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1: 13.12 Products

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2: 7.2 Definitions

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Values at Infinity

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Values at Infinity

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3: 31.11 Expansions in Series of Hypergeometric Functions

… ►The Fuchs-Frobenius solutions at $\mathrm{\infty }$ are ►

31.11.3_1 $$P_j^5=\frac{{\left(\lambda \right)}_j{\left(1-\gamma +\lambda \right)}_j}{{\left(1+\lambda -\mu \right)}_{2j}}{z}^{-\lambda -j}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\lambda +j,1-\gamma +\lambda +j}{1+\lambda -\mu +2j};\frac{1}{z}),$$

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Defines:

$P_j^5$: first Kummer solution at $\mathrm{\infty }$ of generalized hypergeometric differential equation (locally)

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, $j$: nonnegative integer, $\lambda$ and $\mu$

Referenced by:

§31.11(ii), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/31.11.E3_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

Suggested 2022-09-14 by Hans Volkmer

See also:

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

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31.11.3_2 $$P_j^6=\frac{{\left(\lambda -\mu \right)}_{2j}}{{\left(1-\mu \right)}_j{\left(\gamma -\mu \right)}_j}{z}^{-\mu +j}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\mu -j,1-\gamma +\mu -j}{1-\lambda +\mu -2j};\frac{1}{z}).$$

ⓘ

Defines:

$P_j^6$: second Kummer solution at $\mathrm{\infty }$ of generalized hypergeometric differential equation (locally)

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, $j$: nonnegative integer, $\lambda$ and $\mu$

Referenced by:

§31.11(ii), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/31.11.E3_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

Suggested 2022-09-14 by Hans Volkmer

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 $\mathrm{\infty }$. …

4: 31.14 General Fuchsian Equation

… ►The general second-order Fuchsian equation with $N+1$ regular singularities at $z=a_j$, $j=1,2,\mathrm{\dots },N$, and at $\mathrm{\infty }$, is given by …The exponents at the finite singularities $a_j$ are $\{0,1-{\gamma }_j\}$ and those at $\mathrm{\infty }$ are $\{\alpha ,\beta \}$, where …

5: 6.2 Definitions and Interrelations

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Values at Infinity

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6: 31.12 Confluent Forms of Heun’s Equation

… ►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty }$. … ►This has irregular singularities at $z=0$ and $\mathrm{\infty }$, each of rank $1$. … ►This has a regular singularity at $z=0$, and an irregular singularity at $\mathrm{\infty }$ of rank $2$. … ►This has one singularity, an irregular singularity of rank $3$ at $z=\mathrm{\infty }$. …

7: 16.17 Definition

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$L$ is a loop that starts at infinity on a line parallel to the positive real axis, encircles the poles of the $\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-s\right)$ once in the negative sense and returns to infinity on another line parallel to the positive real axis. The integral converges for all $z$ ($\ne 0$) if , and for if $p=q\ge 1$.

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$L$ is a loop that starts at infinity on a line parallel to the negative real axis, encircles the poles of the $\mathrm{\Gamma }\left(1-a_{\mathrm{\ell }}+s\right)$ once in the positive sense and returns to infinity on another line parallel to the negative real axis. The integral converges for all $z$ if $p>q$, and for $|z|>1$ if $p=q\ge 1$.

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8: 16.5 Integral Representations and Integrals

… ►Suppose first that $L$ is a contour that starts at infinity on a line parallel to the positive real axis, encircles the nonnegative integers in the negative sense, and ends at infinity on another line parallel to the positive real axis. …

9: 1.13 Differential Equations

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Transformation of the Point at Infinity

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10: 15.11 Riemann’s Differential Equation

… ►Also, if any of $\alpha$, $\beta$, $\gamma$, is at infinity, then we take the corresponding limit in (15.11.1). …