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1: 31.14 General Fuchsian Equation

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Normal Form

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2: 30.2 Differential Equations

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

3: 31.2 Differential Equations

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§31.2(ii) Normal Form of Heun’s Equation

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4: 3.11 Approximation Techniques

… ►From the equations $\partial S/\partial a_k=0$, $k=0,1,\mathrm{\dots },n$, we derive the normal equations …

5: 29.20 Methods of Computation

… ►The normalization of Lamé functions given in §29.3(v) can be carried out by quadrature (§3.5). ►A second approach is to solve the continued-fraction equations typified by (29.3.10) by Newton’s rule or other iterative methods; see §3.8. …Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. (Equation (29.6.3) serves as a check.) … ►§29.15(i) includes formulas for normalizing the eigenvectors. …

6: 28.7 Analytic Continuation of Eigenvalues

… ►The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). …

7: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

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28.5.5 $${(C_n(q))}^2{\int }_0^{2\pi }{(f_n(x,q))}^{2}dx={(S_n(q))}^2{\int }_0^{2\pi }{(g_n(x,q))}^{2}dx=\pi .$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $q=h^2$: parameter, $n$: integer, $x$: real variable, $f_n(z,q)$: functions, $g_n(z,q)$: functions, $C_n(q)$: factor and $S_n(q)$: factor

Permalink:

http://dlmf.nist.gov/28.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

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8: 3.6 Linear Difference Equations

… ► … ►Let us assume the normalizing condition is of the form $w_0=\lambda$, where $\lambda$ is a constant, and then solve the following tridiagonal system of algebraic equations for the unknowns $w_1^{(N)},w_2^{(N)},\mathrm{\dots },w_{N-1}^{(N)}$; see §3.2(ii). …

9: 18.2 General Orthogonal Polynomials

… ►It is assumed throughout this chapter that for each polynomial $p_n(x)$ that is orthogonal on an open interval $(a,b)$ the variable $x$ is confined to the closure of $(a,b)$ unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.) … ►

18.2.11_2 $$a_{n-1}{c}_n>0,$$ $n\ge 1$.

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Symbols:

$n$: nonnegative integer, $a_n$: real constant and $c_n$: real constant

Referenced by:

§18.2(viii)

Permalink:

http://dlmf.nist.gov/18.2.E11_2

Encodings:

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Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

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18.2.11_6 $${\beta }_n>0,$$ $n\ge 1$.

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Symbols:

$n$: nonnegative integer

Referenced by:

§18.2(viii)

Permalink:

http://dlmf.nist.gov/18.2.E11_6

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

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18.2.11_7 $$q_n(x)=p_n(x)/\sqrt{h_n},$$ $n\ge 0$.

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Symbols:

$p_n(x)$: polynomial of degree $n$ , $q$: real variable, $n$: nonnegative integer, $x$: real variable and $h_n$

Permalink:

http://dlmf.nist.gov/18.2.E11_7

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

… ►Equations (18.14.3_5) and (18.14.8), both for $\alpha =0$, can be seen as special cases of this result for Jacobi and Laguerre polynomials, respectively.

10: 28.12 Definitions and Basic Properties

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28.12.2 $${\lambda }_{\nu }\left(-q\right)={\lambda }_{\nu }\left(q\right)={\lambda }_{-\nu }\left(q\right).$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.12(i), §28.12 and Ch.28

… ►If $q$ is a normal value of the corresponding equation (28.2.16), then these functions are uniquely determined as analytic functions of $z$ and $q$ by the normalization ►

28.12.5 $${\int }_0^{\pi }{\mathrm{me}}_{\nu }(x,q){\mathrm{me}}_{\nu }(-x,q)dx=\pi .$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $q=h^2$: parameter, $x$: real variable and $\nu$: complex parameter

Referenced by:

§28.14

Permalink:

http://dlmf.nist.gov/28.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.12(ii), §28.12 and Ch.28

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