% This file ``txsample.tex''is a sample \AmSTeX file
% for papers in the Proceedings "TM \& SF, Varna'96"
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentstyle{amsppt}
\magnification=1200  %this means all fonts are magnified 1.2 times
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\hcorrection{2truemm}
\baselineskip=18truept
\parindent=18truept
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\font\tenextrab=msbm10
\def\Bbb#1{\text{{\tenextrab #1}}} % or as below
%\def\Bbb#1{{\pmb{#1}}}
% ------------- macro \fnote for acknowledgements footnote
\catcode`@=11    %to allow use of @ as a letter; reset at the end
\def\fnote#1#2{\let\@sf\empty % parameter #2 (the text) is read later
  \ifhmode\edef\@sf{\spacefactor=\the\spacefactor}\/\fi
  {$\,^{#1}$}\@sf\vfootnote{#1}{#2}}
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  \textindent{$\,^{#1}$}#2 \footstrut\vskip 3truept \egroup}
% ----------- layout definitions (titles, theorems, proofs etc)
% due to magnification 1200 \bigskip has been changed to \medskip

% use macro \title to make sections' titles
% like this: \title{\centerline{1. Introduction}}
\def\title#1\par{\vskip 18truept
                 {\bf #1}
                 \vskip 12truept}
\long\outer\def\proposition#1 #2 \par{\medbreak\medskip
    {{\sc Proposition #1.}\enspace {\sl #2}}\par
}
% use macros \proposition for making theorems, lemmas, etc.
% example: \theorem{2.1} Each multiple....
% gives: \par...\bigskip {\sc Theorem 2.1.}\enspace {\sl Each ...}
\long\outer\def\theorem#1 #2 \par{\medbreak\medskip
    {{\sc Theorem #1.}\enspace {\sl #2}}\par
}
\outer\def\lemma#1 #2\par{\medbreak\medskip
    {{\sc Lemma #1.}\enspace {\sl #2}}\par
}
\outer\def\corollary#1 #2 \par{\medbreak\medskip
    {{\sc Corollary #1.}\enspace {\sl #2}}\par
}
% use macros \definition, \example to get:
% \bigskip {\sc Definition 1.2.}\enspace Let $m\ge1$ be...
\long\outer\def\definition#1{\medbreak\medskip
    {\sc Definition #1.}\enspace
}
\long\outer\def\example#1{\medbreak\medskip
    {\sc Example #1.}\enspace
}
% macro \proof
\outer\def\proof{ P r o o f.\enspace}

% ========== END OF DEFINITIONS ===============================
% It follows a part of a sample paper ....
% Attention: Some of the equations ans paragraphs here are surrounded
% by \vskip-*pt, \vskip-*pt to get negative vertical breaks
% ONLY in order to shorten the present sample to two output pages.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\document

% the macro below is to make heading ``TM \& SF, Varna'96''...
\vbox to 4.5truecm {
{\bf
{\head
  \centerline{Transform Methods \& Special Functions, Varna'96}
  \centerline{}
}
\centerline{Proceedings of Second International Workshop, 23 - 30
August 1996}
\centerline{}
\vskip 5truemm
\hrule
}
\vfill
}

% If you have problems with above macro for the heading,
% omit it and produce an empty space of 4.5 cm for the heading
% by the next line command:
%\vbox to 4.5truecm {}

{\bf
\centerline{GENERALIZED FRACTIONAL CALCULUS,}
\vskip 6truept
\centerline{SPECIAL FUNCTIONS AND INTEGRAL TRANSFORMS}
}

\vskip 18truept

\centerline{V. S. Kiryakova$^*$}

\fnote {} {$^*$ Partially supported by the Ministry of Science
and Education under Project MM 433$\slash$94.}

\title{
\centerline{Abstract}
}


In this survey paper we review the main ideas, results and
applications of a {\it generalized fractional calculus\/} developed
in the author's monograph [16]. This generalization of the classical
fractional calculus is based on the {\it essential use of the special
functions (Meijer's $G$- and $H$-functions)} as kernel-functions ...

\vskip 6truept

{\it Mathematics Subject Classification}: 26A33 (main), 33C, 44

{\it Key Words and Phrases}: fractional calculus,
Meijer's $G$- and Fox's $H$-functions

\vskip 6truept

\title{
\centerline{1. Introduction}
}

The generalized fractional calculus presented here is based on
the notion of {\it generalized operators of fractional
integration\/} of Riemann-Liouville and Weyl type
\vskip -22pt
$$
I f(x) = x^{\delta} \int \limits _0^1 \Phi (\sigma) \sigma^{\gamma}
f(x\sigma) d\sigma\ \ ;\ \ W f(x) = x^{\delta} \int \limits_1^{\infty}
\Phi({\frac 1 {\sigma}}) \sigma^{- \gamma -1} f(x\sigma) d\sigma
\tag{1.1}
$$
\vskip -12pt
\noindent
(Kalla [11]), where $\Phi (\sigma)$ is an arbitrary
elementary or special kernel-function ...

\vskip -6pt

\definition{1.1} (see ...) By a {\it Meijer's $G$-function \/}
we mean the generalized hypergeometric function defined by means
of the contour integral
\vskip -14pt
$$
G_{p,q}^{m,n} \left[ \sigma \left|
\matrix (a_k)_1^p \\ (b_k)_1^q \endmatrix
\right.\right] =  \frac 1{2\pi i} \int\limits_{\Cal{L}}
\frac {\prod_{k=1}^m \Gamma(b_k-s)
\prod_{j=1}^n \Gamma\left(1-a_j+s\right)}
{\prod_{k=m+1}^q \Gamma(1-b_k+s)
\prod_{j=n+1}^p \Gamma\left(a_j-s\right)} \sigma^s \, ds ,
\tag{1.2}
$$
\vskip -16pt \noindent
where ....

\vskip -6pt

\definition{1.2} Let $m \ge  1$ be integer, $\beta > 0,
\gamma_1,...,\gamma_m$ and $\delta_1 \ge 0,...,\delta_m \ge 0$
be arbitrary real numbers. By  a {\it generalized
(multiple) Erd\'elyi-Kober operator of integration\/}
of multiorder $\delta = (\delta_1,...,\delta_m)$
we mean an integral operator
\vskip -17pt
$$
I_{\beta,m}^{(\gamma_k),(\delta_k)} f(x) = \int\limits_0^1
G_{m,m}^{m,0} \left[ \sigma \left|
\matrix (\gamma_k+\delta_k)_1^m \\ (\gamma_k)_1^m \endmatrix
\right.\right] f(x\sigma^{\frac 1 {\beta}}) \, d\sigma.
\tag{1.6}
$$
\vskip -15pt \noindent
Then, ....

\title{
\centerline{2. Basic results of the generalized fractional calculus}
}

The main {\it functional spaces} ....

\vskip -15pt

\theorem{2.1} Each multiple E.-K. fractional integral (1.6)
preserves the power functions in $C_{\alpha}, \alpha \ge
\max\limits_k \left[-\beta\left(\gamma_k + 1\right)\right]$
up to a constant multiplier:
\vskip -20pt
$$
I_{\left(\beta_k\right), m}^{\left(\gamma_k\right), \left(\delta_k\right)}
\left\{x^p\right\} = c_p x^p,\ p > \alpha, \quad \text { where }\
c_p = \prod\limits_{k=1}^m {\frac {\Gamma \left(\gamma_k +
{\frac p {\beta_k}} + 1\right)}
{\Gamma \left(\gamma_k + \delta_k + {\frac p {\beta_k}} + 1\right)}}
\tag{2.1}
$$
\vskip -15pt   \noindent
and it is an invertible mapping
$
I_{\left(\beta_k\right), m}^{\left(\gamma_k\right), \left(\delta_k\right)}:
C_{\alpha} \longrightarrow C_{\alpha}^{\left(\eta_1 + \dots + \eta_m\right)}
\subset C_{\alpha}
$ ...

\proof First we verify the correctness of ....


\title{
\centerline{3. Applications to the generalized hypergeometric functions}
\centerline{and Laplace type integral transforms}
}

\vskip -6pt
.......
\vskip -16pt

\corollary{3.5}
Let all the differences $a_k - b_k = \eta_k, k=1,\dots,p$ be
nonnegative integers. Then, the differintegral operator in (3.15)
turns into a differential operator $D_{\eta}$ of integer order
$\eta = \eta_1 + \dots + \eta_k \ge 0$ and of form (1.12), namely:
\vskip -4pt
$$
_pF_p \left(b_1 + \eta_1, \dots, b_p + \eta_p; b_1, \dots, b_p;
x \right) = Q_p (x) \left\{ \exp x\right\}.
\tag{3.18}
$$
\vskip -5pt

Differential representation (3.18) gives an example how differential
formulas for the ``spherical'' g.h.f-s introduced in [16] can be
used for explicit calculation ...

\vskip -8pt

\example{3.8} In particular, for $m = \beta = 2$,
$\gamma_{1,2} = \pm {\frac {\nu} 2}$  ...

\title{
\centerline{References}
}

\leftskip 2truepc
\parindent -2truepc

% example for article
[15] \ {\hbox {V. S. K i r y a k o v a}}, Poisson and Rodrigues
type differential formulas for the $_pF_q$-functions. {\it Atti
Sem. Mat. Fis. Univ. Modena} {\bf 39} (1990), 311-322.

% example for book
[16] \ {\hbox{V. S. K i r y a k o v a}}, \
{\it Generalized Fractional Calculus and Applications}.
Longman, Harlow (1994).

\vskip 1.5truecm

$^{*}$) {\it address follows in 2-3 lines ... }

%example:

% $^{*}$) {\it Institute of Mathematics, Bulgarian Academy of Sciences}

% {\it 1090 Sofia -- BULGARIA}

% {\it e-mail: virginia@math.acad.bg}

\enddocument
\bye
%======== END OF FILE ``txsample.tex''========================