§8.18 Asymptotic Expansions of $\mathop{I_{x}\/}\nolimits\!\left(a,b\right)$

§8.18(i) Large Parameters, Fixed $x$

If $b$ and $x$ are fixed, with $b>0$ and $0, then as $a\to\infty$

 8.18.1 $\mathop{I_{x}\/}\nolimits\!\left(a,b\right)={\mathop{\Gamma\/}\nolimits\!\left% (a+b\right)x^{a}(1-x)^{b-1}}\*\left(\sum_{k=0}^{n-1}\frac{1}{\mathop{\Gamma\/}% \nolimits\!\left(a+k+1\right)\mathop{\Gamma\/}\nolimits\!\left(b-k\right)}% \left(\frac{x}{1-x}\right)^{k}+\mathop{O\/}\nolimits\!\left(\frac{1}{\mathop{% \Gamma\/}\nolimits\!\left(a+n+1\right)}\right)\right),$

for each $n=0,1,2,\dots$. If $b=1,2,3,\dots$ and $n\geq b$, then the $\mathop{O\/}\nolimits$-term can be omitted and the result is exact.

If $b\to\infty$ and $a$ and $x$ are fixed, with $a>0$ and $0, then (8.18.1), with $a$ and $b$ interchanged and $x$ replaced by $1-x$, can be combined with (8.17.4).

§8.18(ii) Large Parameters: Uniform Asymptotic Expansions

Large $a$, Fixed $b$

Let

 8.18.2 $\xi=-\mathop{\ln\/}\nolimits x.$ Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $x$: variable and $\xi$: change of variable Permalink: http://dlmf.nist.gov/8.18.E2 Encodings: TeX, pMML, png See also: Annotations for 8.18(ii)

Then as $a\to\infty$, with $b$ ($>0$) fixed,

 8.18.3 $\mathop{I_{x}\/}\nolimits\!\left(a,b\right)=\frac{\mathop{\Gamma\/}\nolimits\!% \left(a+b\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)}\left(\sum_{k=0}^% {n-1}d_{k}F_{k}+\mathop{O\/}\nolimits\!\left(a^{-n}\right)F_{0}\right),$ Symbols: $\mathop{O\/}\nolimits\!\left(\NVar{x}\right)$: order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathop{I_{\NVar{x}}\/}\nolimits\!\left(\NVar{a},\NVar{b}\right)$: incomplete beta function, $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $n$: nonnegative integer, $a$: parameter, $b$: parameter, $x$: variable, $F_{k}$: functions and $d_{k}$: coefficients Referenced by: Equation (8.18.3) Permalink: http://dlmf.nist.gov/8.18.E3 Encodings: TeX, pMML, png Addition (effective with 1.0.14): Previously this equation appeared without the order estimate as $\mathop{I_{x}\/}\nolimits\!\left(a,b\right)\sim\frac{\mathop{\Gamma\/}% \nolimits\!\left(a+b\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)}\sum_{% k=0}^{\infty}d_{k}F_{k}$. The range of $x$ was extended to include $1$. Reported 2016-11-22 See also: Annotations for 8.18(ii)

uniformly for $x\in(0,1]$. The functions $F_{k}$ are defined by

 8.18.4 $aF_{k+1}=(k+b-a\xi)F_{k}+k\xi F_{k-1},$ Symbols: $k$: nonnegative integer, $a$: parameter, $b$: parameter, $\xi$: change of variable and $F_{k}$: functions Permalink: http://dlmf.nist.gov/8.18.E4 Encodings: TeX, pMML, png See also: Annotations for 8.18(ii)

with

 8.18.5 $\displaystyle F_{0}$ $\displaystyle=a^{-b}\mathop{Q\/}\nolimits\!\left(b,a\xi\right),$ $\displaystyle F_{1}$ $\displaystyle=\frac{b-a\xi}{a}F_{0}+\frac{\xi^{b}e^{-a\xi}}{a\mathop{\Gamma\/}% \nolimits\!\left(b\right)},$

and $\mathop{Q\/}\nolimits\!\left(a,z\right)$ as in §8.2(i). The coefficients $d_{k}$ are defined by the generating function

 8.18.6 $\left(\frac{1-e^{-t}}{t}\right)^{b-1}=\sum_{k=0}^{\infty}d_{k}(t-\xi)^{k}.$

In particular,

 8.18.7 $\displaystyle d_{0}$ $\displaystyle=\left(\frac{1-x}{\xi}\right)^{b-1},$ $\displaystyle d_{1}$ $\displaystyle=\frac{x\xi+x-1}{(1-x)\xi}(b-1)d_{0}.$ Symbols: $b$: parameter, $x$: variable, $\xi$: change of variable and $d_{k}$: coefficients Referenced by: §8.18(ii) Permalink: http://dlmf.nist.gov/8.18.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 8.18(ii)

Compare also §24.16(i). A recurrence relation for the $d_{k}$ can be found in Nemes and Olde Daalhuis (2016).

Symmetric Case

Let

 8.18.8 $x_{0}=a/(a+b).$ Symbols: $a$: parameter, $b$: parameter and $x$: variable Referenced by: §8.18(ii) Permalink: http://dlmf.nist.gov/8.18.E8 Encodings: TeX, pMML, png See also: Annotations for 8.18(ii)

Then as $a+b\to\infty$,

 8.18.9 $\mathop{I_{x}\/}\nolimits\!\left(a,b\right)\sim\tfrac{1}{2}\mathop{\mathrm{% erfc}\/}\nolimits\!\left(-\eta\sqrt{b/2}\right)+\frac{1}{\sqrt{2\pi(a+b)}}\*% \left(\frac{x}{x_{0}}\right)^{a}\left(\frac{1-x}{1-x_{0}}\right)^{b}\sum_{k=0}% ^{\infty}\frac{(-1)^{k}c_{k}(\eta)}{(a+b)^{k}},$

uniformly for $x\in(0,1)$ and $a/(a+b)$, $b/(a+b)\in[\delta,1-\delta]$, where $\delta$ again denotes an arbitrary small positive constant. For $\mathop{\mathrm{erfc}\/}\nolimits$ see §7.2(i). Also,

 8.18.10 $-\tfrac{1}{2}\eta^{2}=x_{0}\mathop{\ln\/}\nolimits\!\left(\frac{x}{x_{0}}% \right)+(1-x_{0})\mathop{\ln\/}\nolimits\!\left(\frac{1-x}{1-x_{0}}\right),$ Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function and $x$: variable Permalink: http://dlmf.nist.gov/8.18.E10 Encodings: TeX, pMML, png See also: Annotations for 8.18(ii)

with $\eta/(x-x_{0})>0$, and

 8.18.11 $c_{0}(\eta)=\frac{1}{\eta}-\frac{\sqrt{x_{0}(1-x_{0})}}{x-x_{0}},$ Symbols: $x$: variable and $c_{k}(\eta)$: coefficients Permalink: http://dlmf.nist.gov/8.18.E11 Encodings: TeX, pMML, png See also: Annotations for 8.18(ii)

with limiting value

 8.18.12 $c_{0}(0)=\frac{1-2x_{0}}{3\sqrt{x_{0}(1-x_{0})}}.$ Symbols: $x$: variable and $c_{k}(\eta)$: coefficients Permalink: http://dlmf.nist.gov/8.18.E12 Encodings: TeX, pMML, png See also: Annotations for 8.18(ii)

For this result, and for higher coefficients $c_{k}(\eta)$ see Temme (1996b, §11.3.3.2). All of the $c_{k}(\eta)$ are analytic at $\eta=0$.

General Case

Let $\widetilde{\Gamma}(z)$ denote the scaled gamma function

 8.18.13 $\widetilde{\Gamma}(z)=(2\pi)^{-1/2}e^{z}z^{(1/2)-z}\mathop{\Gamma\/}\nolimits% \!\left(z\right),$

$\mu=b/a$, and $x_{0}$ again be as in (8.18.8). Then as $a\to\infty$

 8.18.14 $\mathop{I_{x}\/}\nolimits\!\left(a,b\right)\sim\mathop{Q\/}\nolimits\!\left(b,% a\zeta\right)-\frac{(2\pi b)^{-1/2}}{\widetilde{\Gamma}(b)}\left(\frac{x}{x_{0% }}\right)^{a}\left(\frac{1-x}{1-x_{0}}\right)^{b}\sum_{k=0}^{\infty}\frac{h_{k% }(\zeta,\mu)}{a^{k}},$

uniformly for $b\in(0,\infty)$ and $x\in(0,1)$. Here

 8.18.15 $\mu\mathop{\ln\/}\nolimits\zeta-\zeta=\mathop{\ln\/}\nolimits x+\mu\mathop{\ln% \/}\nolimits\!\left(1-x\right)+(1+\mu)\mathop{\ln\/}\nolimits\!\left(1+\mu% \right)-\mu,$ Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $x$: variable, $\mu$ and $\zeta$: variable Permalink: http://dlmf.nist.gov/8.18.E15 Encodings: TeX, pMML, png See also: Annotations for 8.18(ii)

with $(\zeta-\mu)/(x_{0}-x)>0$, and

 8.18.16 $h_{0}(\zeta,\mu)=\mu\left(\frac{1}{\zeta-\mu}-\frac{(1+\mu)^{-3/2}}{x_{0}-x}% \right),$ Symbols: $x$: variable, $\mu$, $\zeta$: variable and $h_{k}(\zeta,\mu)$: coefficients Permalink: http://dlmf.nist.gov/8.18.E16 Encodings: TeX, pMML, png See also: Annotations for 8.18(ii)

with limiting value

 8.18.17 $h_{0}(\mu,\mu)=\frac{1}{3}\left(\frac{1-\mu}{\sqrt{1+\mu}}-1\right).$ Symbols: $\mu$ and $h_{k}(\zeta,\mu)$: coefficients Permalink: http://dlmf.nist.gov/8.18.E17 Encodings: TeX, pMML, png See also: Annotations for 8.18(ii)

For this result and higher coefficients $h_{k}(\zeta,\mu)$ see Temme (1996b, §11.3.3.3). All of the $h_{k}(\zeta,\mu)$ are analytic at $\zeta=\mu$ (corresponding to $x=x_{0}$).

Inverse Function

For asymptotic expansions for large values of $a$ and/or $b$ of the $x$-solution of the equation

 8.18.18 $\mathop{I_{x}\/}\nolimits\!\left(a,b\right)=p,$ $0\leq p\leq 1$,

see Temme (1992b).