About the Project
NIST

q-binomial coefficient

AdvancedHelp

(0.002 seconds)

6 matching pages

1: 26.9 Integer Partitions: Restricted Number and Part Size
26.9.4 [ m n ] q = j = 1 n 1 - q m - n + j 1 - q j , n 0 ,
is the Gaussian polynomial (or q -binomial coefficient); see also §§17.2(i)17.2(ii). …
26.9.5 n = 0 p k ( n ) q n = j = 1 k 1 1 - q j = 1 + m = 1 [ k + m - 1 m ] q q m ,
2: 17.2 Calculus
§17.2(ii) Binomial Coefficients
17.2.27 [ n m ] q = ( q ; q ) n ( q ; q ) m ( q ; q ) n - m = ( q - n ; q ) m ( - 1 ) m q n m - ( m 2 ) ( q ; q ) m ,
17.2.30 [ - n m ] q = [ m + n - 1 m ] q ( - 1 ) m q - m n - ( m 2 ) ,
3: 17.3 q -Elementary and q -Special Functions
17.3.8 A m , s ( q ) = q ( s - m 2 ) + ( s 2 ) j = 0 s ( - 1 ) j q ( j 2 ) [ m + 1 j ] q ( 1 - q s - j ) m ( 1 - q ) m .
17.3.9 a m , s ( q ) = q - ( s 2 ) ( 1 - q ) s ( q ; q ) s j = 0 s ( - 1 ) j q ( j 2 ) [ s j ] q ( 1 - q s - j ) m ( 1 - q ) m .
4: 26.16 Multiset Permutations
26.16.1 [ a 1 + a 2 + + a n a 1 , a 2 , , a n ] q = k = 1 n - 1 [ a k + a k + 1 + + a n a k ] q ,
5: 26.10 Integer Partitions: Other Restrictions
26.10.3 ( 1 - x ) m , n = 0 p m ( k , 𝒟 , n ) x m q n = m = 0 k [ k m ] q q m ( m + 1 ) / 2 x m = j = 1 k ( 1 + x q j ) , | x | < 1 ,
6: 17.5 ϕ 0 0 , ϕ 0 1 , ϕ 1 1 Functions
Euler’s Second Sum
q -Binomial Series
q -Binomial Theorem
Euler’s First Sum
Cauchy’s Sum