# §9.13 Generalized Airy Functions

## §9.13(i) Generalizations from the Differential Equation

Equations of the form

 9.13.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=z^{n}w,$ $n=1,2,3,\ldots,$ ⓘ Defines: $n$: parameter (locally) Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $z$: complex variable and $w$: function Source: Swanson and Headley (1967, (1)) Referenced by: §9.13(i), §9.13(i), §9.13(i) Permalink: http://dlmf.nist.gov/9.13.E1 Encodings: TeX, pMML, png See also: Annotations for 9.13(i), 9.13 and 9

are used in approximating solutions to differential equations with multiple turning points; see §2.8(v). The general solution of (9.13.1) is given by

 9.13.2 $w=z^{1/2}\mathscr{Z}_{p}\left(\zeta\right),$ ⓘ Defines: $w$: function (locally) Symbols: $\mathscr{Z}_{\NVar{\nu}}\left(\NVar{z}\right)$: modified cylinder function, $z$: complex variable, $p$: variable and $\zeta$: variable Source: Swanson and Headley (1967, p. 1401) Permalink: http://dlmf.nist.gov/9.13.E2 Encodings: TeX, pMML, png See also: Annotations for 9.13(i), 9.13 and 9

where

 9.13.3 $\displaystyle p$ $\displaystyle=\frac{1}{n+2}$, $\displaystyle\zeta$ $\displaystyle=\frac{2}{n+2}z^{(n+2)/2}=2pz^{1/(2p)}$, ⓘ Defines: $p$: variable (locally) and $\zeta$: variable (locally) Symbols: $z$: complex variable and $n$: parameter Source: Swanson and Headley (1967, (2)) Permalink: http://dlmf.nist.gov/9.13.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 9.13(i), 9.13 and 9

and $\mathscr{Z}_{p}$ is any linear combination of the modified Bessel functions $I_{p}$ and $e^{p\pi\mathrm{i}}K_{p}$10.25(ii)).

Swanson and Headley (1967) define independent solutions $A_{n}\left(z\right)$ and $B_{n}\left(z\right)$ of (9.13.1) by

 9.13.4 $\displaystyle A_{n}\left(z\right)$ $\displaystyle=(2p/\pi)\sin\left(p\pi\right)z^{1/2}K_{p}\left(\zeta\right)$, $\displaystyle B_{n}\left(z\right)$ $\displaystyle=(pz)^{1/2}\left(I_{-p}\left(\zeta\right)+I_{p}\left(\zeta\right)\right)$,

when $z$ is real and positive, and by analytic continuation elsewhere. (All solutions of (9.13.1) are entire functions of $z$.) When $n=1,$ $A_{n}\left(z\right)$ and $B_{n}\left(z\right)$ become $\mathrm{Ai}\left(z\right)$ and $\mathrm{Bi}\left(z\right)$, respectively.

Properties of $A_{n}\left(z\right)$ and $B_{n}\left(z\right)$ follow from the corresponding properties of the modified Bessel functions. They include:

 9.13.5 $\displaystyle A_{n}\left(0\right)$ $\displaystyle=p^{1/2}B_{n}\left(0\right)=\frac{p^{1-p}}{\Gamma\left(1-p\right)},$ $\displaystyle-A_{n}'\left(0\right)$ $\displaystyle=p^{1/2}B_{n}'\left(0\right)=\frac{p^{p}}{\Gamma\left(p\right)}.$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $A_{\NVar{n}}\left(\NVar{z}\right)$: generalized Airy function, $B_{\NVar{n}}\left(\NVar{z}\right)$: generalized Airy function, $n$: parameter and $p$: variable Source: Swanson and Headley (1967, (4)) Permalink: http://dlmf.nist.gov/9.13.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 9.13(i), 9.13 and 9
 9.13.6 $\displaystyle A_{n}\left(-z\right)$ $\displaystyle=\begin{cases}pz^{1/2}\left(J_{-p}\left(\zeta\right)+J_{p}\left(% \zeta\right)\right),&n\text{ odd},\\ p^{1/2}B_{n}\left(z\right),&n\text{ even},\end{cases}$ 9.13.7 $\displaystyle B_{n}\left(-z\right)$ $\displaystyle=\begin{cases}(pz)^{1/2}\left(J_{-p}\left(\zeta\right)-J_{p}\left% (\zeta\right)\right),&n\text{ odd},\\ p^{-1/2}A_{n}\left(z\right),&n\text{ even}.\end{cases}$
 9.13.8 $\mathscr{W}\left\{A_{n}\left(z\right),B_{n}\left(z\right)\right\}=\frac{2}{\pi% }p^{1/2}\sin\left(p\pi\right).$

As $z\to\infty$

 9.13.9 $\displaystyle A_{n}\left(z\right)$ $\displaystyle=\sqrt{\ifrac{p}{\pi}}\sin\left(p\pi\right)z^{-n/4}e^{-\zeta}% \left(1+O\left(\zeta^{-1}\right)\right),$ $|\operatorname{ph}z|\leq 3p\pi-\delta$, 9.13.10 $\displaystyle A_{n}\left(-z\right)$ $\displaystyle=\begin{cases}2\sqrt{p/\pi}\cos\left(\tfrac{1}{2}p\pi\right)z^{-n% /4}\left(\cos\left(\zeta-\tfrac{1}{4}\pi\right)+e^{|\Im\zeta|}O\left(\zeta^{-1% }\right)\right),&\text{|\operatorname{ph}z|\leq 2p\pi-\delta, n odd},\\ \sqrt{p/\pi}z^{-n/4}e^{\zeta}\left(1+O\left(\zeta^{-1}\right)\right),&\text{|% \operatorname{ph}z|\leq p\pi-\delta, n even},\end{cases}$ 9.13.11 $\displaystyle B_{n}\left(z\right)$ $\displaystyle={\pi}^{-1/2}z^{-n/4}e^{\zeta}\left(1+O\left(\zeta^{-1}\right)% \right),$ $|\operatorname{ph}z|\leq p\pi-\delta$, 9.13.12 $\displaystyle B_{n}\left(-z\right)$ $\displaystyle=\begin{cases}-(\ifrac{2}{\sqrt{\pi}})\sin\left(\tfrac{1}{2}p\pi% \right)z^{-n/4}\left(\sin\left(\zeta-\tfrac{1}{4}\pi\right)+e^{\left|\Im\zeta% \right|}O\left(\zeta^{-1}\right)\right),&\left|\operatorname{ph}z\right|\leq 2% p\pi-\delta,n\text{ odd},\\ (\ifrac{1}{\sqrt{\pi}})\sin\left(p\pi\right)z^{-n/4}e^{-\zeta}\left(1+O\left(% \zeta^{-1}\right)\right),&\left|\operatorname{ph}z\right|\leq 3p\pi-\delta,n% \text{ even}.\end{cases}$

The distribution in $\mathbb{C}$ and asymptotic properties of the zeros of $A_{n}\left(z\right)$, $A_{n}'\left(z\right)$, $B_{n}\left(z\right)$, and $B_{n}'\left(z\right)$ are investigated in Swanson and Headley (1967) and Headley and Barwell (1975).

In Olver (1977a, 1978) a different normalization is used. In place of (9.13.1) we have

 9.13.13 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}t}^{2}}=\tfrac{1}{4}m^{2}t^{m-2}w,$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $w$: function and $m$: index Source: Olver (1977a, (2.01)) Referenced by: §9.13(i), §9.13(i) Permalink: http://dlmf.nist.gov/9.13.E13 Encodings: TeX, pMML, png See also: Annotations for 9.13(i), 9.13 and 9

where $m=3,4,5,\ldots.$ For real variables the solutions of (9.13.13) are denoted by $U_{m}(t)$, $U_{m}(-t)$ when $m$ is even, and by $V_{m}(t)$, $\overline{V}_{m}(t)$ when $m$ is odd. (The overbar has nothing to do with complex conjugates.) Their relations to the functions $A_{n}\left(z\right)$ and $B_{n}\left(z\right)$ are given by

 9.13.14 $\displaystyle m$ $\displaystyle=n+2=1/p$, $\displaystyle t$ $\displaystyle=(\tfrac{1}{2}m)^{-2/m}z=\zeta^{2/m}$, ⓘ Symbols: $z$: complex variable, $n$: parameter, $p$: variable, $\zeta$: variable and $m$: index Permalink: http://dlmf.nist.gov/9.13.E14 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 9.13(i), 9.13 and 9
 9.13.15 $\sqrt{2\pi}\left(\tfrac{1}{2}m\right)^{(m-1)/m}\csc\left(\ifrac{\pi}{m}\right)% A_{n}\left(z\right)=\begin{cases}U_{m}(t),&m\text{ even},\\ V_{m}(t),&m\text{ odd},\end{cases}$
 9.13.16 $\sqrt{\pi}\left(\tfrac{1}{2}m\right)^{(m-2)/(2m)}\csc\left(\ifrac{\pi}{m}% \right)B_{n}\left(z\right)=\begin{cases}U_{m}(-t),&m\text{ even},\\ \overline{V}_{m}(t),&m\text{ odd}.\end{cases}$

Properties and graphs of $U_{m}(t)$, $V_{m}(t)$, $\overline{V}_{m}(t)$ are included in Olver (1977a) together with properties and graphs of real solutions of the equation

 9.13.17 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}t}^{2}}=-\tfrac{1}{4}m^{2}t^{m-2}w,$ $m$ even, ⓘ Defines: $W_{m}$: function (locally) Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $w$: function and $m$: index Source: Olver (1977a, (2.11), p. 131) Referenced by: §9.13(i) Permalink: http://dlmf.nist.gov/9.13.E17 Encodings: TeX, pMML, png See also: Annotations for 9.13(i), 9.13 and 9

which are denoted by $W_{m}(t)$, $W_{m}(-t)$.

In $\mathbb{C}$, the solutions of (9.13.13) used in Olver (1978) are

 9.13.18 $w=U_{m}(te^{-2j\pi i/m}),$ $j=0,\pm 1,\pm 2,\ldots.$

The function on the right-hand side is recessive in the sector $-(2j-1)\pi/m\leq\operatorname{ph}z\leq(2j+1)\pi/m$, and is therefore an essential member of any numerically satisfactory pair of solutions in this region.

Another normalization of (9.13.17) is used in Smirnov (1960), given by

 9.13.19 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}+x^{\alpha}w=0,$ ⓘ Defines: $\alpha$: parameter (locally) and $x$: parameter (locally) Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$ and $w$: function Source: Smirnov (1960, (1.2), p. 1) Permalink: http://dlmf.nist.gov/9.13.E19 Encodings: TeX, pMML, png See also: Annotations for 9.13(i), 9.13 and 9

where $\alpha>-2$ and $x>0$. Solutions are $w=U_{1}(x,\alpha)$, $U_{2}(x,\alpha)$, where

 9.13.20 $U_{1}(x,\alpha)=\frac{1}{(\alpha+2)^{1/(\alpha+2)}}\*\Gamma\left(\frac{\alpha+% 1}{\alpha+2}\right)x^{1/2}J_{-1/(\alpha+2)}\left(\frac{2}{\alpha+2}x^{(\alpha+% 2)/2}\right),$ ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$: gamma function, $\alpha$: parameter, $x$: parameter and $U_{i}$: Smirnov’s generalized Airy function Source: Smirnov (1960, (2.10), p. 10) Referenced by: §9.18(vii) Permalink: http://dlmf.nist.gov/9.13.E20 Encodings: TeX, pMML, png See also: Annotations for 9.13(i), 9.13 and 9
 9.13.21 $U_{2}(x,\alpha)=(\alpha+2)^{1/(\alpha+2)}\*\Gamma\left(\frac{\alpha+3}{\alpha+% 2}\right)x^{1/2}J_{1/(\alpha+2)}\left(\frac{2}{\alpha+2}x^{(\alpha+2)/2}\right),$ ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$: gamma function, $\alpha$: parameter, $x$: parameter and $U_{i}$: Smirnov’s generalized Airy function Source: Smirnov (1960, (2.11), p. 11) Referenced by: §9.18(vii) Permalink: http://dlmf.nist.gov/9.13.E21 Encodings: TeX, pMML, png See also: Annotations for 9.13(i), 9.13 and 9

and $J$ denotes the Bessel function (§10.2(ii)).

When $\alpha$ is a positive integer the relation of these functions to $W_{m}(t)$, $W_{m}(-t)$ is as follows:

 9.13.22 $\displaystyle\alpha$ $\displaystyle=m-2$, $\displaystyle x$ $\displaystyle=(m/2)^{2/m}t$, ⓘ Symbols: $\alpha$: parameter, $x$: parameter and $m$: index Permalink: http://dlmf.nist.gov/9.13.E22 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 9.13(i), 9.13 and 9
 9.13.23 $U_{1}(x,\alpha)=\frac{\pi^{1/2}}{2^{(m+2)/(2m)}\Gamma\left(1/m\right)}\left(W_% {m}(t)+W_{m}(-t)\right),$
 9.13.24 $U_{2}(x,\alpha)=\frac{\pi^{1/2}m^{2/m}}{2^{(m+2)/(2m)}\Gamma\left(-1/m\right)}% \left(W_{m}(t){-}W_{m}(-t)\right).$

For properties of the zeros of the functions defined in this subsection see Laforgia and Muldoon (1988) and references given therein.

## §9.13(ii) Generalizations from Integral Representations

Reid (1972) and Drazin and Reid (1981, Appendix) introduce the following contour integrals in constructing approximate solutions to the Orr–Sommerfeld equation for fluid flow:

 9.13.25 $A_{k}\left(z,p\right)=\frac{1}{2\pi i}\int_{\mathscr{L}_{k}}t^{-p}\exp\left(zt% -\tfrac{1}{3}t^{3}\right)\mathrm{d}t,$ $k=1,2,3$, $p\in\mathbb{C}$, ⓘ Defines: $k$: index (locally) and $p$: parameter (locally) Symbols: $A_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathbb{C}$: complex plane, $\mathrm{d}\NVar{x}$: differential of $x$, $\in$: element of, $\exp\NVar{z}$: exponential function, $\int$: integral, $z$: complex variable and $\mathscr{L}$: integration path Sources: Reid (1972, (A4), p. 362); Drazin and Reid (1981, (A4), p. 466) Referenced by: §9.13(ii) Permalink: http://dlmf.nist.gov/9.13.E25 Encodings: TeX, pMML, png See also: Annotations for 9.13(ii), 9.13 and 9
 9.13.26 $B_{0}\left(z,p\right)=\frac{1}{2\pi i}\int_{\mathscr{L}_{0}}t^{-p}\exp\left(zt% -\tfrac{1}{3}t^{3}\right)\mathrm{d}t,$ $p=0,\pm 1,\pm 2,\ldots,$
 9.13.27 $B_{k}\left(z,p\right)=\int_{\mathscr{I}_{k}}t^{-p}\exp\left(zt-\tfrac{1}{3}t^{% 3}\right)\mathrm{d}t,$ $k=1,2,3$, $p=0,\pm 1,\pm 2,\ldots,$

with $z\in\mathbb{C}$ in all cases. The integration paths $\mathscr{L}_{0}$, $\mathscr{L}_{1}$, $\mathscr{L}_{2}$, $\mathscr{L}_{3}$ are depicted in Figure 9.13.1. $\mathscr{I}_{1}$, $\mathscr{I}_{2}$, $\mathscr{I}_{3}$ are depicted in Figure 9.13.2. When $p$ is not an integer the branch of $t^{-p}$ in (9.13.25) is usually chosen to be $\exp\left(-p(\ln|t|+i\operatorname{ph}t)\right)$ with $0\leq\operatorname{ph}t<2\pi$.

When $p=0$

 9.13.28 $A_{1}\left(z,0\right)=\mathrm{Ai}\left(z\right),$ ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $A_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$: generalized Airy function and $z$: complex variable Sources: Reid (1972, (A10), p. 364, with (A1)); Drazin and Reid (1981, (A10), p. 467, with (A1)) Permalink: http://dlmf.nist.gov/9.13.E28 Encodings: TeX, pMML, png See also: Annotations for 9.13(ii), 9.13 and 9
 9.13.29 $\displaystyle A_{2}\left(z,0\right)$ $\displaystyle=e^{2\pi i/3}\mathrm{Ai}\left(ze^{2\pi i/3}\right)$, $\displaystyle A_{3}\left(z,0\right)$ $\displaystyle=e^{-2\pi i/3}\mathrm{Ai}\left(ze^{-2\pi i/3}\right)$, ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $A_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function and $z$: complex variable Sources: Reid (1972, (A10), p. 364, with (A1)); Drazin and Reid (1981, (A10), p. 467, with (A1)) Permalink: http://dlmf.nist.gov/9.13.E29 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 9.13(ii), 9.13 and 9

and

 9.13.30 $\displaystyle B_{0}\left(z,0\right)$ $\displaystyle=0$, $\displaystyle B_{1}\left(z,0\right)$ $\displaystyle=\pi\mathrm{Hi}\left(z\right)$. ⓘ Symbols: $B_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$: generalized Airy function, $\mathrm{Hi}\left(\NVar{z}\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable Source: Drazin and Reid (1981, p. 470) Permalink: http://dlmf.nist.gov/9.13.E30 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 9.13(ii), 9.13 and 9

Each of the functions $A_{k}\left(z,p\right)$ and $B_{k}\left(z,p\right)$ satisfies the differential equation

 9.13.31 $\frac{{\mathrm{d}}^{3}w}{{\mathrm{d}z}^{3}}-z\frac{\mathrm{d}w}{\mathrm{d}z}+(% p-1)w=0,$ ⓘ Defines: $w$: function (locally) Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $z$: complex variable and $p$: parameter Referenced by: §9.13(ii) Permalink: http://dlmf.nist.gov/9.13.E31 Encodings: TeX, pMML, png See also: Annotations for 9.13(ii), 9.13 and 9

and the difference equation

 9.13.32 $f(p-3)-zf(p-1)+(p-1)f(p)=0.$ ⓘ Defines: $f$: function (locally) Symbols: $z$: complex variable and $p$: parameter Source: Reid (1972, (A9), p. 363) Permalink: http://dlmf.nist.gov/9.13.E32 Encodings: TeX, pMML, png See also: Annotations for 9.13(ii), 9.13 and 9

The $A_{k}\left(z,p\right)$ are related by

 9.13.33 $\displaystyle A_{2}\left(z,p\right)$ $\displaystyle=e^{-2(p-1)\pi i/3}A_{1}\left(ze^{2\pi i/3},p\right)$, $\displaystyle A_{3}\left(z,p\right)$ $\displaystyle=e^{2(p-1)\pi i/3}A_{1}\left(ze^{-2\pi i/3},p\right)$. ⓘ Symbols: $A_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function, $z$: complex variable and $p$: parameter Source: Reid (1972, (A7), p. 363) Permalink: http://dlmf.nist.gov/9.13.E33 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 9.13(ii), 9.13 and 9

Connection formulas for the solutions of (9.13.31) include

 9.13.34 $A_{1}\left(z,p\right)+A_{2}\left(z,p\right)+A_{3}\left(z,p\right)+B_{0}\left(z% ,p\right)=0,$ ⓘ Symbols: $A_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$: generalized Airy function, $B_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$: generalized Airy function, $z$: complex variable and $p$: parameter Source: Reid (1972, (A19), p. 365) Permalink: http://dlmf.nist.gov/9.13.E34 Encodings: TeX, pMML, png See also: Annotations for 9.13(ii), 9.13 and 9
 9.13.35 $B_{2}\left(z,p\right)-B_{3}\left(z,p\right)=2\pi iA_{1}\left(z,p\right),$ ⓘ Symbols: $A_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$: generalized Airy function, $B_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $p$: parameter Source: Drazin and Reid (1981, (A23), p. 471) Permalink: http://dlmf.nist.gov/9.13.E35 Encodings: TeX, pMML, png See also: Annotations for 9.13(ii), 9.13 and 9
 9.13.36 $B_{3}\left(z,p\right)-B_{1}\left(z,p\right)=2\pi iA_{2}\left(z,p\right),$ ⓘ Symbols: $A_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$: generalized Airy function, $B_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $p$: parameter Source: Drazin and Reid (1981, (A23), p. 471) Permalink: http://dlmf.nist.gov/9.13.E36 Encodings: TeX, pMML, png See also: Annotations for 9.13(ii), 9.13 and 9
 9.13.37 $B_{1}\left(z,p\right)-B_{2}\left(z,p\right)=2\pi iA_{3}\left(z,p\right).$ ⓘ Symbols: $A_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$: generalized Airy function, $B_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $p$: parameter Source: Drazin and Reid (1981, (A23), p. 471) Permalink: http://dlmf.nist.gov/9.13.E37 Encodings: TeX, pMML, png See also: Annotations for 9.13(ii), 9.13 and 9

Further properties of these functions, and also of similar contour integrals containing an additional factor $(\ln t)^{q}$, $q=1,2,\ldots,$ in the integrand, are derived in Reid (1972), Drazin and Reid (1981, Appendix), and Baldwin (1985). These properties include Wronskians, asymptotic expansions, and information on zeros.

For further generalizations via integral representations see Chin and Hedstrom (1978), Janson et al. (1993, §10), and Kamimoto (1998).