share/contrib/meijer_g.tex

   1 \documentclass[11pt]{article}
   2 \usepackage{fullpage}
   3
   4 \title{The Meijer G-function\footnote
   5 {Copyright 2007 by Edmond Orignac.
   6 This file is released under the terms of the GNU General Public License, version 2.}}
   7
   8 \begin{document}
   9
  10 \maketitle
  11
  12 \section{Meijer G-function}
  13
  14 The maxima function \texttt{meijer\_gred(x,b,a,c,d)} where
  15 \texttt{x} is an expression, and \texttt{b,a,c,d} are lists
  16 attempts to express the Meijer G-function in terms of simpler
  17 functions, namely generalized hypergeometric functions, bessel
  18 functions and elementary transcendental functions. When no
  19 simplification can be performed, it
  20 returns \texttt{meijer\_g(x,b,a,c,d)}.
  21
  22
  23 \subsection{Some definitions}
  24
  25 The Meijer G-function is defined as a Mellin-Barnes integral\cite{meijer1951,braaksma1962,braaksma1975}:
  26
  27 \begin{equation}
  28   \label{eq:g-function-def}
  29   G^{mn}_{pq}\left(z,\begin{array}{c} a_1\ldots a_n;a_{n+1} \ldots a_p \\ b_1\ldots b_m;b_{m+1} \ldots b_q \end{array} \right) = \frac 1 {2i\pi} \int_C \frac{\prod_{k=1}^m \Gamma(b_k-s) \prod_{j=1}^n \Gamma(1-a_j+s)}{\prod_{k=m+1}^q \Gamma(1-b_k+s) \prod_{j=n+1}^p \Gamma(a_j-s)} z^s ds
  30 \end{equation}
  31
  32 Where the contour $C$ is closed in an appropriate way to ensure the convergence
  33 of the integral. It is also required that no $a_j-b_k$ is an integer.
  34
  35 Many integral transforms of special functions can be expressed in terms of Meijer G-functions.\cite{maloo1966,shah1972,bajpai1968,bajpai1974,dahiya1988}
  36
  37 The Meijer G-function has the following properties\cite{roach1997}:
  38
  39 It is invariant under any permutation of the parameters $a_1,\dots,a_n$ among themselves, any permutation of the parameters $b_1,\ldots, b_m$ among themselves, any permutation of the parameters $a_{n+1},\ldots a_p$ among themselves, and
  40 any permutation of the parameters $b_{m+1},\ldots, b_q$ among themselves as
  41 a trivial consequence of its definition. Furthermore, one has:
  42
  43 \begin{equation}\label{contin}
  44    G^{mn}_{pq}\left(z,\begin{array}{c} a_1\ldots a_n;a_{n+1} \ldots a_p \\ b_1\ldots b_m;b_{m+1} \ldots b_q \end{array} \right)= G^{nm}_{qp}\left(\frac 1 z,\begin{array}{c}1-b_1\ldots 1-b_m;1-b_{m+1} \ldots 1-b_q\\  1-a_1\ldots 1-a_n;1-a_{n+1} \ldots 1-a_p  \end{array} \right)
  45 \end{equation}
  46
  47 \begin{equation} \label{shift}
  48   G^{mn}_{pq}\left(z,\begin{array}{c} a_1+\alpha\ldots a_n+\alpha;a_{n+1}+\alpha \ldots a_p+\alpha \\ b_1+\alpha\ldots b_m+\alpha;b_{m+1}+\alpha \ldots b_q+\alpha \end{array} \right)=z^\alpha  G^{mn}_{pq}\left(z,\begin{array}{c} a_1\ldots a_n;a_{n+1} \ldots a_p \\ b_1\ldots b_m;b_{m+1} \ldots b_q \end{array} \right)
  49 \end{equation}
  50
  51 When no $b_j-b_k$ is an integer, the theorem of L.J. Slater allows the Meijer
  52 G-function to be transformed into a sum of ${}_pF_{q-1}$
  53 generalized hypergeometric functions.\cite{roach1997}
  54
  55 For $p<q$ or for $p=q$ and $|z|<1$ one has:
  56
  57 \begin{eqnarray}\label{slater}
  58 &&G^{mn}_{pq}\left(z,\begin{array}{c} a_1\ldots a_n;a_{n+1} \ldots
  59        a_p \\ b_1\ldots b_m;b_{m+1} \ldots b_q \end{array} \right)= \\[6pt]
  60 &&\sum_{k=1}^m  \frac{ z^{b_k} \prod_{j=1}^n \Gamma(1+b_k-a_j) \prod_{l=1\atop l\ne k}^m \Gamma(b_j-b_k)}{\prod_{j=n+1}^p \Gamma(a_j-b_k) \prod_{j=m+1}^q \Gamma(1+b_k-b_j)} {}_pF_{q-1} (\{1+b_k-a_j\};\{1+b_k-b_l\}_{l\ne k};(-1)^{m+n-p} z)\nonumber
  61 \end{eqnarray}
  62
  63 The case of $q>p$ or $p=q$ and $|z|>1$ can be treated by applying first (\ref{contin}) and then (\ref{slater}). The condition is then that
  64 there are no $a_j-a_k$ that are integer.
  65
  66 \subsection{What the maxima script does}
  67
  68 When one enters \texttt{meijer\_gred(x,b,a,c,d)} with \texttt{a=[a$_1$,\ldots,a$_n$]}, \texttt{b=[b$_1$,...b$_m$]}, \texttt{c=[a$_{n+1}$,...,a$_p$]}, \texttt{d=[b$_{m+1}$,...,b$_q$]}, the maxima script attempts to find a simplified expression for:
  69
  70 \begin{equation}
  71   G^{mn}_{pq}\left(x,\begin{array}{c} a_1\ldots a_n;a_{n+1} \ldots a_p \\ b_1\ldots b_m;b_{m+1} \ldots b_q \end{array} \right)
  72 \end{equation}
  73
  74 By applying first some simple identities,\cite{meijer1951,wille1988} and if that fails applying the Slater  formula (\ref{slater}) after checking that its condition of validity is met.
  75
  76 One can see that the order in which the parameters have to be entered is the following: bottom left, then top left, then top right, then bottom right, i. e. blocks of parameters are read in the anticlockwise sense.
  77
  78
  79 \subsection{Limitations}
  80
  81 No numerical calculation or asymptotic expansion can be
  82  performed by the script, although asymptotic expansions for Meijer G-functions are known.\cite{braaksma1962}
  83
  84 The differentiation formulas for the G-functions are not used
  85 nor defined by the script.\cite{roach1997} This is (in principle)
  86 easy to add with \texttt{gradef} for differentiation with respect to $z$
  87 but more difficult in case of differentiation with respect to the parameters.
  88
  89
  90 Relations between contiguous functions are not used by the script, but
  91 an algorithm exists to take advantage of these.\cite{roach1997}
  92 The relation (\ref{shift}) is not used in simplifications that are not
  93 based on Slater's formula.
  94
  95 Some identities for the G-function have been left out.\cite{meijer1932,meijer1935,meijer1939,meijer1951} Many more could be found in the Bateman Manuscript
  96 project.
  97
  98
  99
 100 \section{MacRobert's E-function}
 101
 102 The MacRobert E-function\cite{macrobert1958,macrobert1959c,macrobert1959b,macrobert1959a,macrobert1959,macrobert1960,macrobert1960a,macrobert1961,macrobert1961a,macrobert1962,macrobert1962a,ragab1962,ragab1993,ragab1962a,ragab1954} is also defined by a Mellin-Barnes integral.
 103 This integral is a particular case of the integral defining
 104 Meijer's G-function, and one has:
 105
 106 \begin{equation}
 107   E(a_1\ldots a_p;b_1\ldots b_q;z)=G^{p1}{q+1,p}\left(z,\begin{array}{c} 1;b_1\ldots b_q\\ a_1\ldots a_p \end{array} \right)
 108 \end{equation}
 109
 110 This identity is used by the function \texttt{macrobert\_ered(a,b,x)} to derive simplified forms of MacRobert's E-function.
 111
 112
 113
 114 %\bibliographystyle{acm}
 115 %\bibliography{maxima}
 116
 117 \begin{thebibliography}{10}
 118
 119 \bibitem{bajpai1968}
 120 {\sc Bajpai, S.~D.}
 121 \newblock A finite integrals involving a confluent hypergemoetric and a meijer
 122   g-function.
 123 \newblock {\em Portugaliae Mathematica 28\/} (1968), 55.
 124 \newblock Available online from National Library of Portugal.
 125
 126 \bibitem{bajpai1974}
 127 {\sc Bajpai, S.~D.}
 128 \newblock Expansion formula for the product of meijer g-function and bessel
 129   functions.
 130 \newblock {\em Portugaliae Mathematica 33\/} (1974), 35.
 131 \newblock Available online from National Library of Portugal.
 132
 133 \bibitem{braaksma1975}
 134 {\sc Braaksma, B.}
 135 \newblock In memoriam c. s. meijer.
 136 \newblock {\em Nieuw Archief voor Wiskunde 23\/} (1975), 95.
 137
 138 \bibitem{braaksma1962}
 139 {\sc Braaksma, B. L.~J.}
 140 \newblock Asymptotic expansion and analytic continuation for a class of
 141   mellin-barnes integrals.
 142 \newblock {\em Compositio Mathematicae 15\/} (1962--1964), 239.
 143 \newblock available from \texttt{http://www.numdam.org}.
 144
 145 \bibitem{dahiya1988}
 146 {\sc Dahiya, R., and Jowhar, I.}
 147 \newblock Two dimensional laplace transforms of generalized hypergeometric
 148   functions.
 149 \newblock {\em Internat. J. Math. \& Math. Sci. 2\/} (1988), 167.
 150
 151 \bibitem{macrobert1958}
 152 {\sc MacRobert, T.}
 153 \newblock Integrals involving e-functions.
 154 \newblock {\em Mathematische Zeitschrift 69}, 234 (1958).
 155 \newblock Available from texttt{http://www.digizeitschriften.de}.
 156
 157 \bibitem{macrobert1959c}
 158 {\sc MacRobert, T.}
 159 \newblock Infinite series of e-functions.
 160 \newblock {\em Mathematische Zeitschrift 71\/} (1959), 143.
 161 \newblock Available from \texttt{http://www.digizeitschriften.de}.
 162
 163 \bibitem{macrobert1959a}
 164 {\sc MacRobert, T.}
 165 \newblock Integrals of products of e-functions.
 166 \newblock {\em Mathematische Annalen 137\/} (1959), 412.
 167 \newblock Available from texttt{http://www.digizeitschriften.de}.
 168
 169 \bibitem{macrobert1959b}
 170 {\sc MacRobert, T.}
 171 \newblock The multiplication formula for the gamma function and e-function
 172   series.
 173 \newblock {\em Mathematische Annalen 139\/} (1959--1960), 133.
 174 \newblock Available from texttt{http://www.digizeitschriften.de}.
 175
 176 \bibitem{macrobert1960a}
 177 {\sc MacRobert, T.}
 178 \newblock Recurrence formulae for the e-functions.
 179 \newblock {\em Mathematische Zeitschrift 73\/} (1960), 254.
 180
 181 \bibitem{macrobert1962a}
 182 {\sc MacRobert, T.}
 183 \newblock Barnes integrals as a sum of e-functions.
 184 \newblock {\em Mathematische Annalen 147\/} (1962), 240.
 185 \newblock Available from \texttt{http://www.digizeitschriften.de}.
 186
 187 \bibitem{macrobert1962}
 188 {\sc MacRobert, T., and Ragab, F.}
 189 \newblock E-function series whose sums are constants.
 190 \newblock {\em Mathematische Zeitschrift 78\/} (1962), 231.
 191
 192 \bibitem{macrobert1959}
 193 {\sc MacRobert, T.~M.}
 194 \newblock product of e-functions.
 195 \newblock {\em Pacific Journal of Mathematics ??\/} (1959), 759.
 196 \newblock available from \texttt{http://www.projecteuclid.org}.
 197
 198 \bibitem{macrobert1960}
 199 {\sc MacRobert, T.~M.}
 200 \newblock product of e-functions.
 201 \newblock {\em Pacific Journal of Mathematics ??\/} (1960), 309.
 202 \newblock available from \texttt{http://www.projecteuclid.org}.
 203
 204 \bibitem{macrobert1961}
 205 {\sc MacRobert, T.~M.}
 206 \newblock Evaluation of an e-function when three of its upper parameters differ
 207   by integral values.
 208 \newblock {\em Pacific Journal of Mathematics ??\/} (1961), 999.
 209 \newblock available from \texttt{http://www.projecteuclid.org}.
 210
 211 \bibitem{macrobert1961a}
 212 {\sc MacRobert, T.~M.}
 213 \newblock Evaluation of an e-function when three of its upper parameters differ
 214   by integral values.
 215 \newblock {\em Pacific Journal of Mathematics ??\/} (1961), 999.
 216 \newblock available from \texttt{http://www.projecteuclid.org}.
 217
 218 \bibitem{maloo1966}
 219 {\sc Maloo, H.}
 220 \newblock Integrals involving products of bessel function and meijer's
 221   g-function.
 222 \newblock {\em Monatshefte f\"ur Mathematik 70\/} (1966), 127.
 223 \newblock available from \texttt{http://www.digizeitschriften.de}.
 224
 225 \bibitem{meijer1932}
 226 {\sc Meijer, C.~S.}
 227 \newblock Asymptotische entwicklungen von besselschen und hankelschen
 228   funktionen f\"ur grosse...
 229 \newblock {\em Math. Ann. 108\/} (1932), 21.
 230 \newblock available from \texttt{http://www.digizeitschriften.de}.
 231
 232 \bibitem{meijer1939}
 233 {\sc Meijer, C.~S.}
 234 \newblock Integraldarstellung fur struvesche und besselche funktionen.
 235 \newblock {\em Compositio Mathematicae 6\/} (1939), 348.
 236 \newblock available from \texttt{http://www.numdam.org}.
 237
 238 \bibitem{meijer1935}
 239 {\sc Meijer, C.~S.}
 240 \newblock Neue integraldarstellungen aus der theorie der whittakerschen und
 241   hankelschen funktionen.
 242 \newblock {\em Mathematische Annalen 112\/} (1939), 469.
 243 \newblock available from \texttt{http://www.digizeitschriften.de}.
 244
 245 \bibitem{meijer1951}
 246 {\sc Meijer, C.~S.}
 247 \newblock Neue integraldarstellung fur besselche funktionen.
 248 \newblock {\em Compositio Mathematicae 8\/} (1951), 49.
 249 \newblock available from \texttt{http://www.numdam.org}.
 250
 251 \bibitem{ragab1954}
 252 {\sc Ragab, F.}
 253 \newblock New integrals involving bessel-functions.
 254 \newblock {\em Mathematische Zeitschrift 61\/} (1954), 386.
 255 \newblock Available from \texttt{http://www.digizeitschriften.de}.
 256
 257 \bibitem{ragab1962}
 258 {\sc Ragab, F.~M.}
 259 \newblock Neue integraldarstellungen aus der theorie der macrobertschen und
 260   besselschen funktionen.
 261 \newblock {\em Mathematische Zeitschrift 79\/} (1962), 147.
 262
 263 \bibitem{ragab1962a}
 264 {\sc Ragab, F.~M.}
 265 \newblock Integrals involving products of bessel functions.
 266 \newblock {\em Mathematische Zeitschrift 80\/} (1962--1963), 177.
 267
 268 \bibitem{ragab1993}
 269 {\sc Ragab, S.~F.}
 270 \newblock Finite and infinite series involving macrobert's e-function.
 271 \newblock {\em Portugaliae Mathematica 50\/} (1993), 507.
 272 \newblock Available online from National Library of Portugal.
 273
 274 \bibitem{roach1997}
 275 {\sc Roach, K.}
 276 \newblock Meijer g-function representations.
 277 \newblock In {\em ISSAC'97 proceedings\/} (New York, 1997), ACM.
 278 \newblock Available from CITESEER.
 279
 280 \bibitem{shah1972}
 281 {\sc Shah, M.}
 282 \newblock On generalized meijer's and associated generlaized legendre
 283   functions.
 284 \newblock {\em Portugaliae Mathematica 31\/} (1972), 55.
 285 \newblock Available online from National Library of Portugal.
 286
 287 \bibitem{wille1988}
 288 {\sc Wille, L.~T.}
 289 \newblock A novel class of g-function integrals.
 290 \newblock {\em Journal of Mathematical Physics 29\/} (1988), 599.
 291
 292 \end{thebibliography}
 293
 294 \end{document}