# rook polynomial

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##### 1: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
###### §31.5 Solutions Analytic at Three Singularities: Heun Polynomials
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. …
##### 3: 24.1 Special Notation
###### Bernoulli Numbers and Polynomials
The origin of the notation $B_{n}$, $B_{n}\left(x\right)$, is not clear. …
###### Euler Numbers and Polynomials
The notations $E_{n}$, $E_{n}\left(x\right)$, 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 $n\times n$ 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 $R(x,B)=\sum_{j=0}^{n}r_{j}(B)\,x^{j}.$
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 $n\times n$ chessboard that avoid the squares in a specified subset $B$ is