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11: 26.3 Lattice Paths: Binomial Coefficients
26.3.2 ( m n ) = 0 , n > m .
26.3.3 n = 0 m ( m n ) x n = ( 1 + x ) m , m = 0 , 1 , ,
26.3.7 ( m + 1 n + 1 ) = k = n m ( k n ) , m n 0 ,
26.3.8 ( m n ) = k = 0 n ( m n 1 + k k ) , m n 0 .
26.3.10 ( m n ) = k = 0 n ( 1 ) n k ( m + 1 k ) , m n 0 ,
12: 21.6 Products
21.6.1 𝒦 = g × h 𝐓 / ( g × h 𝐓 g × h ) ,
that is, 𝒦 is the set of all g × h matrices that are obtained by premultiplying 𝐓 by any g × h matrix with integer elements; two such matrices in 𝒦 are considered equivalent if their difference is a matrix with integer elements. …
21.6.2 𝒟 = | 𝐓 T h / ( 𝐓 T h h ) | ,
that is, 𝒟 is the number of elements in the set containing all h -dimensional vectors obtained by multiplying 𝐓 T on the right by a vector with integer elements. Two such vectors are considered equivalent if their difference is a vector with integer elements. …
13: 34.5 Basic Properties: 6 j Symbol
34.5.8 { j 1 j 2 j 3 l 1 l 2 l 3 } = { j 2 j 1 j 3 l 2 l 1 l 3 } = { j 1 l 2 l 3 l 1 j 2 j 3 } .
34.5.9 { j 1 j 2 j 3 l 1 l 2 l 3 } = { j 1 1 2 ( j 2 + l 2 + j 3 l 3 ) 1 2 ( j 2 l 2 + j 3 + l 3 ) l 1 1 2 ( j 2 + l 2 j 3 + l 3 ) 1 2 ( j 2 + l 2 + j 3 + l 3 ) } ,
34.5.13 E ( j ) = ( ( j 2 ( j 2 j 3 ) 2 ) ( ( j 2 + j 3 + 1 ) 2 j 2 ) ( j 2 ( l 2 l 3 ) 2 ) ( ( l 2 + l 3 + 1 ) 2 j 2 ) ) 1 2 .
34.5.19 l { j 1 j 2 l j 2 j 1 j } = 0 , 2 μ j odd, μ = min ( j 1 , j 2 ) ,
34.5.20 l ( 1 ) l + j { j 1 j 2 l j 1 j 2 j } = ( 1 ) 2 μ 2 j + 1 , μ = min ( j 1 , j 2 ) ,
14: 29.15 Fourier Series and Chebyshev Series
29.15.10 p = 0 n A 2 p + 1 2 = 1 ,
29.15.11 p = 0 n A 2 p + 1 > 0 .
29.15.15 p = 0 n B 2 p + 1 2 = 1 ,
29.15.25 p = 0 n B 2 p + 2 2 = 1 ,
29.15.31 p = 0 n C 2 p + 1 > 0 .
15: 24.14 Sums
In the following two identities, valid for n 2 , the sums are taken over all nonnegative integers j , k , with j + k + = n .
24.14.8 ( 2 n ) ! ( 2 j ) ! ( 2 k ) ! ( 2 ) ! B 2 j B 2 k B 2 = ( n 1 ) ( 2 n 1 ) B 2 n + n ( n 1 2 ) B 2 n 2 ,
24.14.9 ( 2 n ) ! ( 2 j ) ! ( 2 k ) ! ( 2 ) ! E 2 j E 2 k E 2 = 1 2 ( E 2 n E 2 n + 2 ) .
In the next identity, valid for n 4 , the sum is taken over all positive integers j , k , , m with j + k + + m = n .
24.14.10 ( 2 n ) ! ( 2 j ) ! ( 2 k ) ! ( 2 ) ! ( 2 m ) ! B 2 j B 2 k B 2 B 2 m = ( 2 n + 3 3 ) B 2 n 4 3 n 2 ( 2 n 1 ) B 2 n 2 .
16: 26.8 Set Partitions: Stirling Numbers
where the summation is over all nonnegative integers c 1 , c 2 , , c k such that c 1 + c 2 + + c k = n k .
26.8.20 s ( n + 1 , k + 1 ) = n ! j = k n ( 1 ) n j j ! s ( j , k ) ,
26.8.21 s ( n + k + 1 , k ) = j = 0 k ( n + j ) s ( n + j , j ) .
26.8.24 S ( n , k ) = j = k n S ( j 1 , k 1 ) k n j ,
26.8.25 S ( n + 1 , k + 1 ) = j = k n ( n j ) S ( j , k ) ,
17: 27.10 Periodic Number-Theoretic Functions
If k is a fixed positive integer, then a number-theoretic function f is periodic (mod k ) if
27.10.1 f ( n + k ) = f ( n ) , n = 1 , 2 , .
27.10.6 s k ( n ) = d | ( n , k ) f ( d ) g ( k d )
18: 23.18 Modular Transformations
Here e and o are generic symbols for even and odd integers, respectively. In particular, if a 1 , b , c , and d 1 are all even, then …and λ ( τ ) is a cusp form of level zero for the corresponding subgroup of SL ( 2 , ) . … J ( τ ) is a modular form of level zero for SL ( 2 , ) . …
23.18.6 ε ( 𝒜 ) = exp ( π i ( a + d 12 c + s ( d , c ) ) ) ,
19: 27.3 Multiplicative Properties
27.3.1 f ( m n ) = f ( m ) f ( n ) , ( m , n ) = 1 .
27.3.4 J k ( n ) = n k p | n ( 1 p k ) ,
27.3.7 σ α ( m ) σ α ( n ) = d | ( m , n ) d α σ α ( m n d 2 ) ,
27.3.8 ϕ ( m ) ϕ ( n ) = ϕ ( m n ) ϕ ( ( m , n ) ) / ( m , n ) .
27.3.9 f ( m n ) = f ( m ) f ( n ) , m , n = 1 , 2 , .
20: 23.15 Definitions
in which a , b , c , d are integers, with …The set of all bilinear transformations of this form is denoted by SL ( 2 , ) (Serre (1973, p. 77)). A modular function f ( τ ) is a function of τ that is meromorphic in the half-plane τ > 0 , and has the property that for all 𝒜 SL ( 2 , ) , or for all 𝒜 belonging to a subgroup of SL ( 2 , ) , …where c 𝒜 is a constant depending only on 𝒜 , and (the level) is an integer or half an odd integer. …