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21: 13.2 Definitions and Basic Properties

… ►It can be regarded as the limiting form of the hypergeometric differential equation (§15.10(i)) that is obtained on replacing $z$ by $z/b$, letting $b\to \mathrm{\infty }$, and subsequently replacing the symbol $c$ by $b$. … ►In general, $U(a,b,z)$ has a branch point at $z=0$. … ►

13.2.7 $$U(-m,b,z)={(-1)}^m{\left(b\right)}_{m}M(-m,b,z)={(-1)}^m\sum _{s=0}^m\left(\genfrac{}{}{0.0pt}{}{m}{s}\right){\left(b+s\right)}_{m-s}{(-z)}^s.$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $m$: integer, $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.2(i), Erratum (V1.0.10) for Equation (13.2.7)

Permalink:

http://dlmf.nist.gov/13.2.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

The equality $U(-m,b,z)={(-1)}^m{\left(b\right)}_{m}M(-m,b,z)$ has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation. Note also that the notation $a=-n$ has been changed to $a=-m$.

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

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13.2.8 $$U(a,a+n+1,z)=\frac{{(-1)}^n{\left(1-a-n\right)}_n}{z^{a+n}}M(-n,1-a-n,z)=z^{-a}\sum _{s=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{s}\right){\left(a\right)}_s{z}^{-s}.$$

ⓘ

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.2(i), Erratum (V1.0.10) for Equation (13.2.8)

Permalink:

http://dlmf.nist.gov/13.2.E8

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

The equality $U(a,a+n+1,z)=\frac{{(-1)}^n{\left(1-a-n\right)}_n}{z^{a+n}}M(-n,1-a-n,z)$ has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation.

Suggested 2014-02-10 by Adri Olde Daalhuis

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

… ►

§13.2(vii) Connection Formulas

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22: 2.11 Remainder Terms; Stokes Phenomenon

… ►When a rigorous bound or reliable estimate for the remainder term is unavailable, it is unsafe to judge the accuracy of an asymptotic expansion merely from the numerical rate of decrease of the terms at the point of truncation. … ►

§2.11(ii) Connection Formulas

… ►However, on combining (2.11.6) with the connection formula (8.19.18), with $m=1$, we derive … ►However, to enjoy the resurgence property (§2.7(ii)) we often seek instead expansions in terms of the $F$-functions introduced in §2.11(iii), leaving the connection of the error-function type behavior as an implicit consequence of this property of the $F$-functions. … ►In this connection see also Byatt-Smith (2000). …

23: 14.15 Uniform Asymptotic Approximations

… ►Provided that $\mu -\nu \notin \mathbb{Z}$ the corresponding expansions for ${𝖯}_{\nu }^{\mu }\left(x\right)$ and ${𝖰}_{\nu }^{\mp \mu }\left(x\right)$ can be obtained from the connection formulas (14.9.7), (14.9.9), and (14.9.10). … ►The interval is mapped one-to-one to the interval , with the points $x=1$ and $x=\mathrm{\infty }$ corresponding to $\eta =\mathrm{\infty }$ and $\eta =0$, respectively. … ►The points $x={\left(1-{\alpha }^2\right)}^{1/2}$, $x=1$, and $x=\mathrm{\infty }$ are mapped to $y={\alpha }^2$, $y=0$, and $y=-\mathrm{\infty }$, respectively. … ►When $a>0$ the interval is mapped one-to-one to the interval , with the points $x=-a$, $x=a$, and $x=1$ corresponding to $\zeta =-\alpha$, $\zeta =\alpha$, and $\zeta =\mathrm{\infty }$, respectively. When $a=0$ the interval is mapped one-to-one to the interval , with the points $x=-1$, $0$, and $1$ corresponding to $\zeta =-\mathrm{\infty }$, $0$, and $\mathrm{\infty }$, respectively. …