§16.21 Differential Equation

Notes:: See Luke (1969a, Chapter 5).
Keywords:: Meijer -function
Permalink:: http://dlmf.nist.gov/16.21

$w=\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;\mathbf{a};\mathbf{b}\right)$ satisfies the differential equation

16.21.1 $\left((-1)^{{p-m-n}}z(\vartheta-a_{1}+1)\cdots(\vartheta-a_{p}+1)-(\vartheta-b% _{1})\cdots(\vartheta-b_{q})\right)w=0,$

Symbols:: : nonnegative integer, : nonnegative integer, : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters, : solution and $\vartheta$ : differential operator
Referenced by:: §16.21, §16.21
Permalink:: http://dlmf.nist.gov/16.21.E1
Encodings:: TeX, pMML, png

where again $\vartheta=z\ifrac{d}{dz}$ . This equation is of order $\max(p,q)$ . In consequence of (16.19.1) we may assume, without loss of generality, that $p\leq q$ . With the classification of §16.8(i), when the only singularities of (16.21.1) are a regular singularity at and an irregular singularity at $z=\infty$ . When the only singularities of (16.21.1) are regular singularities at , $(-1)^{{p-m-n}}$ , and $\infty$ .

A fundamental set of solutions of (16.21.1) is given by

16.21.2 $\mathop{{G^{{1,p}}_{{p,q}}}\/}\nolimits\!\left(ze^{{(p-m-n-1)\pi i}};{a_{1},% \dots,a_{p}\atop b_{j},b_{1},\dots,b_{{j-1}},b_{{j+1}},\dots,b_{q}}\right),$ $j=1,\dots,q$ .

Symbols:: $\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;{a_{1},\dots,a_{p}\atop b_{1}% ,\dots,b_{q}}\right)$ : Meijer -function, : base of exponential function, : nonnegative integer, : nonnegative integer, : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.21.E2
Encodings:: TeX, pMML, png

For other fundamental sets see Erdélyi et al. (1953a, §5.4) and Marichev (1984).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 16.20 Integrals and Series 16.22 Asymptotic Expansions