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{\rm \centerline{V. S. Kiryakova}}
{\rm \centerline{GENERALIZED FRACTIONAL CALCULUS \dots}}
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\head{GENERALIZED FRACTIONAL CALCULUS, \\ [6pt]
SPECIAL FUNCTIONS AND INTEGRAL TRANSFORMS }

\authors{Virginia S. Kiryakova$^*$ \& Co-Authors$^{**}$}

\fnote {} {$^*$ Partially supported by the Ministry of
Science and Education under Project MM 433$\slash$94;
$^{**}$ Supported by \dots} % if the co-authors are supported by ...

\sect{Abstract} %%% Abstract follows

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

\vskip 6pt

{\it Mathematics Subject Classification}: 26A33 (main), 33C60, 44A20

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

\vskip 3pt

\sect{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
$$
I f(x) = x^{\delta} \int \limits _0^1 \Phi (\sigma) \sigma^{\gamma}
f(x\sigma) d\sigma\ \ ;\ \ W f(x) = x^{\delta} \int \limits_1^{\infty}
\Phi({\frac 1 {\sigma}}) \sigma^{- \gamma -1} f(x\sigma) d\sigma
\eqno{(1.1)}
$$
\noindent
(Kalla [11]), where $\Phi (\sigma)$ is an arbitrary
elementary or special kernel-function ...

%\nopagebreak  \vspace*{-6pt}

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

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

\eqno{(1.6)}
$$
Then, ....

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

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

\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 -18pt
$$
I_{\left(\beta_k\right), m}^{\left(\gamma_k\right), \left(\delta_k\right)}
\left\{x^p\right\} = c_p x^p,\
p > \alpha, \quad  {\rm where\ } c_p = \prod\limits_{k=1}^m
{\frac {\Gamma \left(\gamma_k + {\frac p {\beta_k}} + 1\right)}
{\Gamma \left(\gamma_k + \delta_k + {\frac p {\beta_k}} + 1\right)}}
\eqno{(2.1)}
$$
\vskip -8pt \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 ....

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

$\dots$
\vskip -25pt

\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:
$$
_pF_p \left(b_1 + \eta_1, \dots, b_p + \eta_p;
b_1, \dots, b_p; x \right)
= Q_p (x) \left\{ \exp x\right\}.
\eqno{(3.18)}
$$

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 ...

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

\sect{References}

\leftskip 2pc
\parindent -2pc

[1] \ {\hbox {F i r s t  A u t h o r}}, Title of paper.
{\it Journal \/} {\bf xx}, No x (19xx), xxx-xxx.

\dots

[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.

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

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\it

$^{*}$) Institute of Mathematics \& Informatics
\hfill  Received: Month, Date, Year

Bulgarian Academy of Sciences

1090 Sofia -- BULGARIA

e-mail: virginia@math.bas.bg

\vskip 0.5cm

$^{**}$ Institute of \dots, Town

COUNTRY

e-mail: \dots
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