# Van Vleck theorem

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##### 2: 30.10 Series and Integrals
For an addition theorem, see Meixner and Schäfke (1954, p. 300) and King and Van Buren (1973). …
##### 3: 19.35 Other Applications
###### §19.35(i) Mathematical
Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute $\pi$ to high precision (Borwein and Borwein (1987, p. 26)). …
##### 4: 31.15 Stieltjes Polynomials
The $V(z)$ are called Van Vleck polynomials and the corresponding $S(z)$ Stieltjes polynomials. …
31.15.2 $\sum_{j=1}^{N}\frac{\gamma_{j}/2}{z_{k}-a_{j}}+\sum_{\begin{subarray}{c}j=1\\ j\neq k\end{subarray}}^{n}\frac{1}{z_{k}-z_{j}}=0,$ $k=1,2,\dots,n$.
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. …
##### 5: 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.
• T. M. Apostol (1952) Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.
• 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.
• ##### 6: 28.27 Addition Theorems
Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …
##### 7: 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.
• ##### 8: 25.6 Integer Arguments
25.6.4 $\zeta\left(-2n\right)=0,$ $n=1,2,3,\dots$.
25.6.8 $\zeta\left(2\right)=3\sum_{k=1}^{\infty}\frac{1}{k^{2}\genfrac{(}{)}{0.0pt}{}{% 2k}{k}}.$
25.6.9 $\zeta\left(3\right)=\frac{5}{2}\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k^{3}% \genfrac{(}{)}{0.0pt}{}{2k}{k}}.$
25.6.10 $\zeta\left(4\right)=\frac{36}{17}\sum_{k=1}^{\infty}\frac{1}{k^{4}\genfrac{(}{% )}{0.0pt}{}{2k}{k}}.$
##### 9: 6.17 Physical Applications
• N. G. de Bruijn (1937) Integralen voor de $\zeta$-functie van Riemann. Mathematica (Zutphen) B5, pp. 170–180 (Dutch).