# Van Vleck polynomials

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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
Table 18.3.1 provides the definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and normalization (§§18.2(i) and 18.2(iii)). … For exact values of the coefficients of the Jacobi polynomials $P^{(\alpha,\beta)}_{n}\left(x\right)$, the ultraspherical polynomials $C^{(\lambda)}_{n}\left(x\right)$, the Chebyshev polynomials $T_{n}\left(x\right)$ and $U_{n}\left(x\right)$, the Legendre polynomials $P_{n}\left(x\right)$, the Laguerre polynomials $L^{(n)}\left(x\right)$, and the Hermite polynomials $H_{n}\left(x\right)$, see Abramowitz and Stegun (1964, pp. 793–801). … For another version of the discrete orthogonality property of the polynomials $T_{n}\left(x\right)$ see (3.11.9). … Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …
##### 5: 31.15 Stieltjes Polynomials
###### §31.15 Stieltjes Polynomials
The $V(z)$ are called Van Vleck polynomials and the corresponding $S(z)$ 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 $z_{1}^{\prime},z_{2}^{\prime},\dots,z_{n-1}^{\prime}$ are the zeros of $S^{\prime}(z)$ (the derivative of $S(z)$), then $t_{k}$ is either a zero of $S^{\prime}(z)$ or a solution of the equation … See Marden (1966), Alam (1979), and Al-Rashed and Zaheer (1985) for further results on the location of the zeros of Stieltjes and Van Vleck polynomials. …
##### 6: Bibliography V
• A. L. Van Buren, R. V. Baier, S. Hanish, and B. J. King (1972) Calculation of spheroidal wave functions. J. Acoust. Soc. Amer. 51, pp. 414–416.
• A. L. Van Buren and J. E. Boisvert (2007) Accurate calculation of the modified Mathieu functions of integer order. Quart. Appl. Math. 65 (1), pp. 1–23.
• Van Buren (website) Mathieu and Spheroidal Wave Functions: Fortran Programs for their Accurate Calculation
• H. C. van de Hulst (1957) Light Scattering by Small Particles. John Wiley and Sons. Inc., New York.
• H. C. van de Hulst (1980) Multiple Light Scattering. Vol. 1, Academic Press, New York.
• ##### 7: Bibliography
• M. Abramowitz (1949) Asymptotic expansions of spheroidal wave functions. J. Math. Phys. Mass. Inst. Tech. 28, pp. 195–199.
• A. M. Al-Rashed and N. Zaheer (1985) Zeros of Stieltjes and Van Vleck polynomials and applications. J. Math. Anal. Appl. 110 (2), pp. 327–339.
• M. Alam (1979) Zeros of Stieltjes and Van Vleck polynomials. Trans. Amer. Math. Soc. 252, pp. 197–204.
• R. Askey and J. Wilson (1985) Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.
• R. Askey (1989) Continuous $q$-Hermite Polynomials when $q>1$ . In $q$-series and Partitions (Minneapolis, MN, 1988), IMA Vol. Math. Appl., Vol. 18, pp. 151–158.
• ##### 8: Tom H. Koornwinder
… … Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC. … Koornwinder has been active as an officer in the SIAM Activity Group on Special Functions and Orthogonal Polynomials. …
##### 10: 27.18 Methods of Computation: Primes
A practical version is described in Bosma and van der Hulst (1990). The AKS (Agrawal–Kayal–Saxena) algorithm is the first deterministic, polynomial-time, primality test. …