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1: 31.13 Asymptotic Approximations

§31.13 Asymptotic Approximations

►For asymptotic approximations for the accessory parameter eigenvalues $q_m$, see Fedoryuk (1991) and Slavyanov (1996). …

2: 31.18 Methods of Computation

… ►The computation of the accessory parameter for the Heun functions is carried out via the continued-fraction equations (31.4.2) and (31.11.13) in the same way as for the Mathieu, Lamé, and spheroidal wave functions in Chapters 28–30.

3: 31.4 Solutions Analytic at Two Singularities: Heun Functions

… ►For an infinite set of discrete values $q_m$, $m=0,1,2,\mathrm{\dots }$, of the accessory parameter $q$, the function $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ is analytic at $z=1$, and hence also throughout the disk . … ►

31.4.1 $$(0,1){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z),$$ $m=0,1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$(s_1,s_2){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§31.6

Permalink:

http://dlmf.nist.gov/31.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.4 and Ch.31

►The eigenvalues $q_m$ satisfy the continued-fraction equation ►

31.4.2 $$q=\frac{a\gamma P_1}{Q_1+q-}\frac{R_1{P}_2}{Q_2+q-}\frac{R_2{P}_3}{Q_3+q-}\mathrm{\cdots },$$

ⓘ

Symbols:

$\gamma$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $P_j$: coefficient, $Q_j$: coefficient and $R_j$: coefficient

Referenced by:

§31.18

Permalink:

http://dlmf.nist.gov/31.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.4 and Ch.31

… ►

31.4.3 $$(s_1,s_2){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z),$$ $m=0,1,2,\mathrm{\dots }$,

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Symbols:

$(s_1,s_2){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§31.4, §31.6

Permalink:

http://dlmf.nist.gov/31.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.4 and Ch.31

…

4: 31.14 General Fuchsian Equation

… ►

31.14.1 $$\frac{d^{2}w}{{dz}^2}+\left(\sum _{j=1}^N\frac{{\gamma }_j}{z-a_j}\right)\frac{dw}{dz}+\left(\sum _{j=1}^N\frac{q_j}{z-a_j}\right)w=0,$$ ${\sum }_{j=1}^N{q}_j=0$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities

Referenced by:

§31.15(i)

Permalink:

http://dlmf.nist.gov/31.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.14(i), §31.14 and Ch.31

… ►

$\alpha \beta ={\displaystyle \sum _{j=1}^N}{a}_j{q}_j.$

►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 $q_j$. With $a_1=0$ and $a_2=1$ the total number of free parameters is $3N-3$. … ►

31.14.3 $$w(z)=\left(\prod _{j=1}^N{(z-a_j)}^{-{\gamma }_j/2}\right)W(z),$$

ⓘ

Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $W(z)$: solution

Permalink:

http://dlmf.nist.gov/31.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

…

5: 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. The total number of free parameters is six. … ►Then (suppressing the parameter $k$) … ►Except for the identity automorphism, each alters the parameters.

6: 31.16 Mathematical Applications

… ►It describes the monodromy group of Heun’s equation for specific values of the accessory parameter. … ►

31.16.1 $${\mathrm{𝐻𝑝}}_{n,m}\left(x\right){\mathrm{𝐻𝑝}}_{n,m}\left(y\right)=\sum _{j=0}^n{A}_j{\sin }^{2j}\theta P_{n-j}^{(\gamma +\delta +2j-1,ϵ-1)}\left(\cos \left(2\theta \right)\right)P_j^{(\delta -1,\gamma -1)}\left(\cos \left(2\varphi \right)\right),$$

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Symbols:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\cos z$: cosine function, $\sin z$: sine function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $j$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\beta$: real or complex parameter, $\theta$: angle, $\varphi$: angle, $x$: variable, $y$: variable and $A_j$: coefficients

Referenced by:

§31.16(ii), §31.16(ii)

Permalink:

http://dlmf.nist.gov/31.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

… ►

31.16.3 $$A_0=\frac{n!}{{\left(\gamma +\delta \right)}_n}{\mathrm{𝐻𝑝}}_{n,m}\left(1\right),Q_0{A}_0+R_0{A}_1=0,$$

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Symbols:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\beta$: real or complex parameter, $A_j$: coefficients, $Q_j$ and $R_j$

Referenced by:

§31.16(ii), Erratum (V1.0.21) for Equations (31.16.2) and (31.16.3), Erratum (V1.0.21) for Equations (31.16.2) and (31.16.3)

Permalink:

http://dlmf.nist.gov/31.16.E3

Encodings:

pMML, png, TeX

Correction (effective with 1.0.21):

The initial data $A_0$, previously missing, has now been included.

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

… ►

31.16.5 $$P_j=\frac{(ϵ-j+n)j(\beta +j-1)(\gamma +\delta +j-2)}{(\gamma +\delta +2j-3)(\gamma +\delta +2j-2)},$$

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Defines:

$P_j$ (locally)

Symbols:

$\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

►

31.16.6 $$Q_j=-aj(j+\gamma +\delta -1)-q+\frac{(j-n)(j+\beta )(j+\gamma )(j+\gamma +\delta -1)}{(2j+\gamma +\delta )(2j+\gamma +\delta -1)}+\frac{(j+n+\gamma +\delta -1)j(j+\delta -1)(j-\beta +\gamma +\delta -1)}{(2j+\gamma +\delta -1)(2j+\gamma +\delta -2)},$$

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Defines:

$Q_j$ (locally)

Symbols:

$\gamma$: real or complex parameter, $\delta$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.16.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

…

7: 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

… ►

31.11.6 $$K_j=\frac{(j+\alpha -\mu -1)(j+\beta -\mu -1)(j+\gamma -\mu -1)(j-\mu )}{(2j+\lambda -\mu -1)(2j+\lambda -\mu -2)},$$

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Symbols:

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

Referenced by:

Erratum (V1.1.7) for Equation (31.11.6)

Permalink:

http://dlmf.nist.gov/31.11.E6

Encodings:

pMML, png, TeX

Correction (effective with 1.1.7):

The sign has been corrected and the final term in the numerator $(j+\lambda -1)$ has been corrected to be $(j-\mu )$.

Suggested 2022-09-14 by Hans Volkmer

See also:

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

►

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

ⓘ

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:

pMML, png, TeX

See also:

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

… ►

31.11.12 $$P_j^5=\frac{{\left(\alpha \right)}_j{\left(1-\gamma +\alpha \right)}_j}{{\left(1+\alpha -\beta +ϵ\right)}_{2j}}{z}^{-\alpha -j}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha +j,1-\gamma +\alpha +j}{1+\alpha -\beta +ϵ+2j};\frac{1}{z}),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $\gamma$: real or complex parameter, $ϵ$: real or complex parameter, $j$: nonnegative integer, $\alpha$: real or complex parameter, $\beta$: real or complex parameter and $P_j^5$: first Kummer solution at $\mathrm{\infty }$ of generalized hypergeometric differential equation

Referenced by:

§31.11(iii), §31.11(iii), §31.11(iii), Erratum (V1.1.7) for Subsection 31.11(iii)

Permalink:

http://dlmf.nist.gov/31.11.E12

Encodings:

pMML, png, TeX

Notational Change (effective with 1.1.7):

We have rewritten the gamma functions in the prefactor more concisely using Pochhammer symbols.

See also:

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

… ►In this case the accessory parameter $q$ is a root of the continued-fraction equation …

8: 31.15 Stieltjes Polynomials

… ►If $z_1,z_2,\mathrm{\dots },z_n$ are the zeros of an $n$th degree Stieltjes polynomial $S(z)$, then every zero $z_k$ is either one of the parameters $a_j$ or a solution of the system of equations … ►

31.15.6 $j=1,2,\mathrm{\dots },N-1$,

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Symbols:

$j$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities

Referenced by:

§31.15(iii)

Permalink:

http://dlmf.nist.gov/31.15.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

… ►In this case the accessory parameters $q_j$ are given by ►

31.15.7 $$q_j={\gamma }_j\sum _{k=1}^n\frac{1}{z_k-a_j},$$ $j=1,2,\mathrm{\dots },N$.

ⓘ

Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities

Permalink:

http://dlmf.nist.gov/31.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

… ►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 $j=1,2,\mathrm{\dots },N-1$. …

9: 20 Theta Functions

Chapter 20 Theta Functions

…

10: Guide to Searching the DLMF

… ►If you don’t specify the font style or font accessories in the query, the style and accessories won’t matter in the search, but if you specify them, they will matter. …