Liouville%E2%80%93Green%20approximation%20theorem

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21: 30.2 Differential Equations

… ►The Liouville normal form of equation (30.2.1) is …

22: 18.36 Miscellaneous Polynomials

… ►They are related to Hermite–Padé approximation and can be used for proofs of irrationality or transcendence of interesting numbers. … ►Orthogonality of the the classical OP’s with respect to a positive weight function, as in Table 18.3.1 requires, via Favard’s theorem, $A_n{A}_{n-1}{C}_n>0$ for $n\ge 1$ as per (18.2.9_5). … ►These results are proven in Everitt et al. (2004), via construction of a self-adjoint Sturm–Liouville operator which generates the $L_n^{(-k)}(x)$ polynomials, self-adjointness implying both orthogonality and completeness. … ►The $y(x)={\widehat{L}}_n^{(k)}\left(x\right)$ satisfy a second order Sturm–Liouville eigenvalue problem of the type illustrated in Table 18.8.1, as satisfied by classical OP’s, but now with rational, rather than polynomial coefficients: … ►In §18.39(i) it is seen that the functions, $\sqrt{w(x)}{\widehat{H}}_{n+3}\left(x\right)$, are solutions of a Schrödinger equation with a rational potential energy; and, in spite of first appearances, the Sturm oscillation theorem, Simon (2005c, Theorem 3.3, p. 35), is satisfied. …

23: 2.8 Differential Equations with a Parameter

… ►In Case III $f(z)$ has a simple pole at $z_0$ and ${(z-z_0)}^{2}g(z)$ is analytic at $z_0$. … ►First we apply the Liouville transformation (§1.13(iv)) to (2.8.1). … ►In Case III the approximating equation is … ►For connection formulas for Liouville–Green approximations across these transition points see Olver (1977b, a, 1978). … ►For examples of uniform asymptotic approximations in terms of Whittaker functions with fixed second parameter see §18.15(i) and §28.8(iv). …

24: 15.16 Products

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25: 3.7 Ordinary Differential Equations

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§3.7(iv) Sturm–Liouville Eigenvalue Problems

… ►The Sturm–Liouville eigenvalue problem is the construction of a nontrivial solution of the system …

26: 1.9 Calculus of a Complex Variable

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DeMoivre’s Theorem

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Jordan Curve Theorem

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Cauchy’s Theorem

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Liouville’s Theorem

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Dominated Convergence Theorem

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27: 18.38 Mathematical Applications

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§18.38(i) Classical OP’s: Numerical Analysis

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Approximation Theory

… ►For these results and applications in approximation theory see §3.11(ii) and Mason and Handscomb (2003, Chapter 3), Cheney (1982, p. 108), and Rivlin (1969, p. 31). … ►The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. … ►

18.38.3 $$\sum _{m=0}^n{P}_m^{(\alpha ,0)}\left(x\right)=\frac{{\left(\alpha +2\right)}_n}{n!}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-n,n+\alpha +2,\frac{1}{2}(\alpha +1)}{\alpha +1,\frac{1}{2}(\alpha +3)};\frac{1}{2}(1-x))\ge 0,$$ $x\ge -1$, $\alpha \ge -2$, $n=0,1,\mathrm{\dots }$,

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Askey and Gasper (1976, Theorem 3, p.717)(proved)

Referenced by:

§18.14(iv), §18.18(viii), Erratum (V1.2.0) for Equation (18.38.3)

Permalink:

http://dlmf.nist.gov/18.38.E3

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

This equation was updated to include the value of the sum in terms of the ${}_3{}^{}F_2^{}$ function. Also the constraint was previously $\alpha >-1$.

See also:

Annotations for §18.38(ii), §18.38(ii), §18.38 and Ch.18

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28: 1.15 Summability Methods

… ►For $\mathrm{\Re }\alpha >0$ and $x\ge 0$, the Riemann-Liouville fractional integral of order $\alpha$ is defined by … ►

§1.15(viii) Tauberian Theorems

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29: 2.6 Distributional Methods

§2.6 Distributional Methods

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§2.6(ii) Stieltjes Transform

… ►Corresponding results for the generalized Stieltjes transform … ►The Riemann–Liouville fractional integral of order $\mu$ is defined by … ►For rigorous derivations of these results and also order estimates for ${\delta }_n(x)$, see Wong (1979) and Wong (1989, Chapter 6).

30: 27.3 Multiplicative Properties

… ►Examples are $\lfloor 1/n\rfloor$ and $\lambda \left(n\right)$, and the Dirichlet characters, defined in §27.8. …