difference equations on variable

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1: 18.22 Hahn Class: Recurrence Relations and Differences

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§18.22(ii) Difference Equations in $x$

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2: Mathematical Introduction

… ►These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). …

3: 9.13 Generalized Airy Functions

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§9.13(i) Generalizations from the Differential Equation

►Equations of the form … ►In Olver (1977a, 1978) a different normalization is used. … ► … ►and the difference equation …

4: 18.28 Askey–Wilson Class

… ► $y$) such that $P_n(z)=p_n(\frac{1}{2}(z+z^{-1}))$ in the Askey–Wilson case, and $P_n(y)=p_n(q^{-y}+c{q}^{y+1})$ in the $q$-Racah case, and both are eigenfunctions of a second order $q$-difference operator similar to (18.27.1). … ►

$q$-Difference Equation

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18.28.6_1 $$(L{R}_n)(z)=(q^{-n}+abcd{q}^{n-1})R_n(z),$$

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Symbols:

$z$: complex variable, $q$: real variable and $n$: nonnegative integer

Referenced by:

§18.28(ii), §18.28(ii), Erratum (V1.2.0) Section 18.28

Permalink:

http://dlmf.nist.gov/18.28.E6_1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

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18.28.6_3 $$(z+z^{-1})R_n(z)=a_n\left(R_{n+1}(z)-R_n(z)\right)+c_n\left(R_{n-1}(z)-R_n(z)\right)+(a+a^{-1})R_n(z),$$

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Symbols:

$z$: complex variable and $n$: nonnegative integer

Referenced by:

§18.28(ii)

Permalink:

http://dlmf.nist.gov/18.28.E6_3

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

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5: 28.2 Definitions and Basic Properties

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28.2.1 $$w^{\prime \prime }+(a-2q\cos \left(2z\right))w=0.$$

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Symbols:

$\cos z$: cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution

A&S Ref:

20.3.1 20.1.1 (in slightly different form)

Referenced by:

§1.18(vii), Figure 28.17.1, Figure 28.17.1, §28.17, §28.17, §28.2(ii), §28.2(ii), §28.2(iii), §28.2(iv), §28.20(i), §28.29(i), §28.32(i), §28.32(i), 1st item, 2nd item, §28.33(iii), item (a), item (c), item (c), §28.5(i), §28.8(iv), §28.8(iv), §28.8(iv), §30.2(iii)

Permalink:

http://dlmf.nist.gov/28.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(i), §28.2 and Ch.28

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28.2.3 $$(1-{\zeta }^2)w^{\prime \prime }-\zeta w^{\prime }+\left(a+2q-4q{\zeta }^2\right)w=0.$$

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Symbols:

$q=h^2$: parameter, $a$: parameter, $w(z)$: Mathieu’s equation solution and $\zeta$: change of variable

A&S Ref:

20.1.7 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(i), §28.2 and Ch.28

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6: 28.20 Definitions and Basic Properties

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28.20.1 $$w^{\prime \prime }-\left(a-2q\cosh \left(2z\right)\right)w=0,$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution

A&S Ref:

20.1.2 (in slightly different form) 20.8.6

Referenced by:

§28.20(iii), §28.32(i), 1st item, 2nd item, item (b), §28.8(iv), §28.8(iv)

Permalink:

http://dlmf.nist.gov/28.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(i), §28.20 and Ch.28

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28.20.6 $${\mathrm{Fe}}_n(z,q)=\mp \mathrm{i}{\mathrm{fe}}_n(\pm \mathrm{i}z,q),$$ $n=0,1,\mathrm{\dots }$,

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Defines:

${\mathrm{Fe}}_n(z,q)$: modified Mathieu function

Symbols:

${\mathrm{fe}}_n(z,q)$: second solution, Mathieu’s equation, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $n$: integer and $z$: complex variable

A&S Ref:

20.6.1 (in slightly different form) 20.6.2 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.20.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(ii), §28.20 and Ch.28

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28.20.7 $${\mathrm{Ge}}_n(z,q)={\mathrm{ge}}_n(\pm \mathrm{i}z,q),$$ $n=1,2,\mathrm{\dots }$.

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Defines:

${\mathrm{Ge}}_n(z,q)$: modified Mathieu function

Symbols:

${\mathrm{ge}}_n(z,q)$: second solution, Mathieu’s equation, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $n$: integer and $z$: complex variable

A&S Ref:

20.6.1 (in slightly different form) 20.6.2 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.20.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(ii), §28.20 and Ch.28

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

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28.5.1 $${\mathrm{fe}}_n(z,q)=C_n(q)\left(z{\mathrm{ce}}_n(z,q)+f_n(z,q)\right),$$

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Defines:

${\mathrm{fe}}_n(z,q)$: second solution, Mathieu’s equation

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $f_n(z,q)$: functions and $C_n(q)$: factor

A&S Ref:

20.3.6 (in different form)

Referenced by:

§28.5(i), §28.5(i)

Permalink:

http://dlmf.nist.gov/28.5.E1

Encodings:

pMML, png, TeX

See also:

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

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28.5.2 $${\mathrm{ge}}_n(z,q)=S_n(q)\left(z{\mathrm{se}}_n(z,q)+g_n(z,q)\right),$$

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Defines:

${\mathrm{ge}}_n(z,q)$: second solution, Mathieu’s equation

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $g_n(z,q)$: functions and $S_n(q)$: factor

A&S Ref:

20.3.7 (in different form)

Referenced by:

§28.5(i), §28.5(i)

Permalink:

http://dlmf.nist.gov/28.5.E2

Encodings:

pMML, png, TeX

See also:

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

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8: 30.14 Wave Equation in Oblate Spheroidal Coordinates

§30.14 Wave Equation in Oblate Spheroidal Coordinates

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§30.14(i) Oblate Spheroidal Coordinates

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§30.14(iv) Separation of Variables

… ►Equation (30.14.7) can be transformed to equation (30.2.1) by the substitution $z=\pm \mathrm{i}\xi$. … ►

9: 28.12 Definitions and Basic Properties

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§28.12(i) Eigenvalues ${\lambda }_{\nu +2n}\left(q\right)$

… ►When $q=0$ Equation (28.2.16) has simple roots, given by … … ►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 …They have the following pseudoperiodic and orthogonality properties: …

10: 28.10 Integral Equations

§28.10 Integral Equations

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§28.10(i) Equations with Elementary Kernels

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§28.10(ii) Equations with Bessel-Function Kernels

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§28.10(iii) Further Equations

… ►For relations with variable boundaries see Volkmer (1983).