# §13.8 Asymptotic Approximations for Large Parameters

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

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

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

For fixed $a$ and $z$ in $\Complex$

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

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

 13.8.3 $\left(e^{t}-1\right)^{a-1}\mathop{\exp\/}\nolimits\!\left(t+z(1-e^{-t})\right)% =\sum_{s=0}^{\infty}q_{s}(z,a)t^{s+a-1}.$

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 $\zeta=\sqrt{2(\lambda-1-\mathop{\ln\/}\nolimits\lambda)}$ with $\mathop{\mathrm{sign}\/}\nolimits\!\left(\zeta\right)=\mathop{\mathrm{sign}\/}% \nolimits\!\left(\lambda-1\right)$. Then

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

and

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

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

Special cases are

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

and

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

To obtain approximations for $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ and $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ 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 $\mathop{U\/}\nolimits\!\left(a,b,x\right)=\frac{2e^{\frac{1}{2}x}}{\mathop{% \Gamma\/}\nolimits\!\left(a\right)}\left(\sqrt{\frac{2}{\beta}\mathop{\tanh\/}% \nolimits\!\left(\frac{w}{2}\right)}\left(\frac{1-e^{-w}}{\beta}\right)^{-b}% \beta^{1-b}\mathop{K_{1-b}\/}\nolimits\!\left(2\beta a\right)+a^{-1}\left(% \frac{a^{-1}+\beta}{1+\beta}\right)^{1-b}e^{-2\beta a}\mathop{O\/}\nolimits\!% \left(1\right)\right),$

where $w=\mathop{\mathrm{arccosh}\/}\nolimits\!\left(1+(2a)^{-1}x\right)$, and $\beta=\ifrac{(w+\mathop{\sinh\/}\nolimits w)}{2}$. (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 $\mathop{M\/}\nolimits\!\left(a,b,x\right)=\mathop{\Gamma\/}\nolimits\!\left(b% \right)e^{\frac{1}{2}x}\left((\tfrac{1}{2}b-a)x\right)^{\frac{1}{2}-\frac{1}{2% }b}\*\left(\mathop{J_{b-1}\/}\nolimits\!\left(\sqrt{2x(b-2a)}\right)+\mathop{% \mathrm{env}\mskip-2.0mu J_{b-1}\/}\nolimits\!\left(\sqrt{2x(b-2a)}\right)% \mathop{O\/}\nolimits\!\left(\left|a\right|^{-\frac{1}{2}}\right)\right),$

and

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

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

For asymptotic approximations to $\mathop{M\/}\nolimits\!\left(a,b,x\right)$ and $\mathop{U\/}\nolimits\!\left(a,b,x\right)$ 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 $\left|\mathop{\mathrm{ph}\/}\nolimits a\right|\leq\pi-\delta$ and $b$ and $z$ fixed,

 13.8.11 $\mathop{U\/}\nolimits\!\left(a,b,z\right)\sim 2\left(z/a\right)^{(1-b)/2}\frac% {e^{z/2}}{\mathop{\Gamma\/}\nolimits\!\left(a\right)}\*\left(\mathop{K_{b-1}\/% }\nolimits\!\left(2\sqrt{az}\right)\sum_{s=0}^{\infty}\frac{p_{s}(z)}{a^{s}}+% \sqrt{z/a}\mathop{K_{b}\/}\nolimits\!\left(2\sqrt{az}\right)\sum_{s=0}^{\infty% }\frac{q_{s}(z)}{a^{s}}\right),$ Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function, $e$: base of exponential function, $\mathop{K_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: modified Bessel function of the second kind, $\sim$: asymptotic equality, $s$: 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. Reported 2015-10-27
 13.8.12 $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)\sim\left(z/a\right)^{(1-b% )/2}\frac{e^{z/2}\mathop{\Gamma\/}\nolimits\!\left(1+a-b\right)}{\mathop{% \Gamma\/}\nolimits\!\left(a\right)}\*\left(\mathop{I_{b-1}\/}\nolimits\!\left(% 2\sqrt{az}\right)\sum_{s=0}^{\infty}\frac{p_{s}(z)}{a^{s}}-\sqrt{z/a}\mathop{I% _{b}\/}\nolimits\!\left(2\sqrt{az}\right)\sum_{s=0}^{\infty}\frac{q_{s}(z)}{a^% {s}}\right),$ Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Olver’s confluent hypergeometric function, $e$: base of exponential function, $\mathop{I_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: modified Bessel function of the first kind, $\sim$: asymptotic equality, $s$: 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. Reported 2015-10-27
 13.8.13 $\mathop{{\mathbf{M}}\/}\nolimits\!\left(-a,b,z\right)\sim\left(z/a\right)^{(1-% b)/2}\frac{e^{z/2}\mathop{\Gamma\/}\nolimits\!\left(1+a\right)}{\mathop{\Gamma% \/}\nolimits\!\left(a+b\right)}\*\left(\mathop{J_{b-1}\/}\nolimits\!\left(2% \sqrt{az}\right)\sum_{s=0}^{\infty}\frac{p_{s}(z)}{(-a)^{s}}-\sqrt{z/a}\mathop% {J_{b}\/}\nolimits\!\left(2\sqrt{az}\right)\sum_{s=0}^{\infty}\frac{q_{s}(z)}{% (-a)^{s}}\right),$ Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Olver’s confluent hypergeometric function, $e$: base of exponential function, $\sim$: asymptotic equality, $s$: 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. Reported 2015-10-27
 13.8.14 $\mathop{U\/}\nolimits\!\left(-a,b,z\right)\sim\left(z/a\right)^{(1-b)/2}e^{z/2% }\mathop{\Gamma\/}\nolimits\!\left(1+a\right)\*\left(C_{b-1}(a,2\sqrt{az})\sum% _{s=0}^{\infty}\frac{p_{s}(z)}{(-a)^{s}}-\sqrt{z/a}C_{b}(a,2\sqrt{az})\sum_{s=% 0}^{\infty}\frac{q_{s}(z)}{(-a)^{s}}\right),$ Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function, $e$: base of exponential function, $(\NVar{a},\NVar{b})$: open interval, $\sim$: asymptotic equality, $s$: 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. Reported 2015-10-27

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

 13.8.15 $\displaystyle p_{k}(z)$ $\displaystyle=\sum_{s=0}^{k}\binom{k}{s}\left(1-b+s\right)_{k-s}z^{s}c_{k+s}(z),$ $\displaystyle q_{k}(z)$ $\displaystyle=\sum_{s=0}^{k}\binom{k}{s}\left(2-b+s\right)_{k-s}z^{s}c_{k+s+1}% (z)$ Symbols: $\left(\NVar{a}\right)_{\NVar{n}}$: Pochhammer’s symbol (or shifted factorial), $\binom{\NVar{m}}{\NVar{n}}$: 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. Reported 2015-10-27

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

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

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