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\input lathatmb
\begin{document}

%% {1} = {Title of Paper}
\head{Generalized Fractional Calculus \\ [4pt]
Special Functions and Integral Transforms
\footnote{Partially supported by Grant No ....}
}
%% {2} = Author(s) Name(s), if more than one authors,
%% names are numbered and the same for address at paper's end
{Virginia S. Kiryakova}
%% {3} = {Presented by ...}
{Presented by P. Kenderov}
%% {4} = {Short Title for the Running Heads}
{Generalized Fractional Calculus, \dots}
%% {5} = Author(s) Name(s), Initials only, without numbers
%% if more than one author
{V. Kiryakova}

\abst{
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 ...
}

%%%% Commands like "\vskip -18pt" below are only used to obtain
%%%% as short as possible sample version of a paper for "MB"
%%%% For macros not used here, see definitions in "lathatmb.tex",
%%%% like macros for theorems, proofs, lemmas, remarks, etc.

\bigskip
\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
%\vskip -18pt
$$
I f(x) = x^{\delta} \int \limits _0^1 \Phi (\sigma) \sigma^{\gamma}
f(x\sigma) d\sigma\ \ ;\ \ W f(x) = x^{\delta} \int \limits_1^{\infty}
\Phi({\frac 1 {\sigma}}) \sigma^{- \gamma -1} f(x\sigma) d\sigma
\eqno{(1.1)}
$$
%\vskip -10pt
\noindent
(Kalla [11]), where $\Phi (\sigma)$ is an arbitrary
elementary or special kernel-function ...

%\vskip -6pt

\defi{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 -16pt
$$
G_{p,q}^{m,n} \left[ \sigma \left|
\begin{array} {c}
(a_k)_1^p \\ (b_k)_1^q \\ \end{array} \right.\right] = \frac
1{2\pi i} \int\limits_{\cal{L}}{ \frac {\displaystyle \prod_{k=1}^m
\Gamma(b_k-s) \prod_{j=1}^n \Gamma\left(1-a_j+s\right)}
{\displaystyle \prod_{k=m+1}^q \Gamma(1-b_k+s) \prod_{j=n+1}^p
\Gamma\left(a_j-s\right)}} \sigma^s \, ds , \eqno{(1.2)} $$
%\vskip -14pt \noindent
where ....

%\vskip -6pt

\defi{Definition 1.2.} Let $m \ge 1$ be integer, $\beta > 0,
\gamma_1,...,\gamma_m$ and $\delta_1 \ge 0,$ $\dots,\delta_m \ge 0$
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 -15pt
$$
I_{\beta,m}^{(\gamma_k),(\delta_k)} f(x) = \int\limits_0^1
G_{m,m}^{m,0} \left[ \sigma \left|
\begin{array} {c}
(\gamma_k+\delta_k)_1^m \\ (\gamma_k)_1^m \\
\end{array}
\right.\right] f(x\sigma^{\frac 1 {\beta}}) \, d\sigma.
\eqno{(1.6)}
$$
%\vskip -12pt \noindent
Then, ....

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

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

%\vskip -4pt

\bth{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}
$ ...
\eth

\dok First we verify the correctness of ....
\dokend %%% to make a black box for the end of proof

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

%\vskip -6pt
.......
%\vskip -6pt

\bcor{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 -15pt
$$
_pF_p \left(b_1 + \eta_1, \dots, b_p + \eta_p;
b_1, \dots, b_p; x \right)
= Q_p (x) \left\{ \exp x\right\}.
\eqno{(3.18)}
$$
\ecor

%\vskip -15pt

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 $m = \beta = 2$,
$\gamma_{1,2} = \pm {\frac {\nu} 2}$  ...

%%%%%%%%%%%%%%% References

\sect{References}

\leftskip 2pc
\parindent -2pc

\dotfill    %%% here is only extract of two items...,
%%% the first for books and the second for articles in journals

[15] \sperr{V. Kiryakova.} {\it Generalized Fractional Calculus
and Applications}, Longman, Harlow, 1994.

[16] V. K\,i\,r\,y\,a\,k\,o\,v\,a.
All the special functions are fractional differintegrals of
elementary functions, {\it J. Phys. A: Math. \& Gen.},
{\bf 30}, No 14, 1997, 5085-5103.

\vskip 0.5cm

%%% Author's Address(es); if more than one authors,
%%% addresses are numbered like authors in beginning of paper

{\it
Institute of Mathematics and Informatics
\hfill  Received xx.xx.199x  %%% specified by Edit. Board

Bulgarian Academy of Sciences

Sofia 1090, BULGARIA

E-MAIL: virginia@math.acad.bg
}

\end{document}

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