exponent parameters

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1: 31.8 Solutions via Quadratures

… ►For half-odd-integer values of the exponent parameters: … ►The curve $\mathrm{\Gamma }$ reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for $m_j\in \mathbb{Z}$. …

2: 31.14 General Fuchsian Equation

… ►The three sets of parameters comprise the singularity parameters $a_j$, the exponent parameters $\alpha ,\beta ,{\gamma }_j$, and the $N-2$ free accessory parameters $q_j$. …

3: 31.2 Differential Equations

… ►All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, $ℂ\cup \{\mathrm{\infty }\}$, can be transformed into (31.2.1). ►The parameters play different roles: $a$ is the singularity parameter; $\alpha ,\beta ,\gamma ,\delta ,ϵ$ are exponent parameters; $q$ is the accessory parameter. …

4: 27.2 Functions

… ►is the sum of the $\alpha$th powers of the divisors of $n$, where the exponent $\alpha$ can be real or complex. …

5: 31.15 Stieltjes Polynomials

… ►If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index $𝐦=(m_1,m_2,\mathrm{\dots },m_{N-1})$, where each $m_j$ is a nonnegative integer, there is a unique Stieltjes polynomial with $m_j$ zeros in the open interval $(a_j,a_{j+1})$ for each $j=1,2,\mathrm{\dots },N-1$. …

6: 30.2 Differential Equations

… ►

30.2.1 $$\frac{d}{dz}\left((1-z^2)\frac{dw}{dz}\right)+\left(\lambda +{\gamma }^2(1-z^2)-\frac{{\mu }^2}{1-z^2}\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $w$: solution to DE, $\lambda$: real parameter, ${\gamma }^2$: real parameter and $\mu$: real parameter

Referenced by:

§30.11(i), §30.12, §30.12, §30.13(iv), §30.14(iv), §30.16(ii), §30.2(ii), §30.2(iii), §30.3(i), §30.4(i), §30.5, §30.6, §30.6

Permalink:

http://dlmf.nist.gov/30.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.2(i), §30.2 and Ch.30

►This equation has regular singularities at $z=\pm 1$ with exponents $\pm \frac{1}{2}\mu$ and an irregular singularity of rank 1 at $z=\mathrm{\infty }$ (if $\gamma \ne 0$). The equation contains three real parameters $\lambda$, ${\gamma }^2$, and $\mu$. … ►With $\zeta =\gamma z$ Equation (30.2.1) changes to ►

30.2.4 $$({\zeta }^2-{\gamma }^2)\frac{d^{2}w}{{d\zeta }^2}+2\zeta \frac{dw}{d\zeta }+\left({\zeta }^2-\lambda -{\gamma }^2-\frac{{\gamma }^2{\mu }^2}{{\zeta }^2-{\gamma }^2}\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $w$: solution to DE, $\lambda$: real parameter, ${\gamma }^2$: real parameter, $\mu$: real parameter and $\zeta$: parameter

Referenced by:

§30.2(iii)

Permalink:

http://dlmf.nist.gov/30.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.2(ii), §30.2 and Ch.30

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7: 31.3 Basic Solutions

… ► $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ denotes the solution of (31.2.1) that corresponds to the exponent $0$ at $z=0$ and assumes the value $1$ there. … ►Similarly, if $\gamma \ne 1,2,3,\mathrm{\dots }$, then the solution of (31.2.1) that corresponds to the exponent $1-\gamma$ at $z=0$ is … ►Solutions of (31.2.1) corresponding to the exponents $0$ and $1-\delta$ at $z=1$ are respectively, … ►Solutions of (31.2.1) corresponding to the exponents $0$ and $1-ϵ$ at $z=a$ are respectively, … ►Solutions of (31.2.1) corresponding to the exponents $\alpha$ and $\beta$ at $z=\mathrm{\infty }$ are respectively, …

8: 31.11 Expansions in Series of Hypergeometric Functions

… ►

31.11.3 $$\lambda +\mu =\gamma +\delta -1=\alpha +\beta -ϵ.$$

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Defines:

$\lambda$ (locally) and $\mu$ (locally)

Symbols:

$\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

… ►

31.11.7 $$L_j=a(\lambda +j)(\mu -j)-q+\frac{(j+\alpha -\mu )(j+\beta -\mu )(j+\gamma -\mu )(j+\lambda )}{(2j+\lambda -\mu )(2j+\lambda -\mu +1)}+\frac{(j-\alpha +\lambda )(j-\beta +\lambda )(j-\gamma +\lambda )(j-\mu )}{(2j+\lambda -\mu )(2j+\lambda -\mu -1)},$$

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Symbols:

$\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter, $\lambda$ , $\mu$ and $L_j$: coefficients

Permalink:

http://dlmf.nist.gov/31.11.E7

Encodings:

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See also:

Annotations for §31.11(ii), §31.11 and Ch.31

… ►Then the Fuchs–Frobenius solution at $\mathrm{\infty }$ belonging to the exponent $\alpha$ has the expansion (31.11.1) with … ►For example, consider the Heun function which is analytic at $z=a$ and has exponent $\alpha$ at $\mathrm{\infty }$. …In this case the accessory parameter $q$ is a root of the continued-fraction equation …

9: 28.29 Definitions and Basic Properties

… ►

28.29.9 $$2\cos \left(\pi \nu \right)=w_{\text{I}}(\pi ,\lambda )+w_{\text{II}}^{\prime }(\pi ,\lambda ).$$

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Symbols:

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

Referenced by:

§28.29(ii), §28.29(ii), §28.29(ii)

Permalink:

http://dlmf.nist.gov/28.29.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.29(ii), §28.29 and Ch.28

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10: 28.2 Definitions and Basic Properties

… ►

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:

pMML, png, TeX

See also:

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

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