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1: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
§31.5 Solutions Analytic at Three Singularities: Heun Polynomials
31.5.2 𝐻𝑝 n , m ( a , q n , m ; n , β , γ , δ ; z ) = H ( a , q n , m ; n , β , γ , δ ; z )
is a polynomial of degree n , and hence a solution of (31.2.1) that is analytic at all three finite singularities 0 , 1 , a . These solutions are the Heun polynomials. …
2: 35.4 Partitions and Zonal Polynomials
§35.4 Partitions and Zonal Polynomials
Normalization
Orthogonal Invariance
Summation
Mean-Value
3: 24.1 Special Notation
Bernoulli Numbers and Polynomials
The origin of the notation B n , B n ( x ) , is not clear. …
Euler Numbers and Polynomials
The notations E n , E n ( x ) , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
4: 18.3 Definitions
§18.3 Definitions
For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). …For explicit power series coefficients up to n = 12 for these polynomials and for coefficients up to n = 6 for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). …
Bessel polynomials
Bessel polynomials are often included among the classical OP’s. …
5: 26.15 Permutations: Matrix Notation
The matrix represents the placement of n nonattacking rooks on an n × n chessboard, that is, rooks that share neither a row nor a column with any other rook. … Let r j ( B ) be the number of ways of placing j nonattacking rooks on the squares of B . …The rook polynomial is the generating function for r j ( B ) :
26.15.3 R ( x , B ) = j = 0 n r j ( B ) x j .
26.15.4 R ( x , B ) = R ( x , B 1 ) R ( x , B 2 ) .
6: 24.18 Physical Applications
§24.18 Physical Applications
Bernoulli polynomials appear in statistical physics (Ordóñez and Driebe (1996)), in discussions of Casimir forces (Li et al. (1991)), and in a study of quark-gluon plasma (Meisinger et al. (2002)). Euler polynomials also appear in statistical physics as well as in semi-classical approximations to quantum probability distributions (Ballentine and McRae (1998)).
7: 26.18 Counting Techniques
With the notation of §26.15, the number of placements of n nonattacking rooks on an n × n chessboard that avoid the squares in a specified subset B is
26.18.3 n ! + t = 1 n ( 1 ) t r t ( B ) ( n t ) ! .
8: 24.3 Graphs
See accompanying text
Figure 24.3.1: Bernoulli polynomials B n ( x ) , n = 2 , 3 , , 6 . Magnify
See accompanying text
Figure 24.3.2: Euler polynomials E n ( x ) , n = 2 , 3 , , 6 . Magnify
9: 18.4 Graphics
See accompanying text
Figure 18.4.1: Jacobi polynomials P n ( 1.5 , 0.5 ) ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify
See accompanying text
Figure 18.4.2: Jacobi polynomials P n ( 1.25 , 0.75 ) ( x ) , n = 7 , 8 . … Magnify
See accompanying text
Figure 18.4.4: Legendre polynomials P n ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify
See accompanying text
Figure 18.4.5: Laguerre polynomials L n ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify
See accompanying text
Figure 18.4.7: Monic Hermite polynomials h n ( x ) = 2 n H n ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify
10: 18.7 Interrelations and Limit Relations
§18.7 Interrelations and Limit Relations
Chebyshev, Ultraspherical, and Jacobi
Legendre, Ultraspherical, and Jacobi
§18.7(ii) Quadratic Transformations
§18.7(iii) Limit Relations