On the q -binomial coefficients and binomial congruences q -series - PowerPoint PPT Presentation

On the q -binomial coefficients and binomial congruences q -series seminar University of Illinois at Urbana–Champaign Armin Straub November 15, 2012 University of Illinois at Urbana–Champaign On the q -binomial coefficients and binomial congruences Armin Straub 1 / 35

Our goal today • Following a question of Andrews we seek a q -analog of: � ap � � a � THM mod p 3 For primes p � 5 : ≡ Ljunggren bp b 1952 George Andrews q -analogs of the binomial coefficient congruences of Babbage, Wolstenholme and Glaisher Discrete Mathematics 204, 1999 On the q -binomial coefficients and binomial congruences Armin Straub 2 / 35

Basic q -analogs • The natural number n has the q -analog: [ n ] q = q n − 1 q − 1 = 1 + q + . . . q n − 1 In the limit q → 1 a q -analog reduces to the classical object. On the q -binomial coefficients and binomial congruences Armin Straub 3 / 35

Basic q -analogs • The natural number n has the q -analog: [ n ] q = q n − 1 q − 1 = 1 + q + . . . q n − 1 In the limit q → 1 a q -analog reduces to the classical object. • The q -factorial: [ n ] q ! = [ n ] q [ n − 1] q · · · [1] q • The q -binomial coefficient: � n � [ n ] q ! = D1 k [ k ] q ! [ n − k ] q ! q On the q -binomial coefficients and binomial congruences Armin Straub 3 / 35

Basic q -analogs • The natural number n has the q -analog: [ n ] q = q n − 1 q − 1 = 1 + q + . . . q n − 1 In the limit q → 1 a q -analog reduces to the classical object. • The q -factorial: [ n ] q ! = [ n ] q [ n − 1] q · · · [1] q • The q -binomial coefficient: For q -series fans: � n � [ n ] q ! ( q ; q ) n = [ k ] q ! [ n − k ] q ! = D1 k ( q ; q ) k ( q ; q ) n − k q On the q -binomial coefficients and binomial congruences Armin Straub 3 / 35

A q -binomial coefficient EG � 6 � = 6 · 5 = 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 On the q -binomial coefficients and binomial congruences Armin Straub 4 / 35

A q -binomial coefficient EG � 6 � = 6 · 5 = 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 On the q -binomial coefficients and binomial congruences Armin Straub 4 / 35

A q -binomial coefficient EG � 6 � = 6 · 5 = 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 ( q ) =[3] q =[5] q • The cyclotomic polynomial Φ 6 ( q ) becomes 1 for q = 1 and hence invisible in the classical world On the q -binomial coefficients and binomial congruences Armin Straub 4 / 35

Cyclotomic polynomials The n th cyclotomic polynomial: � ( q − ζ k ) where ζ = e 2 πi/n Φ n ( q ) = 1 � k<n ( k,n )=1 • This is an irreducible polynomial with integer coefficients. irreducibility due to Gauss — nontrivial On the q -binomial coefficients and binomial congruences Armin Straub 5 / 35

Cyclotomic polynomials The n th cyclotomic polynomial: � ( q − ζ k ) where ζ = e 2 πi/n Φ n ( q ) = 1 � k<n ( k,n )=1 • This is an irreducible polynomial with integer coefficients. irreducibility due to Gauss — nontrivial • [ n ] q = q n − 1 � q − 1 = Φ d ( q ) For primes: [ p ] q = Φ p ( q ) 1 <d � n d | n On the q -binomial coefficients and binomial congruences Armin Straub 5 / 35

Some cyclotomic polynomials exhibited EG Φ 2 ( q ) = q + 1 Φ 3 ( q ) = q 2 + q + 1 Φ 6 ( q ) = q 2 − q + 1 Φ 9 ( q ) = q 6 + q 3 + 1 Φ 21 ( q ) = q 12 − q 11 + q 9 − q 8 + q 6 − q 4 + q 3 − q + 1 . . . Φ 102 ( q ) = q 32 + q 31 − q 29 − q 28 + q 26 + q 25 − q 23 − q 22 + q 20 + q 19 − q 17 − q 16 − q 15 + q 13 + q 12 − q 10 − q 9 + q 7 + q 6 − q 4 − q 3 + q + 1 On the q -binomial coefficients and binomial congruences Armin Straub 6 / 35

Some cyclotomic polynomials exhibited EG Φ 2 ( q ) = q + 1 Φ 3 ( q ) = q 2 + q + 1 Φ 6 ( q ) = q 2 − q + 1 Φ 9 ( q ) = q 6 + q 3 + 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 On the q -binomial coefficients and binomial congruences Armin Straub 6 / 35

Back to q -binomials • [ n ] q = q n − 1 � q − 1 = Φ d ( q ) 1 <d � n d | n � n � [ n ] q [ n − 1] q · · · [ n − k + 1] q • = k [ k ] q [ k − 1] q · · · [1] q q • How often does Φ d ( q ) appear in this? � n � � n − k � � k � • It appears − − times d d d On the q -binomial coefficients and binomial congruences Armin Straub 7 / 35

Back to q -binomials • [ n ] q = q n − 1 � q − 1 = Φ d ( q ) 1 <d � n d | n � n � [ n ] q [ n − 1] q · · · [ n − k + 1] q • = k [ k ] q [ k − 1] q · · · [1] q q • How often does Φ d ( q ) appear in this? � n � � n − k � � k � • It appears − − times d d d • Obviously nonnegative: the q -binomials are indeed polynomials • Also at most one: square-free � n � • always contains Φ n ( q ) if 0 < k < n . k q • Good way to compute q -binomials and even get them factorized for free On the q -binomial coefficients and binomial congruences Armin Straub 7 / 35

The coefficients of q -binomial coefficients • Here’s some q -binomials in expanded form: � 6 � EG = q 8 + q 7 + 2 q 6 + 2 q 5 + 3 q 4 + 2 q 3 + 2 q 2 + q + 1 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 degree of the q -binomial is k ( n − k ) . • All coefficients are positive! • In fact, the coefficients are unimodal . Sylvester, 1878 On the q -binomial coefficients and binomial congruences Armin Straub 8 / 35

q -binomials: Pascal’s triangle The q -binomials can be build from the q -Pascal rule: � n � � n − 1 � � n − 1 � D2 + q k = k k − 1 k q q q On the q -binomial coefficients and binomial congruences Armin Straub 9 / 35

q -binomials: Pascal’s triangle The q -binomials can be build from the q -Pascal rule: � n � � n − 1 � � n − 1 � D2 + q k = k k − 1 k q q q 1 1 1 1 1 + q 1 (1 + q ) + q 2 1 1 + q (1 + q ) 1 . . . EG � 4 � = 1 + q + q 2 + q 2 (1 + q + q 2 ) = 1 + q + 2 q 2 + q 3 + q 4 2 q On the q -binomial coefficients and binomial congruences Armin Straub 9 / 35

q -binomials: combinatorial � n � � � q w ( S ) = where w ( S ) = s j − j k D3 q S ∈ ( n k ) j w ( S ) = “normalized sum of S ” 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 On the q -binomial coefficients and binomial congruences Armin Straub 10 / 35

q -binomials: combinatorial � n � � � q w ( S ) = where w ( S ) = s j − j k D3 q S ∈ ( n k ) j w ( S ) = “normalized sum of S ” 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 counts the number of k q • k -element subsets of n whose normalized sum is m On the q -binomial coefficients and binomial congruences Armin Straub 10 / 35

q -binomials: combinatorial � n � � � q w ( S ) = where w ( S ) = s j − j k D3 q S ∈ ( n k ) j w ( S ) = “normalized sum of S ” 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 counts the number of k q • 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 On the q -binomial coefficients and binomial congruences Armin Straub 10 / 35

q -Chu-Vandermonde Different representations make different properties apparent! � m + n � � m �� � n � • Chu-Vandermonde: = k j k − j j On the q -binomial coefficients and binomial congruences Armin Straub 11 / 35

q -Chu-Vandermonde Different representations make different properties apparent! � m + n � � m �� � n � • Chu-Vandermonde: = k j k − j j • Purely from the combinatorial representation: � m + n � � S − k ( k +1) / 2 � = q k q S ∈ ( m + n k ) On the q -binomial coefficients and binomial congruences Armin Straub 11 / 35

q -Chu-Vandermonde Different representations make different properties apparent! � m + n � � m �� � n � • Chu-Vandermonde: = k j k − j j • Purely from the combinatorial representation: � m + n � � S − k ( k +1) / 2 � = q k q S ∈ ( m + n k ) � S 1 + � S 2 +( k − j ) m − k ( k +1) / 2 � � � = q j S 1 ∈ ( m j ) S 2 ∈ ( n k − j ) On the q -binomial coefficients and binomial congruences Armin Straub 11 / 35