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1: 26.17 The Twelvefold Way
Table 26.17.1: The twelvefold way.
elements of N elements of K f unrestricted f one-to-one f onto
labeled labeled k n ( k n + 1 ) n k ! S ( n , k )
unlabeled labeled ( k + n 1 n ) ( k n ) ( n 1 n k )
labeled unlabeled S ( n , 1 ) + S ( n , 2 ) + + S ( n , k ) { 1 n k 0 n > k S ( n , k )
unlabeled unlabeled p k ( n ) { 1 n k 0 n > k p k ( n ) p k 1 ( n )
2: 31.14 General Fuchsian Equation
The general second-order Fuchsian equation with N + 1 regular singularities at z = a j , j = 1 , 2 , , N , and at , is given by
31.14.1 d 2 w d z 2 + ( j = 1 N γ j z a j ) d w d z + ( j = 1 N q j z a j ) w = 0 , j = 1 N q j = 0 .
The exponents at the finite singularities a j are { 0 , 1 γ j } and those at are { α , β } , where …With a 1 = 0 and a 2 = 1 the total number of free parameters is 3 N 3 . …
31.14.3 w ( z ) = ( j = 1 N ( z a j ) γ j / 2 ) W ( z ) ,
3: Preface
Lozier directed the NIST research, technical, and support staff associated with the project, administered grants and contracts, together with Boisvert compiled the Software sections for the Web version of the chapters, conducted editorial and staff meetings, represented the project within NIST and at professional meetings in the United States and abroad, and together with Olver carried out the day-to-day development of the project. … Among the research, technical, and support staff at NIST these are B. …Zelen. …
4: 26.5 Lattice Paths: Catalan Numbers
26.5.1 C ( n ) = 1 n + 1 ( 2 n n ) = 1 2 n + 1 ( 2 n + 1 n ) = ( 2 n n ) ( 2 n n 1 ) = ( 2 n 1 n ) ( 2 n 1 n + 1 ) .
26.5.3 C ( n + 1 ) = k = 0 n C ( k ) C ( n k ) ,
26.5.4 C ( n + 1 ) = 2 ( 2 n + 1 ) n + 2 C ( n ) ,
26.5.5 C ( n + 1 ) = k = 0 n / 2 ( n 2 k ) 2 n 2 k C ( k ) .
26.5.7 lim n C ( n + 1 ) C ( n ) = 4 .
5: 31.15 Stieltjes Polynomials
If t k is a zero of the Van Vleck polynomial V ( z ) , corresponding to an n th degree Stieltjes polynomial S ( z ) , and z 1 , z 2 , , z n 1 are the zeros of S ( z ) (the derivative of S ( z ) ), then t k is either a zero of S ( z ) or a solution of the equation … The zeros z k , k = 1 , 2 , , n , of the Stieltjes polynomial S ( z ) are the critical points of the function G , that is, points at which G / ζ k = 0 , k = 1 , 2 , , n , where … then there are exactly ( n + N 2 N 2 ) polynomials S ( z ) , each of which corresponds to each of the ( n + N 2 N 2 ) ways of distributing its n zeros among N 1 intervals ( a j , a j + 1 ) , j = 1 , 2 , , N 1 . … If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index 𝐦 = ( m 1 , m 2 , , m N 1 ) , where each m j is a nonnegative integer, there is a unique Stieltjes polynomial with m j zeros in the open interval ( a j , a j + 1 ) for each j = 1 , 2 , , N 1 . … Let S 𝐦 ( z ) and S 𝐥 ( z ) be Stieltjes polynomials corresponding to two distinct multi-indices 𝐦 = ( m 1 , m 2 , , m N 1 ) and 𝐥 = ( 1 , 2 , , N 1 ) . …
6: 26.8 Set Partitions: Stirling Numbers
s ( n , k ) denotes the Stirling number of the first kind: ( 1 ) n k times the number of permutations of { 1 , 2 , , n } with exactly k cycles. … where ( x ) n is the Pochhammer symbol: x ( x + 1 ) ( x + n 1 ) . … For n 1 , … uniformly for n = o ( k 1 / 2 ) . For asymptotic approximations for s ( n + 1 , k + 1 ) and S ( n , k ) that apply uniformly for 1 k n as n see Temme (1993) and Temme (2015, Chapter 34). …
7: 24.5 Recurrence Relations
24.5.1 k = 0 n 1 ( n k ) B k ( x ) = n x n 1 , n = 2 , 3 , ,
24.5.5 k = 0 n ( n k ) 2 k E n k + E n = 2 .
24.5.6 k = 2 n ( n k 2 ) B k k = 1 ( n + 1 ) ( n + 2 ) B n + 1 , n = 2 , 3 , ,
24.5.7 k = 0 n ( n k ) B k n + 2 k = B n + 1 n + 1 , n = 1 , 2 , ,
24.5.8 k = 0 n 2 2 k B 2 k ( 2 k ) ! ( 2 n + 1 2 k ) ! = 1 ( 2 n ) ! , n = 1 , 2 , .
8: Customize DLMF
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9: 26.6 Other Lattice Path Numbers
D ( m , n ) is the number of paths from ( 0 , 0 ) to ( m , n ) that are composed of directed line segments of the form ( 1 , 0 ) , ( 0 , 1 ) , or ( 1 , 1 ) . … M ( n ) is the number of lattice paths from ( 0 , 0 ) to ( n , n ) that stay on or above the line y = x and are composed of directed line segments of the form ( 2 , 0 ) , ( 0 , 2 ) , or ( 1 , 1 ) . … N ( n , k ) is the number of lattice paths from ( 0 , 0 ) to ( n , n ) that stay on or above the line y = x , are composed of directed line segments of the form ( 1 , 0 ) or ( 0 , 1 ) , and for which there are exactly k occurrences at which a segment of the form ( 0 , 1 ) is followed by a segment of the form ( 1 , 0 ) . … r ( n ) is the number of paths from ( 0 , 0 ) to ( n , n ) that stay on or above the diagonal y = x and are composed of directed line segments of the form ( 1 , 0 ) , ( 0 , 1 ) , or ( 1 , 1 ) . …
26.6.10 D ( m , n ) = D ( m , n 1 ) + D ( m 1 , n ) + D ( m 1 , n 1 ) , m , n 1 ,
10: 24.15 Related Sequences of Numbers
24.15.7 B n = k = 0 n ( 1 ) k ( n + 1 k + 1 ) S ( n + k , k ) / ( n + k k ) ,
24.15.8 k = 0 n ( 1 ) n + k s ( n + 1 , k + 1 ) B k = n ! n + 1 .
24.15.10 2 n 1 4 n p 2 B 2 n S ( p + 2 n , p 1 ) ( mod p 3 ) , 2 2 n p 3 .
The Fibonacci numbers are defined by u 0 = 0 , u 1 = 1 , and u n + 1 = u n + u n 1 , n 1 . The Lucas numbers are defined by v 0 = 2 , v 1 = 1 , and v n + 1 = v n + v n 1 , n 1 . …