difference equations

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21: 15.4 Special Cases

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15.4.34 $$F(3a,a;2a;{\mathrm{e}}^{\mathrm{i}\pi /3})=\sqrt{\pi }{\mathrm{e}}^{\mathrm{i}\pi a/2}\frac{2^{2a}\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{3^{(3a+1)/2}}\left(\frac{1}{\mathrm{\Gamma }\left(\frac{1}{3}+a\right)\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{1}{\mathrm{\Gamma }\left(\frac{2}{3}+a\right)\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $a$: real or complex parameter

Proof sketch:

From §15.5(ii) one obtains that $w(a)={\mathrm{e}}^{-\mathrm{i}\pi a/2}{2}^{-2a}{3}^{(3a+1)/2}F(3a,a;2a;{\mathrm{e}}^{\mathrm{i}\pi /3})\mathrm{\Gamma }\left(\frac{1}{3}+a\right)/\mathrm{\Gamma }\left(\frac{1}{2}+a\right)$ is a solution of the difference equation $(3a+5)w(a+2)-6(a+1)w(a+1)+(3a+1)w(a)=0$. Two linearly independent solutions of this difference equation are $w_1(a)=1$ and $w_2(a)=\mathrm{\Gamma }\left(\frac{1}{3}+a\right)/\mathrm{\Gamma }\left(\frac{2}{3}+a\right)$. Combining (15.6.1) with Laplace’s method (see §2.4(iii)) we obtain that $w(a)\sim \sqrt{\pi }\left(\frac{1}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{a^{-1/3}}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right)$, as $a\to \mathrm{\infty }$. Hence, $w(a)=\sqrt{\pi }\left(\frac{1}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{w_2(a)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right)$.

Notes:

See §15.2(ii) for treatment of the case when the third parameter is a nonpositive integer.

Referenced by:

§15.4(iii), §15.4(iii), Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/15.4.E34

Encodings:

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Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §15.4(iii), §15.4 and Ch.15

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22: Bibliography R

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Research Institute for Symbolic Computation, Hagenberg im Mühlkreis, Austria.

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Notes:

Software packages for hypergeometric and $q$-hypergeometric summation, sequences and power series, permutation groups, partition analysis, and difference equations.

External Links:

RISC Combinatorics Group

Referenced by:

6th item.

Encodings:

BibTeX

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23: 18.28 Askey–Wilson Class

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$q$-Difference Equation

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24: Bibliography B

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An Introduction to Linear Difference Equations. Dover Publications Inc., New York.

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Notes:

Unaltered republication of the original 1927 Harvard University edition.

External Links:

MathReview, ZentralBlatt

Referenced by:

§8.1, Notations P, Notations Q.

Encodings:

BibTeX

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25: 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: …

26: 2.7 Differential Equations

… ►when $s=1,2,3,\mathrm{\dots }$. …

27: 18.27 $q$-Hahn Class

… ►For other formulas, including $q$-difference equations, recurrence relations, duality formulas, special cases, and limit relations, see Koekoek et al. (2010, Chapter 14). …

28: 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:

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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:

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See also:

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

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28.2.5

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Symbols:

$q=h^2$: parameter, $a$: parameter, $w(z)$: Mathieu’s equation solution, $w_{\text{I}}(z;a,q)$: fundamental solution and $w_{\text{II}}(z;a,q)$: fundamental solution

A&S Ref:

20.3.9 (in different notation)

Referenced by:

§28.29(ii), item (a), item (c)

Permalink:

http://dlmf.nist.gov/28.2.E5

Encodings:

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See also:

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

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28.2.16 $$\cos \left(\pi \nu \right)=w_{\text{I}}(\pi ;a,q)=w_{\text{I}}(\pi ;a,-q).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $q=h^2$: parameter, $\nu$: complex parameter, $a$: parameter and $w_{\text{I}}(z;a,q)$: fundamental solution

A&S Ref:

20.3.10 (in slightly different form)

Referenced by:

§28.12(i), §28.12(i), §28.12(ii), §28.2(iii), §28.2(iv), §28.2(v), item (c), §28.34(i)

Permalink:

http://dlmf.nist.gov/28.2.E16

Encodings:

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See also:

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

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29: 31.17 Physical Applications

… ►For applications of Heun’s equation and functions in astrophysics see Debosscher (1998) where different spectral problems for Heun’s equation are also considered. …

30: 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:

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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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