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1: 26.14 Permutations: Order Notation
§26.14(ii) Generating Functions
26.14.4 n , k = 0 n k x k t n n ! = 1 x exp ( ( x 1 ) t ) x , | x | < 1 , | t | < 1 .
2: 26.5 Lattice Paths: Catalan Numbers
Table 26.5.1: Catalan numbers.
n C ( n ) n C ( n ) n C ( n )
6 132 13 7 42900 20 65641 20420
§26.5(ii) Generating Function
3: 26.10 Integer Partitions: Other Restrictions
§26.10(ii) Generating Functions
where the last right-hand side is the sum over m 0 of the generating functions for partitions into distinct parts with largest part equal to m . …
§26.10(vi) Bessel-Function Expansion
where I 1 ( x ) is the modified Bessel function10.25(ii)), and …
26.10.19 f ( h , k ) = j = 1 k [ [ 2 j 1 2 k ] ] [ [ h ( 2 j 1 ) k ] ] ,
4: 14.30 Spherical and Spheroidal Harmonics
§14.30 Spherical and Spheroidal Harmonics
Most mathematical properties of Y l , m ( θ , ϕ ) can be derived directly from (14.30.1) and the properties of the Ferrers function of the first kind given earlier in this chapter. …
Herglotz generating function
The following is the Herglotz generating function
5: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
Table 26.4.1: Multinomials and partitions.
n m λ M 1 M 2 M 3
5 2 2 1 , 3 1 10 20 10
5 3 1 2 , 3 1 20 20 10
§26.4(ii) Generating Function
6: 26.3 Lattice Paths: Binomial Coefficients
Table 26.3.1: Binomial coefficients ( m n ) .
m n
6 1 6 15 20 15 6 1
Table 26.3.2: Binomial coefficients ( m + n m ) for lattice paths.
m n
3 1 4 10 20 35 56 84 120 165
§26.3(ii) Generating Functions
7: Bibliography G
  • W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.
  • W. Gautschi (2004) Orthogonal Polynomials: Computation and Approximation. Numerical Mathematics and Scientific Computation, Oxford University Press, New York.
  • A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
  • S. Goldstein (1927) Mathieu functions. Trans. Camb. Philos. Soc. 23, pp. 303–336.
  • Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.
  • 8: Bibliography B
  • W. N. Bailey (1938) The generating function of Jacobi polynomials. J. London Math. Soc. 13, pp. 8–12.
  • A. Bañuelos and R. A. Depine (1980) A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.
  • K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.
  • W. G. Bickley and J. Nayler (1935) A short table of the functions Ki n ( x ) , from n = 1 to n = 16 . Phil. Mag. Series 7 20, pp. 343–347.
  • S. Bochner (1952) Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.
  • 9: Bibliography C
  • B. C. Carlson (1985) The hypergeometric function and the R -function near their branch points. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 63–89.
  • R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.
  • H. S. Cohl (2013b) On a generalization of the generating function for Gegenbauer polynomials. Integral Transforms Spec. Funct. 24 (10), pp. 807–816.
  • M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
  • M. D. Cooper, R. H. Jeppesen, and M. B. Johnson (1979) Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.
  • 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 ,