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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: 18.27 q -Hahn Class
18.27.4 y = 0 N Q n ( q y ) Q m ( q y ) [ N y ] q ( α q ; q ) y ( β q ; q ) N y ( α q ) y = h n δ n , m , n , m = 0 , 1 , , N ,
18.27.4_1 h n = ( α q ) n N 1 α β q 2 n + 1 ( α β q n + 1 ; q ) N + 1 ( β q ; q ) n [ N n ] q ( α q ; q ) n .
7: 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