � �� � becomes invisible =Φ 6 ( q ) =[3] q =[5] q Negative thinking and polynomial analogs Armin Straub 2 / 40

Cyclotomic polynomials q -binomial coefficients The n th cyclotomic polynomial: DEF � ( q − ζ k ) where ζ = e 2 πi/n Φ n ( q ) = 1 � k<n ( k,n )=1 irreducible polynomial (nontrivial; Gauss!) with integer coefficients • [ n ] q = q n − 1 � q − 1 = Φ d ( q ) For primes: [ p ] q = Φ p ( q ) 1 <d � n d | n EG Φ 5 ( q ) = q 4 + q 3 + q 2 + q + 1 Φ 21 ( q ) = q 12 − q 11 + q 9 − q 8 + q 6 − q 4 + q 3 − q + 1 Φ 105 ( q ) = q 48 + q 47 + q 46 − q 43 − q 42 − 2 q 41 − q 40 − q 39 + q 36 + q 35 + q 34 + q 33 + q 32 + q 31 − q 28 − q 26 − q 24 − q 22 − q 20 + q 17 + q 16 + q 15 + q 14 + q 13 + q 12 − q 9 − q 8 − 2 q 7 − q 6 − q 5 + q 2 + q + 1 Negative thinking and polynomial analogs Armin Straub 3 / 40

q -binomials: factored and expanded q -binomial coefficients LEM n � n � [ n ] q ! Φ d ( q ) ⌊ n/d ⌋ − ⌊ k/d ⌋ − ⌊ ( n − k ) /d ⌋ � factored = [ k ] q ![ n − k ] q ! = k ∈ { 0 , 1 } q d =2 proof n n � � � Φ d ( q ) ⌊ n/d ⌋ [ n ] q ! = Φ d ( q ) = m =1 d | m d =2 d> 1 • In particular, the q -binomial is a polynomial. (of degree k ( n − k ) ) Negative thinking and polynomial analogs Armin Straub 4 / 40

q -binomials: factored and expanded q -binomial coefficients LEM n � n � [ n ] q ! Φ d ( q ) ⌊ n/d ⌋ − ⌊ k/d ⌋ − ⌊ ( n − k ) /d ⌋ � factored = [ k ] q ![ n − k ] q ! = k ∈ { 0 , 1 } q d =2 proof n n � � � Φ d ( q ) ⌊ n/d ⌋ [ n ] q ! = Φ d ( q ) = m =1 d | m d =2 d> 1 • In particular, the q -binomial is a polynomial. (of degree k ( n − k ) ) � 6 � EG = q 8 + q 7 + 2 q 6 + 2 q 5 + 3 q 4 + 2 q 3 + 2 q 2 + q + 1 expanded 2 q � 9 � = q 18 + q 17 + 2 q 16 + 3 q 15 + 4 q 14 + 5 q 13 + 7 q 12 3 q + 7 q 11 + 8 q 10 + 8 q 9 + 8 q 8 + 7 q 7 + 7 q 6 + 5 q 5 + 4 q 4 + 3 q 3 + 2 q 2 + q + 1 • The coefficients are positive and unimodal . Sylvester, 1878 Negative thinking and polynomial analogs Armin Straub 4 / 40

q -binomials: combinatorial q -binomial coefficients � n � “normalized sum of Y ” THM � � q w ( Y ) = where w ( Y ) = y j − j k D2 q j Y The sum is over all k -element subsets Y of { 1 , 2 , . . . , n } . EG { 1 , 2 } , { 1 , 3 } , { 1 , 4 } , { 2 , 3 } , { 2 , 4 } , { 3 , 4 } → 0 → 1 → 2 → 2 → 3 → 4 � 4 � = 1 + q + 2 q 2 + q 3 + q 4 2 q Negative thinking and polynomial analogs Armin Straub 5 / 40

q -binomials: combinatorial q -binomial coefficients � n � “normalized sum of Y ” THM � � q w ( Y ) = where w ( Y ) = y j − j k D2 q j Y The sum is over all k -element subsets Y of { 1 , 2 , . . . , n } . EG { 1 , 2 } , { 1 , 3 } , { 1 , 4 } , { 2 , 3 } , { 2 , 4 } , { 3 , 4 } → 0 → 1 → 2 → 2 → 3 → 4 � 4 � = 1 + q + 2 q 2 + q 3 + q 4 2 q � n � The coefficient of q m in q counts the number of k • k -element subsets of n whose normalized sum is m , Negative thinking and polynomial analogs Armin Straub 5 / 40

q -binomials: combinatorial q -binomial coefficients � n � “normalized sum of Y ” THM � � q w ( Y ) = where w ( Y ) = y j − j k D2 q j Y The sum is over all k -element subsets Y of { 1 , 2 , . . . , n } . EG { 1 , 2 } , { 1 , 3 } , { 1 , 4 } , { 2 , 3 } , { 2 , 4 } , { 3 , 4 } → 0 → 1 → 2 → 2 → 3 → 4 � 4 � = 1 + q + 2 q 2 + q 3 + q 4 2 q � n � The coefficient of q m in q counts the number of k • k -element subsets of n whose normalized sum is m , • partitions λ of m whose Ferrer’s diagram fits in a k × ( n − k ) box. Negative thinking and polynomial analogs Armin Straub 5 / 40

q -binomials: three characterizations q -binomial coefficients The q -binomial satisfies the q -Pascal rule : THM � n � � n − 1 � � n − 1 � D3 + q k = k k − 1 k q q q Negative thinking and polynomial analogs Armin Straub 6 / 40

q -binomials: three characterizations q -binomial coefficients The q -binomial satisfies the q -Pascal rule : THM � n � � n − 1 � � n − 1 � D3 + q k = k k − 1 k q q q � n � THM = number of k -dim. subspaces of F n D4 q k q Negative thinking and polynomial analogs Armin Straub 6 / 40

q -binomials: three characterizations q -binomial coefficients The q -binomial satisfies the q -Pascal rule : THM � n � � n − 1 � � n − 1 � D3 + q k = k k − 1 k q q q � n � THM = number of k -dim. subspaces of F n D4 q k q THM Suppose yx = qxy (and that q commutes with x, y ). Then: n � n � � ( x + y ) n = x k y n − k D5 k q k =0 Negative thinking and polynomial analogs Armin Straub 6 / 40

q -calculus q -binomial coefficients The q -derivative : DEF D q f ( x ) = f ( qx ) − f ( x ) qx − x D q x n = ( qx ) n − x n = q n − 1 EG q − 1 x n − 1 = [ n ] q x n − 1 qx − x Negative thinking and polynomial analogs Armin Straub 7 / 40

q -calculus q -binomial coefficients • D q e x q = e x The q -derivative : DEF q • e x q · e y q = e x + y D q f ( x ) = f ( qx ) − f ( x ) q provided that yx = qxy qx − x q · e − x • e x 1 /q = 1 D q x n = ( qx ) n − x n = q n − 1 EG q − 1 x n − 1 = [ n ] q x n − 1 qx − x ∞ x n ∞ ( x (1 − q )) n 1 � � • The q -exponential : e x q = [ n ] q ! = = ( q ; q ) n ( x (1 − q ); q ) ∞ n =0 n =0 Negative thinking and polynomial analogs Armin Straub 7 / 40

q -calculus q -binomial coefficients • D q e x q = e x The q -derivative : DEF q • e x q · e y q = e x + y D q f ( x ) = f ( qx ) − f ( x ) q provided that yx = qxy qx − x q · e − x • e x 1 /q = 1 D q x n = ( qx ) n − x n = q n − 1 EG q − 1 x n − 1 = [ n ] q x n − 1 qx − x ∞ x n ∞ ( x (1 − q )) n 1 � � • The q -exponential : e x q = [ n ] q ! = = ( q ; q ) n ( x (1 − q ); q ) ∞ n =0 n =0 • The q -integral : from formally inverting D q � x ∞ � q n xf ( q n x ) f ( x ) d q x := (1 − q ) 0 n =0 Negative thinking and polynomial analogs Armin Straub 7 / 40

q -calculus q -binomial coefficients • D q e x q = e x The q -derivative : DEF q • e x q · e y q = e x + y D q f ( x ) = f ( qx ) − f ( x ) q provided that yx = qxy qx − x q · e − x • e x 1 /q = 1 D q x n = ( qx ) n − x n = q n − 1 EG q − 1 x n − 1 = [ n ] q x n − 1 qx − x ∞ x n ∞ ( x (1 − q )) n 1 � � • The q -exponential : e x q = [ n ] q ! = = ( q ; q ) n ( x (1 − q ); q ) ∞ n =0 n =0 • The q -integral : from formally inverting D q � x ∞ � q n xf ( q n x ) f ( x ) d q x := (1 − q ) 0 n =0 • The q -gamma function : D6 � ∞ x s − 1 e − qx • Γ q ( s + 1) = [ s ] q Γ q ( s ) Γ q ( s ) = 1 /q d q x • Γ q ( n + 1) = [ n ] q ! 0 Can similarly define q -beta via a q -Euler integral. Negative thinking and polynomial analogs Armin Straub 7 / 40