# §13.8 Asymptotic Approximations for Large Parameters

## §13.8(i) Large $|b|$, Fixed $a$ and $z$

If $b\to\infty$ in $\mathbb{C}$ in such a way that $\left|b+n\right|\geq\delta>0$ for all $n=0,1,2,\dots$, then

 13.8.1 $M\left(a,b,z\right)=\sum_{s=0}^{n-1}\frac{{\left(a\right)_{s}}}{{\left(b\right% )_{s}}s!}z^{s}+O\left(|b|^{-n}\right).$

For fixed $a$ and $z$ in $\mathbb{C}$

 13.8.2 $M\left(a,b,z\right)\sim\frac{\Gamma\left(b\right)}{\Gamma\left(b-a\right)}\sum% _{s=0}^{\infty}{\left(a\right)_{s}}q_{s}(z,a)b^{-s-a},$

as $b\to\infty$ in $|\operatorname{ph}b|\leq\pi-\delta$, where $q_{0}(z,a)=1$ and

 13.8.3 $\left(e^{t}-1\right)^{a-1}\exp\left(t+z(1-e^{-t})\right)=\sum_{s=0}^{\infty}q_% {s}(z,a)t^{s+a-1}.$ ⓘ Symbols: $\exp\NVar{z}$: exponential function, $\mathrm{e}$: base of natural logarithm, $s$: nonnegative integer and $z$: complex variable Referenced by: §13.8(i) Permalink: http://dlmf.nist.gov/13.8.E3 Encodings: TeX, pMML, png See also: Annotations for §13.8(i), §13.8 and Ch.13

When the foregoing results are combined with Kummer’s transformation (13.2.39), an approximation is obtained for the case when $|b|$ is large, and $|b-a|$ and $|z|$ are bounded.

## §13.8(ii) Large $b$ and $z$, Fixed $a$ and $b/z$

Let $\lambda=z/b>0$ and $\zeta=\sqrt{2(\lambda-1-\ln\lambda)}$ with $\operatorname{sign}\left(\zeta\right)=\operatorname{sign}\left(\lambda-1\right)$. Then

 13.8.4 $M\left(a,b,z\right)\sim b^{\frac{1}{2}a}e^{\frac{1}{4}\zeta^{2}b}\left(\lambda% \left(\frac{\lambda-1}{\zeta}\right)^{a-1}U\left(a-\tfrac{1}{2},-\zeta\sqrt{b}% \right)+\left(\lambda\left(\frac{\lambda-1}{\zeta}\right)^{a-1}-\left(\frac{% \zeta}{\lambda-1}\right)^{a}\right)\frac{U\left(a-\tfrac{3}{2},-\zeta\sqrt{b}% \right)}{\zeta\sqrt{b}}\right)$

and

 13.8.5 $U\left(a,b,z\right)\sim b^{-\frac{1}{2}a}e^{\frac{1}{4}\zeta^{2}b}\left(% \lambda\left(\frac{\lambda-1}{\zeta}\right)^{a-1}U\left(a-\tfrac{1}{2},\zeta% \sqrt{b}\right)-\left(\lambda\left(\frac{\lambda-1}{\zeta}\right)^{a-1}-\left(% \frac{\zeta}{\lambda-1}\right)^{a}\right)\frac{U\left(a-\tfrac{3}{2},\zeta% \sqrt{b}\right)}{\zeta\sqrt{b}}\right)$

as $b\to\infty$, uniformly in compact $\lambda$-intervals of $(0,\infty)$ and compact real $a$-intervals. For the parabolic cylinder function $U$ see §12.2, and for an extension to an asymptotic expansion see Temme (1978).

Special cases are

 13.8.6 $M\left(a,b,b\right)=\sqrt{\pi}\left(\frac{b}{2}\right)^{\frac{1}{2}a}\left(% \frac{1}{\Gamma\left(\frac{1}{2}(a+1)\right)}+\frac{(a+1)\sqrt{8/b}}{3\Gamma% \left(\frac{1}{2}a\right)}+O\left(\frac{1}{b}\right)\right),$

and

 13.8.7 $U\left(a,b,b\right)=\sqrt{\pi}\left(2b\right)^{-\frac{1}{2}a}\left(\frac{1}{% \Gamma\left(\frac{1}{2}(a+1)\right)}-\frac{(a+1)\sqrt{8/b}}{3\Gamma\left(\frac% {1}{2}a\right)}+O\left(\frac{1}{b}\right)\right).$

To obtain approximations for $M\left(a,b,z\right)$ and $U\left(a,b,z\right)$ that hold as $b\to\infty$, with $a>\tfrac{1}{2}-b$ and $z>0$ combine (13.14.4), (13.14.5) with §13.20(i).

Also, more complicated—but more powerful—uniform asymptotic approximations can be obtained by combining (13.14.4), (13.14.5) with §§13.20(iii) and 13.20(iv).

For other asymptotic expansions for large $b$ and $z$ see López and Pagola (2010).

For more asymptotic expansions for the cases $b\to\pm\infty$ see Temme (2015, §§10.4 and 22.5)

## §13.8(iii) Large $a$

For the notation see §§10.2(ii), 10.25(ii), and 2.8(iv).

When $a\to+\infty$ with $b$ ($\leq 1$) fixed,

 13.8.8 $U\left(a,b,x\right)=\frac{2e^{\frac{1}{2}x}}{\Gamma\left(a\right)}\left(\sqrt{% \frac{2}{\beta}\tanh\left(\frac{w}{2}\right)}\left(\frac{1-e^{-w}}{\beta}% \right)^{-b}\beta^{1-b}K_{1-b}\left(2\beta a\right)+a^{-1}\left(\frac{a^{-1}+% \beta}{1+\beta}\right)^{1-b}e^{-2\beta a}O\left(1\right)\right),$

where $w=\operatorname{arccosh}\left(1+(2a)^{-1}x\right)$, and $\beta=\ifrac{(w+\sinh w)}{2}$. (13.8.8) holds uniformly with respect to $x\in[0,\infty)$. For the case $b>1$ the transformation (13.2.40) can be used.

For an extension to an asymptotic expansion complete with error bounds see Temme (1990b), and for related results see §13.21(i).

When $a\to-\infty$ with $b$ ($\geq 1$) fixed,

 13.8.9 $M\left(a,b,x\right)=\Gamma\left(b\right)e^{\frac{1}{2}x}\left((\tfrac{1}{2}b-a% )x\right)^{\frac{1}{2}-\frac{1}{2}b}\*\left(J_{b-1}\left(\sqrt{2x(b-2a)}\right% )+\mathrm{env}\mskip-2.0mu J_{b-1}\left(\sqrt{2x(b-2a)}\right)O\left({\left|a% \right|^{-\frac{1}{2}}}\right)\right),$

and

 13.8.10 $U\left(a,b,x\right)=\Gamma\left(\tfrac{1}{2}b-a+\tfrac{1}{2}\right)e^{\frac{1}% {2}x}x^{\frac{1}{2}-\frac{1}{2}b}\*\left(\cos\left(a\pi\right)J_{b-1}\left(% \sqrt{2x(b-2a)}\right)-\sin\left(a\pi\right)Y_{b-1}\left(\sqrt{2x(b-2a)}\right% )+\mathrm{env}\mskip-2.0mu Y_{b-1}\left(\sqrt{2x(b-2a)}\right)O\left({\left|a% \right|^{-\frac{1}{2}}}\right)\right),$

uniformly with respect to bounded positive values of $x$ in each case.

For asymptotic approximations to $M\left(a,b,x\right)$ and $U\left(a,b,x\right)$ as $a\to-\infty$ that hold uniformly with respect to $x\in(0,\infty)$ and bounded positive values of $(b-1)/\left|a\right|$, combine (13.14.4), (13.14.5) with §§13.21(ii), 13.21(iii).

When $a\to\infty$ in $\left|\operatorname{ph}a\right|\leq\pi-\delta$ and $b$ and $z$ fixed,

 13.8.11 $U\left(a,b,z\right)\sim 2\left(z/a\right)^{(1-b)/2}\frac{e^{z/2}}{\Gamma\left(% a\right)}\*\left(K_{b-1}\left(2\sqrt{az}\right)\sum_{s=0}^{\infty}\frac{p_{s}(% z)}{a^{s}}+\sqrt{z/a}K_{b}\left(2\sqrt{az}\right)\sum_{s=0}^{\infty}\frac{q_{s% }(z)}{a^{s}}\right),$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $K_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the second kind, $s$: nonnegative integer and $z$: complex variable Referenced by: §13.8(iii) Permalink: http://dlmf.nist.gov/13.8.E11 Encodings: TeX, pMML, png Addition (effective with 1.0.11): The whole paragraph containing this equation has been added. See also: Annotations for §13.8(iii), §13.8 and Ch.13
 13.8.12 ${\mathbf{M}}\left(a,b,z\right)\sim\left(z/a\right)^{(1-b)/2}\frac{e^{z/2}% \Gamma\left(1+a-b\right)}{\Gamma\left(a\right)}\*\left(I_{b-1}\left(2\sqrt{az}% \right)\sum_{s=0}^{\infty}\frac{p_{s}(z)}{a^{s}}-\sqrt{z/a}I_{b}\left(2\sqrt{% az}\right)\sum_{s=0}^{\infty}\frac{q_{s}(z)}{a^{s}}\right),$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Olver’s confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $I_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the first kind, $s$: nonnegative integer and $z$: complex variable Permalink: http://dlmf.nist.gov/13.8.E12 Encodings: TeX, pMML, png Addition (effective with 1.0.11): The whole paragraph containing this equation has been added. See also: Annotations for §13.8(iii), §13.8 and Ch.13
 13.8.13 ${\mathbf{M}}\left(-a,b,z\right)\sim\left(z/a\right)^{(1-b)/2}\frac{e^{z/2}% \Gamma\left(1+a\right)}{\Gamma\left(a+b\right)}\*\left(J_{b-1}\left(2\sqrt{az}% \right)\sum_{s=0}^{\infty}\frac{p_{s}(z)}{(-a)^{s}}-\sqrt{z/a}J_{b}\left(2% \sqrt{az}\right)\sum_{s=0}^{\infty}\frac{q_{s}(z)}{(-a)^{s}}\right),$ ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$: gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Olver’s confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $s$: nonnegative integer and $z$: complex variable Referenced by: (13.8.13) Permalink: http://dlmf.nist.gov/13.8.E13 Encodings: TeX, pMML, png Addition (effective with 1.0.11): The whole paragraph containing this equation has been added. Expansion (13.8.13) is (10.3.58) in Temme (2015). In that reference the $(-1)^{s}$ is missing. See also: Annotations for §13.8(iii), §13.8 and Ch.13
 13.8.14 $U\left(-a,b,z\right)\sim\left(z/a\right)^{(1-b)/2}e^{z/2}\Gamma\left(1+a\right% )\*\left(C_{b-1}(a,2\sqrt{az})\sum_{s=0}^{\infty}\frac{p_{s}(z)}{(-a)^{s}}-% \sqrt{z/a}C_{b}(a,2\sqrt{az})\sum_{s=0}^{\infty}\frac{q_{s}(z)}{(-a)^{s}}% \right),$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $(\NVar{a},\NVar{b})$: open interval, $s$: nonnegative integer and $z$: complex variable Referenced by: (13.8.14) Permalink: http://dlmf.nist.gov/13.8.E14 Encodings: TeX, pMML, png Addition (effective with 1.0.11): The whole paragraph containing this equation has been added. Expansion (13.8.14) is (10.3.68) in Temme (2015). In that reference the $(-1)^{s}$ is missing. See also: Annotations for §13.8(iii), §13.8 and Ch.13

where $C_{\nu}\left(a,\zeta\right)=\cos\left(\pi a\right)J_{\nu}\left(\zeta\right)+% \sin\left(\pi a\right)Y_{\nu}\left(\zeta\right)$ and

 13.8.15 $\displaystyle p_{k}(z)$ $\displaystyle=\sum_{s=0}^{k}\genfrac{(}{)}{0.0pt}{}{k}{s}{\left(1-b+s\right)_{% k-s}}z^{s}c_{k+s}(z),$ $\displaystyle q_{k}(z)$ $\displaystyle=\sum_{s=0}^{k}\genfrac{(}{)}{0.0pt}{}{k}{s}{\left(2-b+s\right)_{% k-s}}z^{s}c_{k+s+1}(z)$ ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $s$: nonnegative integer and $z$: complex variable Referenced by: (13.8.15) Permalink: http://dlmf.nist.gov/13.8.E15 Encodings: TeX, TeX, pMML, pMML, png, png Addition (effective with 1.0.11): The whole paragraph containing this equation has been added. Expansion (13.8.15) is (10.3.41) in Temme (2015). In that reference there are typographical errors in the Pochhammer symbols. See also: Annotations for §13.8(iii), §13.8 and Ch.13

where $c_{0}(z)=1$ and

13.8.16 $(k+1)c_{k+1}(z)+\sum_{s=0}^{k}\left(\frac{bB_{s+1}}{(s+1)!}+\frac{z(s+1)B_{s+2% }}{(s+2)!}\right)c_{k-s}(z)=0,$
$k=0,1,2,\dots$.

For the Bernoulli numbers $B_{k}$ see §24.2(i) and for proofs and similar results in which $z$ can also be unbounded see Temme (2015, Chapters 10 and 27)