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

Keywords:: Bessel functions, Hankel functions, Mittag-Leffler function, modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.46

The function $\mathop{\phi\/}\nolimits\!\left(\rho,\beta;z\right)$ is defined by

10.46.1 $\mathop{\phi\/}\nolimits\!\left(\rho,\beta;z\right)=\sum _{{k=0}}^{\infty}\frac{z^{k}}{k!\mathop{\Gamma\/}\nolimits\!\left(\rho k+\beta\right)},$ $\rho>-1$ .

Defines:: $\mathop{\phi\/}\nolimits\!\left(\rho,\beta;z\right)$ : generalized Bessel function
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $!$ : $n!$ : factorial, $k$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/10.46.E1
Encodings:: TeX, pMML, png

From (10.25.2)

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

Symbols:: $\mathop{\phi\/}\nolimits\!\left(\rho,\beta;z\right)$ : generalized Bessel function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $z$ : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.46.E2
Encodings:: TeX, pMML, png

For asymptotic expansions of $\mathop{\phi\/}\nolimits\!\left(\rho,\beta;z\right)$ 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 $\mathop{\phi\/}\nolimits\!\left(\rho,\beta;z\right)$ can be expressed in terms of the Mittag-Leffler function:

10.46.3 $\mathop{E_{{a,b}}\/}\nolimits\!\left(z\right)=\sum _{{k=0}}^{\infty}\frac{z^{k}}{\mathop{\Gamma\/}\nolimits\!\left(ak+b\right)},$ $a>0$ .

Defines:: $\mathop{E_{{a,b}}\/}\nolimits\!\left(z\right)$ : Mittag-Leffler function
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $k$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/10.46.E3
Encodings:: TeX, pMML, png

See Paris (2002c). This reference includes exponentially-improved asymptotic expansions for $\mathop{E_{{a,b}}\/}\nolimits\!\left(z\right)$ 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) and Paris and Kaminski (2001, §5.1.4).

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