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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 $b$. … ►Although $M(a,b,z)$ does not exist when $b=-n$, $n=0,1,2,\mathrm{\dots }$, many formulas containing $M(a,b,z)$ continue to apply in their limiting form. … ►

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(iii) Limiting Forms as $z\to 0$

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§13.2(iv) Limiting Forms as $z\to \mathrm{\infty }$

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22: 5.11 Asymptotic Expansions

… ►For explicit formulas for $g_k$ in terms of Stirling numbers see Nemes (2013a), and for asymptotic expansions of $g_k$ as $k\to \mathrm{\infty }$ see Boyd (1994) and Nemes (2015a). …

23: 21.5 Modular Transformations

… ►The modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which $\xi (𝚪)$ is determinate: … ►For explicit results in the case $g=1$, see §20.7(viii).