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1: 10.56 Generating Functions
§10.56 Generating Functions
2: 10.12 Generating Function and Associated Series
§10.12 Generating Function and Associated Series
3: 10.35 Generating Function and Associated Series
§10.35 Generating Function and Associated Series
4: 18.12 Generating Functions
§18.12 Generating Functions
Jacobi
Ultraspherical
Legendre
Laguerre
5: 27.4 Euler Products and Dirichlet Series
The Riemann zeta function is the prototype of series of the form
27.4.4 F ( s ) = n = 1 f ( n ) n - s ,
The function F ( s ) is a generating function, or more precisely, a Dirichlet generating function, for the coefficients. The following examples have generating functions related to the zeta function: …
6: Howard S. Cohl
Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and q -series. …
7: 26.5 Lattice Paths: Catalan Numbers
§26.5(ii) Generating Function
8: 14.21 Definitions and Basic Properties
§14.21(iii) Properties
The generating function expansions (14.7.19) (with P replaced by P ) and (14.7.22) apply when | h | < min | z ± ( z 2 - 1 ) 1 / 2 | ; (14.7.21) (with P replaced by P ) applies when | h | > max | z ± ( z 2 - 1 ) 1 / 2 | .
9: 26.15 Permutations: Matrix Notation
The rook polynomial is the generating function for r j ( B ) : … N ( x , B ) is the generating function:
26.15.6 N ( x , B ) = k = 0 n N k ( B ) x k ,
26.15.7 N ( x , B ) = k = 0 n r k ( B ) ( n - k ) ! ( x - 1 ) k .
26.15.8 N 0 ( B ) N ( 0 , B ) = k = 0 n ( - 1 ) k r k ( B ) ( n - k ) ! .
10: 26.6 Other Lattice Path Numbers
§26.6(ii) Generating Functions
For sufficiently small | x | and | y | , …
26.6.6 n = 0 D ( n , n ) x n = 1 1 - 6 x + x 2 ,
26.6.7 n = 0 M ( n ) x n = 1 - x - 1 - 2 x - 3 x 2 2 x 2 ,
26.6.8 n , k = 1 N ( n , k ) x n y k = 1 - x - x y - ( 1 - x - x y ) 2 - 4 x 2 y 2 x ,