# §10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

The function $\phi\left(\rho,\beta;z\right)$ is defined by

 10.46.1 $\phi\left(\rho,\beta;z\right)=\sum_{k=0}^{\infty}\frac{z^{k}}{k!\Gamma\left(% \rho k+\beta\right)},$ $\rho>-1$. ⓘ Defines: $\phi\left(\NVar{\rho},\NVar{\beta};\NVar{z}\right)$: generalized Bessel function Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable Permalink: http://dlmf.nist.gov/10.46.E1 Encodings: TeX, pMML, png See also: Annotations for §10.46 and Ch.10

From (10.25.2)

 10.46.2 $I_{\nu}\left(z\right)=\left(\tfrac{1}{2}z\right)^{\nu}\phi\left(1,\nu+1;\tfrac% {1}{4}z^{2}\right).$

For asymptotic expansions of $\phi\left(\rho,\beta;z\right)$ as $z\to\infty$ in various sectors of the complex $z$-plane for fixed real values of $\rho$ and fixed real or complex values of $\beta$, see Wright (1935) when $\rho>0$, and Wright (1940b) when $-1<\rho<0$. For exponentially-improved asymptotic expansions in the same circumstances, together with smooth interpretations of the corresponding Stokes phenomenon (§§2.11(iii)2.11(v)) see Wong and Zhao (1999b) when $\rho>0$, and Wong and Zhao (1999a) when $-1<\rho<0$.

The Laplace transform of $\phi\left(\rho,\beta;z\right)$ can be expressed in terms of the Mittag-Leffler function:

 10.46.3 $E_{a,b}\left(z\right)=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma\left(ak+b\right)},$ $a>0$. ⓘ Defines: $E_{\NVar{a},\NVar{b}}\left(\NVar{z}\right)$: Mittag-Leffler function Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $k$: nonnegative integer and $z$: complex variable Permalink: http://dlmf.nist.gov/10.46.E3 Encodings: TeX, pMML, png See also: Annotations for §10.46 and Ch.10

See Paris (2002c). This reference includes exponentially-improved asymptotic expansions for $E_{a,b}\left(z\right)$ when $|z|\to\infty$, together with a smooth interpretation of Stokes phenomena. See also Wong and Zhao (2002a), and for further information on the Mittag-Leffler function see Erdélyi et al. (1955, §18.1), Paris and Kaminski (2001, §5.1.4), and Haubold et al. (2011).

For incomplete modified Bessel functions and Hankel functions, including applications, see Cicchetti and Faraone (2004).