\documentclass[oneside]{article}
\usepackage{amsmath}
\renewcommand{\baselinestretch}{1.2}
\pagestyle{myheadings}
\hoffset=-0.55cm
\voffset=-0.8cm
\oddsidemargin=-0.4mm
\textwidth=16cm
\topmargin=-1.9mm
\headheight=0.4cm
\headsep=0.5cm
\textheight=25cm
\makeatletter
\renewcommand{\@oddhead}{\hfil --- \large{\thepage} --- \hfil}
\renewcommand{\@oddfoot}{}
\makeatother
\newcommand{\p}[2]{\genfrac{}{}{0pt}{}{#1}{#2}}
\newcommand{\de}{\delta}
\newcommand{\be}{\beta}
\newcommand{\al}{\alpha}
\newcommand{\dsp}{\displaystyle}
\newcommand{\an}[2]{\langle #1 | #2 \rangle}
\newcommand{\ga}{\gamma}
\newcommand{\tz}{\genfrac{}{}{0pt}{}{;}{;}}
\newcommand{\dv}{\genfrac{}{}{0pt}{}{:}{:}}
\begin{document}
\thispagestyle{empty}
\vspace*{1.5cm}
\begin{center}
{\large{\bf O r d e r\quad o f \quad L e n i n}} \\[1mm]
{\large{\bf KELDYSH INSTITUTE OF APPLIED MATHEMATICS}} \\[1mm]
{\large{\bf R u s s i a n \quad A c a d e m y \quad o f \quad
S c i e n c e }} \\
\vspace{2cm}
{\Large
A.W. Niukkanen\\[1mm]
I.B. Shchenkov, G.B. Efimov}\\
\vspace{1.5cm}
{\LARGE
OPERATOR FACTORIZATION\\
TECHNIQUE OF FORMULA\\
DERIVATION IN THE THEORY\\
OF SIMPLE AND MULTIPLE\\
HYPERGEOMETRIC\\
FUNCTIONS OF ONE AND\\
SEVERAL VARIABLES\\
}
\vspace{2cm}
{\large{\bf
\vspace{2cm}
Moscow -- 2003}}
\end{center}
\newpage
{\Large
\noindent
А.В. Ниукканен, И.Б. Щенков, Г.Б. Ефимов.
{\bf Метод операторной факторизации и новая техника вывода формул
в теории гипергеометрических рядов от одного и нескольких
переменных}\\[-6mm]
{\bf Аннотация}
Показано, что вычислительные приемы метода операторной
факторизации дают простую и универсальную основу для новой теории
гипергеометрических рядов от любого числа переменных. Даны
примеры, показывающие как работает метод на практике. Обсуждены
перспективы развития метода, включая необходимую модернизацию
системы Сантра~2. Дан предварительный анализ возможности
использования надстройки Сантра~3 над алгоритмическим языком
Рефал в качестве основы для создания глобально-универсальной
программы, способной выполнять полный набор операций,
составляющих ядро метода факторизации.
Работа поддержана Российским фондом фундаментальных
исследований, грант 03-01-00708.\\[-4mm]
\noindent
A.W. Niukkanen, I.B. Shchenkov, G.B. Efimov.
{\bf Operator factorization technique of formula derivation in
the theory of simple and multiple hypergeometric functions of one
and several variables} \\[-6mm]
{\bf Abstract}
It is shown that computation technique of the operator
factorization method provides a simple and universal foundation
for a new theory of hypergeometric series in any number
of variables. Examples showing how the method works in practice
are given. We also discuss the prospects of the method including
the necessary modernization of Santra~2 system. We also give a
preliminary analysis of the potentialities of using the
superstructure Santra~3 over the Refal language as a basis for
computer implementation of a globally universal program capable
to perform the complete set of operations inherent in the
factorization method.
The work was supported by Russian Foundation for Basic Research
under grant 03-01-00708.
}
\newpage
\begin{center}
{\Large{\bf Contents }}
\end{center}
%\vspace{0.2cm}
{\Large
\noindent
\lefteqn{{\bf 1.\,Introduction
}
\quad........................................................................%
.............
}\hspace{16cm}4 \\[-6mm]
\noindent
\lefteqn{{\bf 2.\, General \;\, outline \;\, of \;\, the \;\,
problem}
\quad...................................................
}\hspace{16cm}6\\[-4mm]
\noindent
\hspace*{1cm}\parbox{14cm}{
2.1.\,Researcher and computer\,
2.2.\,Sturm und Drang?\,
2.3.\,What kind of knowledge do we need\,
2.4.\,The NIST project: what is about computer algebra\,
2.5.\,Underestimated role of symbolic computing\,
2.6.\,Misuse of symbolic manipulation
} \\[-1mm]
\noindent
\lefteqn{{\bf 3.\, Hypergeometric \;\, series}
\quad...................................................................
}\hspace{16cm}8\\[-4mm]
\noindent
\hspace*{1cm}\parbox{14cm}{
3.1.\,Where the series arise\,
3.2.\,Why we can not use "standard" methods\,
3.3.\,Hypergeometric series: notation
} \\ [-2mm]
\noindent
\lefteqn{{\bf 4.\, Operator \;\, factorization \;\,
method}
\quad....................................................
}\hspace{16cm}11\\[-4mm]
\noindent
\hspace*{1cm}\parbox{14cm}{
4.1.\,$\Omega$-multiplication is a fundamental
operation underlying the factorization method\,
4.2.\,Operator factorization principle: an illustrative example
of general power series\,
4.3.\,$\Omega$-representability of multiplication of
power series coefficients\,
4.4.\,Illustrative example of total factorization\,
4.5.\,Factorization formulas\,
4.6.\,General concepts, which constitute the structural
basis of the method\,
4.7.\,How the method works\,
4.8.\,Different computational modes
}\\[-1mm]
\noindent
\lefteqn{{\bf 5.\, Formula \;\, derivation \;\, based \;\, on \;\,
the \;\, use \;\, of \;\, the \;\, operator \;\, }}\\
\hspace*{4mm}\lefteqn{{\bf fac\-to\-ri\-zation \;\, method}
\quad......................................................................
}\hspace{15.5cm}16\\[-4mm]
\noindent
\hspace*{1cm}\parbox{14cm}{
5.1.\,Passing to examples\,
5.2.\,Transformation of $F_4$\,
5.3.\,Analytical corollaries
}\\[-2mm]
\noindent
\lefteqn{{\bf 6.\, Computer \;\, generation \;\, of \;\,
formulas \;\, using \;\, bounded \;\, uni- \;\,}}\\
\hspace*{4mm}\lefteqn{{\bf ver\-sal programs}
\quad....................................................................%
.............
}\hspace{15.5cm}19\\[-3.5mm]
\noindent
\hspace*{1cm}\parbox{14cm}{
6.1.\,Programs\,
6.2.\,Example of computer generation: the case of
the Appell function $F_4$
} \\[-2mm]
\noindent
\lefteqn{{\bf 7.\,Prospects \;\, of \;\, the \;\,
globally-universal \;\, approach \;\, within \;\, the \;\,}}\\
\hspace*{4mm}\lefteqn{{\bf system \;\, of \;\, analytical \;\,
transformations \;\, Santra~3}
\quad.........................
}\hspace{15.5cm}21\\[-3.5mm]
\noindent
\lefteqn{{\bf Summary }
\quad.............................................................%
.................................
}\hspace{15.9cm}24\\[-4mm]
\noindent
\lefteqn{{\bf References
\quad.............................................................%
...............................
}}\hspace{15.9cm}25
}
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ref2
{\Large
\begin{center}
{\Large {\bf 1.\,Introduction \footnotemark[1]{ }}}
\end{center}
\footnotetext[1]{\large
The work was supported by Russian Foundation of
Basic Research under Grant 03-01-00708
}
%%%%%%%%%%%%%%%%%%%%%%%% subject3
Mathematical models which play the role of theoretical
substitutes for underlying natural or technological phenomena
including those under supervision of computer--aided control
systems give us an information about the phenomena in terms
of functions having specialized structures. The vast majority,
up to 95\%, of the special functions involved in description of
fundamentally important processes are connected with
hypergeometric series which thereby give us a key to a
substantial part of applied mathematics really needed by numerous
users.
There is a further simple motive for our interest to the
hypergeometric series. Following elementary functions these
series present the most important class of functions naturally
arising from simple operations over elementary functions. It
suggests that along with elementary (EL) functions hypergeometric
(HYP) series will inevitably turn into an obligatory component
of modern software especially for the coming computer generation
with prevailing role of "intellectual" interactive symbolic
(SYMB) analysis over purely numerical (NUM) computations. Looking
at the vertices (EL, NUM), (HYP, NUM), (EL, SYMB) and (HYP, SYMB)
in functions-methods space one can say, loosely, that the first
vertex pertains to the past, the next two symbolize the present
and the last one relates to the future. It is just the
future the present work is aimed at. For better visualization of
what was said above we remind that Lozier and Olver maintain that
even when one moves from (EL, NUM) to (HYP, NUM) "enormous gaps
remain for functions having variable parameters in addition to
the argument". On our move from (EL, SYMB) to (HYP, SYMB) the
gaps would have been "twice as enormous" if we had not had the
operator factorization method in our disposal.
{\bf The operator factorization method} \cite{fpm1}--\cite{nima}
greatly
facilitating solution of many mathematical problems including
the study of multiple hypergeometric series is
the main object of our interest aimed at verification of
possibility to use the main operations of the method for
construction a superstructure over the system of analytical
transformations SANTRA \cite{1}--\cite{prep2}.
The check--up is based on a detailed
overveiw of the formal structure of the method. We show that
the method uses a {\bf new fundamental operation} over power
series, {\bf new analytical technique} and {\bf new form of
presenting results}. There is a strong evidence that the
set of the basic operations of the method is complete within
a wide range of problems including analysis of multiple
hypergeometric series.
%The theory of hypergeometric series and
%special functions does not only have now its own subject but
%also disposes of an investigation method of its own.
In Sec.2 we formulate some general statements which being
trivial on their own can help us to better understand the
position occupied by this method amongst other scientific
methods.
In Sec.3 we describe a compact and transparent
notational system that allows the structure of any multiple
hypergeometric series to be comprehended easily in full detail.
In Sec.4 a brief overveiw of the operator factorization method
is given. An $\Omega$-multiplication operation, factorization
formulas and general concepts underlying the method are discussed
in brief. It is shown that the method permits us to use different
analytical schemes as a basis for working out different
computational algorithms.
The main factor
determining the value of the method is that it makes use of {\bf
a limited set of operations} (see Sec.2 in Ref.\cite{prep2})
sufficient to derive
any formula in the theory of simple and multiple hypergeometric
series. Moreover, as compared with indefinitely large number of
conventional approaches this gives us the most simple and direct
way to the desired result. Technically, there are 3
computational modes to reach the goal. First, we can use the
set of operations manually, with pen and paper, to cover
all the way to the final result without use of computer (see
an example in Sec.5). Second, we can make an attempt to use the
manual mode to obtain a set of macro--operations playing the role
of basis relations for a given class of formulas and then
generate all relations belonging to the class by computer--aided
combining of the basis macro--operations. An example of such
"bounded--universal" approach is given in Sec.6.
Up to now the set of main operations was not implemented in the
form of computer commands. Numerous examples of successful
applications of the first and the second computational modes
suggest that an interactive program using the complete set of
operations of factorization method may give us a globally
universal computer--aided "formula constructor" playing the role
of a central core with respect to the partially universal
programs and allowing the researcher to obtain the desired
relation without any cease of the user session. Preliminary
examination of feasibility of the globally universal
computational mode with the help of the system of analytical
transformations SANTRA is one of the main goal of the present
work (see Sec.7).
%%%%%%%%%%%%%%%%%%%%%%%%%% human4
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ref2
\begin{center}
{\Large {\bf 2.\,
General outline of the problem}}
\end{center}
\noindent
{\large {\bf 2.1
Researcher and computer} }
The main body of scientific knowledge had been built before
computer became a full member of scientific process. That's why
scientific knowledge generated by centuries of human
intellectual activity bears an indelible imprint of human
ingenuity and human failings. Both these extremes put obstacles
in our way to computerizable scientific knowledge.
Putting
aside innumerable minor human drawbacks we stress that
{\bf results} are most important from {\bf an\-tro\-po\-cent\-ric}
standpoint. On the contrary the {\bf derivation rules}
play the primary role for {\bf computercentric} science. This
difference ser\-ves as the
main obstacle for teaching computer human tricks. \\[-5mm]
\noindent
{\large{\bf 2.2. Sturm und Drang?} }
After principles governing a given domain of science begin
to work their way there comes the time of {\it Sturm und Drang}
epoch. The {\bf negative results of this impetuous activity}
being out of any reasonable control are not at all less than
positive results.
The most grave consequence of the rush is that the domain
being allegedly conquered by science is in fact conquered by
{\bf narrow layer of elite} capable to discern the main
constitutive features of a new theory uprising from the primary
intellectual chaos. As for the scientific community as
a whole and even for the most part of the "elite" the domain
transforms from a "blank spot" into a "black spot" of
intellectual disorder. Only deep systematic revision of the
domain may give us a hope to rectify the situation. A good theory
is a simple theory requiring a will and industry
rather than great abilities. \\[-5mm]
\noindent
{\large{\bf 2.3. What kind of knowledge do we need
}}
Specialization divides knowledge into scores of subfields
all having methods of their own. The knowledge we need is that
which helps different fields to combine and merge rather than
come apart and disintegrate.
Mathematics is the language of Science. Analysis is the heart
of mathematics and the concept of function is the heart of analysis.
Therefore the {\bf mathematical reference data} containing the
most commonly used knowledge seems to present primary interest
for researchers.
Computer-aided accumulating, processing and generating of
mathematical knowledge varies from passive data bases to
sophisticated knowledge bases which can produce information not
present explicitly in the bases. Unfortunately, the use of
computer algebra methods which can help us to
build effective knowledge bases is limited by the fact that we
meet a severe want of new sufficiently simple and
universal analytical methods which would give us an effective
basis for {\bf symbolic manipulations}. \\[-5mm]
\noindent
{\large{\bf 2.4. The NIST project: what is about computer algebra }}
Several years ago the National Institute of Standards
and Technology (NIST, USA) initiated the project of a Digital
Library of Mathematical Functions (DLMF) \cite{loz}.
In an earlier web description of the project
(http:/\!/math.nist.gov/Di\-gi\-talMathLib/\!/pub\-li\-ca\-ti\-ons/%
nistir6297/) there had been stressed that
"standardization of mathematical knowledge requires
growing use of {\bf symbolic} and numerical software", "in the
intervening decades {\bf computer algebra
and symbolics} have come into wide use" and
"many users will want easy-to-use support ... in
numerical and {\bf symbolic} computation".
Then it is said that "the role of symbolic
computation in the DLMF is still being discussed"\,(!).
The statement of P.~Paule (DLMF associate editor)
that "the goal is a presentation of computer algebra
concepts" with "a new chapter on computer algebra", in view of
what was said above, sounds quite unexpectedly.
{\bf Where has the computer algebra gone?} \\[-5mm]
\noindent
{\large{\bf 2.5.
Underestimated role of symbolic computing }}
Without {\bf intensive practical use of computer algebra} methods
any reference database would lose most of its value.
The less objects are included in the base (34 in DLMF!) the less
interest it presents for user. On the contrary, the giant
book-like site would be crammed with misprints and authors'
mistakes. Data level in the field of multiple hypergeometric
series may exceed the capacity of a medium-size website. The
giant site would confront with cross-referencing and search
difficulties.
{\bf The knowledge created by computer at will of user is what we
should aim at}. \\[-5mm]
\noindent
{\large{\bf 2.6. Misuse of symbolic manipulation }
}
An attempt to attack the problem of symbolic manipulation
of {\bf simple} hypergeometric series by brute force method
has been
undertaken by C.~Krattenthaler in his HYP and HYPQ packages.
The results do not seem to completely
justify the efforts. The formal combination of occasional
formulas can hardly serve us as a reliable way
to {\it interesting} new results. Moreover the efforts needed to
obtain the new formulas seem to be rather tedious. "The
philosophy of this package is: {\it Do it yourself}\,!\," The
disappointing idea implies that "you should be able to {\it
control each step} in a series of manipulations by yourself"
What we really need is {\bf an integrated complete set of
commands} rather than a disorderly collection of {\bf occasional
tools}.
\begin{center}
{\Large {\bf 3.\,
Hypergeometric series}}
\end{center}
\noindent
{\large{\bf 3.1. Where the series arise
}}
Hypergeometric series is probably what an applied
ma\-thematician or a theoretical physicist {\bf more often runs
into} when making his calculation work. Sometimes it happens
{\bf even without knowing it}!
The hypergeometric series are ubiquitous.
They appear, under different guises, in elementary
functions, differential equations, heat conduction, solid state
dynamics, hydraulics, atomic and molecular physics, quantum
mechanics, elementary particles (Feynman diagrams),
combinatorics, group representations, algebraic geometry, etc.
The late corresponding member of Academy of Science of the USSR,
professor K.I. Babenko counted up to 1500 special function of
hypergeometric type.
I.M.~Gelfand and his co-authors remark that {\bf hypergeometric
series play an outstanding unifying role in science} and assert
that "impetuous development of the theory of hypergeometric
functions begun in the eighties of present century". In course
of time the role of hypergeometric functions will become more
significant due to their state--of--the--art and, especially,
potential importance for applied sciences and engineering.
\\[-5mm]
\noindent
{\large{\bf 3.2. Why we can not use "standard" methods
}}
The present level of standard methods in the theory of
hypergeometric series can be clearly seen from Richard Askey's
statement made in his preface to "Special Functions" (Reidel,
1984). He wrote "There are {\bf many examples} (of special
functions - A.N.) {\bf and no single way} of looking at them that can
illuminate all examples or even all the important properties of
a single example of a special function".\\[-4mm]
The Askey's statement does not hold any more. The operator
{\bf factorization method just gives us a single way} of looking at
scores of thousands of special functions and multiple
hypergeometric series. Moreover {\bf it allows us to "computerize" the
theory} of these functions in a two-fold way. Our main goal is to
discuss these ways and their relation to the existing computer -
aided approaches to accumulating, processing and generating of
scientific knowledge. By the example of the operator factorization
method one can see one of the ways making the structure of
knowledge easily accessible to computer. \\[-5mm]
\noindent
{\large{\bf 3.3. Hypergeometric series: notation
}} \\[-3mm]
\noindent
{\bf 1.
Generalized hypergeometric series in one variable of type
$A/\!/B$:}
$$
F\left[\p{a^1,\ldots,a^A}{b^1,\ldots,b^B}\p{;\,x}{}\right] =
{\,} \sum_{i=0}^{\infty}
\frac{(a^1,i)\,\ldots\,(a^A,i)}{(b^1,i)\,\ldots\,(b^B,i)}
\:\frac{x^i}{i\,!}
$$\\[-0.4cm]
where Pochhammer symbol $(a,i)$ is
$$
(a,i) = a(a+1)\cdots(a+i-1) = \Gamma(a+i)/\Gamma(a).
$$
{\bf 2. Contracted notation}
$$
F^A_B[{\bf d};x]
=\sum_{i=0}^{\infty}
({\bf d},i)\:\frac{x^i}{i\,!}
= \sum_{i=0}^{\infty}
\frac{({\bf a},i)}{({\bf b},i)}\:\frac{x^i}{i\,!}.
$$
Numerator and denominator sets of parameters are
$
{\bf a}\, =
[a^1,\,\ldots\, ,a^A]\,\,,
${}
$
{\bf b}\, =
[b^1,\,\ldots\, ,b^B]\;
$
and ${\bf d} = {\bf a}/\!/{\bf b}$ is a double set of parameters.
Except of the above definitions explicit summations are not used
in the method.\\[-3mm]
%%%%%%%%%%%%%%%%%%%%%%% 15.notaII
\noindent
{\bf 3. Multiple series: conventions}\\[-3mm]
\noindent
{\bf a. Complex parameter} $<\alpha\mid m_1,\,\ldots\,m_N>$
corresponds to complex Pochhammer symbol
$
(\alpha,m_1i_1+\,\ldots\,+m_Ni_N)\equiv
(\alpha,{\bf m}\cdot{\bf i})
$
in the coefficient of multiple series.\\[-3mm]
\noindent
{\bf b. Glueing complex parameter}
$
\langle\alpha\mid m_1,\,\ldots\,m_N\rangle
$
contains {\bf several} non-zero
{\bf integer spectral components} $m_n$. \\[-3mm]
\noindent
{\bf c. Individual "complex" parameter}
$
\langle\alpha|0,\cdots,0,m_n,0,\cdots,0\rangle
$
contains only one non-zero spectral number. \\[-3mm]
%%%%%%%%%%%%%%%%%%%%%%% 16.notaIII
\noindent
{\bf 4. Multiple series: more conventions} \\[-3mm]
\noindent
{\bf d. The colon (:)} will serve us as delimiter between the glueing
and individual parameters. The latters will be put to the right of
the colon sequentially, according to the positions of their non-zero
components. \\[-3mm]
\noindent
{\bf e. The simple parameters} of the form
$\langle a|0,\cdots,0,1,0,\cdots,0\rangle$ and
$<\alpha\mid 1,\,\ldots{\,},1>$
are included in the glueing and the individual lists as
$\alpha$ and $a$ where spectral components are omitted. \\[-3mm]
\noindent
{\bf f. For brevity:}\, ${\bar m}_i = -m_i,\,\,\, m_{1\bar 2}
= m_1-m_2$, etc. \\[-3mm]
\noindent
{\bf g. Empty set of parameters ${\bf *}$} corresponds to
formal Pochhammer symbol $(*,i)\equiv~1$. \\[-3mm]
%%%%%%%%%%%%%%%%%%%%%%% 17.notaIV
\noindent
{\bf 5. Multiple series: general notation}
$$
^NF[L;{\bf x}]\equiv
^NF\left[\begin{array}{lcl}\langle\alpha^1|{\bf m}^1
\rangle, &\ldots &,\langle\alpha^A|{\bf m}^A\rangle;\,{\bf x}\\
\langle\beta^1|\,{\bf l}^1\rangle, &\ldots &, \langle\beta^B|\,{\bf
l}^B\rangle\end{array}\right]
$$
$$
= \sum^{\infty}_{\bf i=0}L({\bf
i})\frac{{\bf x}^{\bf i}}{{\bf i}{\,}!} =
\sum_{i_1,\ldots,i_N}L(i_1,\ldots,i_N)
\frac{x_1^{i_1}\cdots x_N^{i_N}}{i_1!\cdots i_N!},
$$
$$
L({\bf i}) = \frac{(\alpha^1,\,{\bf m}^1\cdot{\bf i})\ldots(\alpha^A,\,
{\bf m}^A\cdot{\bf i})}{(\beta^1,\,{\bf l}^1\cdot{\bf
i})\ldots(\beta^B,\,{\bf l}^B\cdot{\bf i})}\,.
$$
Any parameter can be transferred from numerator to denominator
and {\it vice versa} with the help of identity $(\alpha,i) =
(-1)^i(1-\alpha,i)^{-1}$.
In special cases it is of use to differ between glueing and
individual parameters and numerator and denominator parameters.
\\[-3mm]
%%%%%%%%%%%%%%%%%%%%%%%%% 18.notat3f
\noindent
{\bf 6. Example of $^3F$} gives us an instructive way to
understand what is implied by the above notation:
\begin{multline}
^3F\left[
\p{\an{a}{1,{\bar2},0}\an{b}{2,1,{\bar{1}}},c}{d,\an{e}{0,1,1}}%
\dv\p{g}{*}\tz\p{\an{h}{2}}{k}\tz\p{*}{l}\p{;\,x_1,x_2,x_3}{}
\right]
\nonumber \\
= \sum_{i_1,i_2,i_3}
\frac{(a,i_1-2i_2)(b,2i_1+i_2-i_3)(c,i_1+i_2+i_3)}%
{(d,i_1+i_2+i_3)(e,i_2+i_3)}
\nonumber \\ \times\,
\frac{(g,i_1)(h,2i_2)}{(k,i_2)(l,i_3)}\,
\frac{x_1^{i_1}}{i_1!}\,
\frac{x_2^{i_2}}{i_2!}\,
\frac{x_3^{i_3}}{i_3!}\,.
\nonumber
\end{multline}
\begin{center}
{\Large {\bf 4.\,
Operator factorization method}}
\end{center}
\noindent
{\large{\bf 4.1. $\Omega$-multiplication is a fundamental
operation underlying the factorization method
}}
$\Omega$-product $u * v$ of functions $u(x_1,\ldots,x_N)$ and
$v(x_1,\ldots,x_N)$ is defined as
$$
\langle u * v|x_1,\,\ldots\,,x_N\rangle =
u\left(\frac{d}{ds_1},\ldots,\frac{d}{ds_N}\right)
v(x_1s_1,\,\ldots\,,x_Ns_N)\mid_{\forall s_n=0}.
$$
For the $\Omega$-product some important general properties
are fulfilled: for example, commutation property $u*v = v*u$;
association property $w*(u*v)=(w*u)*v$\,;
coupling rule, etc. See subsection COMMANDS(II) in \cite{prep2},
Sec.2\,. \\[-5mm]
\noindent
{\large{\bf 4.2. Operator factorization principle: an illustrative example
of general power series}}
Introduce the notation for arbitrary power series
$$
F[A;\,x] = \sum^{\infty}_{i=0}A(i)\frac{x^i}{i\,!}\,,\quad
%F[B;\,x] = \sum^{\infty}_{i=0}B(i)\frac{x^i}{i\,!}\,,\quad
F[A,B;\,x] = \sum^{\infty}_{i=0}A(i)B(i)\frac{x^i}{i\,!}\,,
$$
etc. Then factorization formula holds:
$$
F[A,B;\,x] = F\left[A;\frac{d}{ds}\right]\,F[B;\,xs]|_{s=0}\,.
$$
The condition $s=0$ should be introduced after fulfillment of
term by term differentiation in the right hand side of the
factorization formula. \\[-5mm]
\noindent
{\large{\bf 4.3. $\Omega$-representability of multiplication of
power series coefficients}}
\noindent
The above formula can be written as
$$
\qquad F[A\times B;\,x] = \langle F[A]\,*\,F[B]\,|\,x\rangle.
$$
This notation makes it obvious the {\bf property of
$\Omega$-representability} of multiplication operation over
coefficients of an {\it arbitrary power series}.
Suppose we multiply two series with coefficients $A(i)$ and
$B(i)$, respectively. It is well known that the operation which
should be applied to $A$ and $B$ to produce coefficients of the
resultant series is the convolution operation. Now we multiply
coefficients of the series. The operation which should be
applied to the two initial series to produce the resultant
series with coefficients $A(i)\times B(i)$ is just the
$\Omega$-multiplication operation. \\[-5mm]
\noindent
{\large{\bf 4.4. Illustrative example of total factorization}}
Apart from the above simple exercise in calculus, what
practical use can be reached with the help of the
$\Omega$-multiplication. Can we give a clear-cut illustrative
example? Here it is
\begin{multline}
F\left[\p{a^1,\ldots,a^A}{b^1,\ldots,b^B}\p{;\,x}{}\right] \\
= F^1_0\left[\p{a^1}{*}\right]*F^1_0\left[\p{a^2}{*}\right]
*\cdots*F^1_0\left[\p{a^A}{*}\right]*%
F^0_1\left[\p{*}{b^1}\right]*F^0_1\left[\p{*}{b^2}\right]%
*\cdots*F^0_1\left[\p{*}{b^B}\right].
\nonumber
\end{multline}
We see that any property of generalized hypergeometric series
is a corollary of the properties of the binomial series $F^1_0$
and the Bessel-type series $F^0_1$. The technique necessary for
derivations of this kind will be given later. \\[-5mm]
\noindent
{\large{\bf 4.5. Factorization formulas }}
\noindent
Factorization formulas for generalized hypergeometric series
in one variable have the form:
$$
\left.
F[{\bf d}_1\,;\:x_1\,\frac{d}{ds}]\:F[{\bf d}_2;x_2s]
\right|_{s=0} =
F[{\bf d}_1,{\bf d}_2\,;\:x_1\,x_2]\; ,
$$
$$
\left.
F[{\bf d}_1\,;\:x_1\,\frac{d}{ds}]\:F[{\bf d}_2;x_2s^m]
\right|_{s=0} =
F[<{\bf d}_1\mid m>,\,{\bf d}_2\,;\:x^m_1\,x_2]\; ,
$$
where ${\bf d}_1$ and ${\bf d}_2$ are double sets of numerator and
denominator parameters.
The above formulas allow complicated series to be expressed as
$\Omega$-products of simpler series; {\em vice versa}, an
$\Omega$-product can be expressed in an algebraic form of a more
complicated series. Moreover the formulas permit us to introduce
very useful general concepts (see subsection 4.6).\\[-6mm]
\noindent
{\bf General factorization formula} for multiple (not
necessarily hypergeometric) series is
\begin{multline}
^NF[L_1,L_2\,;\:x_1,\,\ldots\,,x_N] = \\[1.5mm]
= ^NF\left[L_1\,;\:\frac{d}{ds_1},\,\ldots\,,\frac{d}{ds_N}\right]\,%
\left.
^NF[L_2\,;\:x_1s_1,\,\ldots\,,x_Ns_N]\,\right|_{\forall s_n=0}\; ,
\nonumber
\end{multline}
where $L_k(i_1,\,\dots\,,i_N),\, k=1,2$ are {\bf arbitrary}
coefficients. It is a direct paraphrase of the factorization formula
for $F[{\bf d}_1,{\bf d}_2;\,x]$.\\[-6mm]
\noindent
In hypergeometric case the coefficients
$({\bf d}, m_1i_1 + \,\cdots\, + m_Ni_N)$ or
$({\bf d}, i_1 + \,\cdots\, + i_N)$ have a {\bf special}
dependence om summation variables. This permits us to use the
{\bf special factorization formulas}:
\begin{multline}
^NF[<{\bf d}\mid m_1,\,\ldots\,,m_N>,\,L\,;\:x_1,\,\ldots\,,x_N]
= \\[1.5mm]
\left.{ } = F\left[{\bf d}\,;\:\frac{d}{ds}\right]\,
^NF[L\,;\:x_1s^{m_1},\,\ldots\,,x_Ns^{m_N}]\, \right|_{s=0}\; ,
\nonumber
\end{multline}
$$
^NF[{\bf d},L;x_1,\,\ldots\,,x_N] =
\left. ^1F\left[{\bf d};\frac{d}{ds}\right]\,
^NF[L;x_1s,\,\ldots\,,x_Ns] \right|_{s=0}\; ,
$$
where, contrary to the general factorization formula, the first
multipliers in the $\Omega$-products are {\bf simple}
hypergeometric series. \\[-4mm]
\noindent
{\large{\bf 4.6. General concepts, which constitute the structural
basis of the method
}} \\[-4mm]
{\bf 1. $\Omega$-identical expressions}
By analogy with {\bf arithmetically identical expressions}
the algebraic expressions connected by finite number of
arithmetic operations {\bf and $\Omega$-multiplication
operations} will be called
{\bf $\Omega$-identical expressions}. \\[-5mm]
{\bf 2. $\Omega$-equivalent operators}
The operators $F_1$ and $F_2$ are called $\Omega$-equivalent
operators
($F_1\rightleftharpoons F_2$) if the identity
$$
F_1\left(\frac{d}{ds},s\right) \Psi(xs)\bigl|_{s=0} =
F_2\left(\frac{d}{ds},s\right) \Psi(xs)\bigl|_{s=0}
$$
holds for an arbitrary function $\Psi$.
Note that the $\Omega$-equivalent operators are not
necessarily identical to one another. \\[-5mm]
\noindent
{\bf 3. $\Omega$-equivalent relations}
\noindent
The functional relation $f\,*\,f_1=f\,*\,f_2$ or
\begin{multline}
f\left(d/ds_1,\,\ldots\,,d/ds_N\right)
f_1(x_1s_1,\,\ldots\,,x_Ns_N)\bigl|_{\forall s_n=0} \\
= f\left(d/ds_1,\,\ldots\,,d/ds_N\right) \nonumber
f_2(x_1s_1,\,\ldots\,,x_Ns_N)\bigl|_{\forall s_n=0}
\end{multline}
\noindent
where $f$ is an arbitrary function of $N$ variables will be called
$\Omega$-equivalent to the relation \\
\hspace*{2.5cm} $f_1(x_1,\,\ldots\,,x_N) =
f_2(x_1,\,\ldots\,,x_N)$. \\[-5mm]
\noindent
{\bf 4. $\Omega$-equivalent classes and proto-relations}
\noindent
In a class of $\Omega$-equivalent relations a {\bf simplest relation}
can be chosen to serve as a {\bf proto-relation} underlying the class
and giving rise to all its members. Having proved the proto-relation
we {\bf prove all formulas} belonging to the class. \\[-5mm]
\noindent
{\large{\bf 4.7. How the method works}} \\ [-4mm]
\noindent
{\bf (I) A standard four step scheme} \\ [-6mm]
{\bf 1. Analysis.}\,
An initial series is decomposed into an $\Omega$-product of
simpler series.
{\bf 2. Basic transformations.}\,
The known properties of the simpler series are used to transform
the factorized expression to the desired form.
{\bf 3. Auxiliary transformations.}\,
A finite number of auxiliary transformations is employed to
convert the resultant expression to the form permitting the use
of a factorization formula.
{\bf 4. Synthesis.}\,
The appropriate factorization formula is applied to turn the
operator expression into an algebraic form. \\ [-4mm]
%%%%%%%%%%%%%% 29.howwork2
\noindent
{\bf (II) Scheme based on the concept of $\Omega$-equivalent
relations} \\ [-6mm]
The usefulness of the relation
$f\,*\,f_1 = f\,*\,f_2$ depends on $f_1,f_2$ and $f$.
To have a very simple illustrative example we use the relation
$F[a;x]=(1-x)^{-a}$ to convert $(1-x)^{-a}=(1-x)(1-x)^{-a-1}$
into
$$
F^1_0[a;x]=F^1_0[a+1;x]-x\,F^1_0[a+1;x]\; .
$$
Applying the operator
$F^0_2[{\bf *}\,//\,a,a+1;zd/dx]\bigr|_{x=0}$ to both sides of
the binomial identity and using relations from the COMMANDS list
(Ref.\cite{prep2}, Sec.2)
we obtain the recurrence relation for the Bessel-type series
$$
F^0_1\left[\begin{array}{lc}{\bf *}&;\:z\\b&{}\end{array}\right] =
F^0_1\left[\begin{array}{lc}{\bf *}&;\:z\\b-1&{}\end{array}\right] -
\frac{z}{b(b-1)}\:
F^0_1\left[\begin{array}{lc}{\bf *}&;\:z\\b+1&{}\end{array}\right]\; .
$$ \\[-3mm]
\noindent
{\large{\bf 4.8. Different computational modes
}} \\[-3mm]
{\bf 1.} Factorization method gives us a {\bf limited set} of
operations sufficient to obtain manually, with pen and paper,
any property of an arbitrary series {(\bf manual mode)}.\\[-5mm]
{\bf 2.} If the operations were implemented in the form of computer
commands we could obtain a superstructure over an existent
computer algebra system capable to derive any formula in an
interactive mode {(\bf globally universal computer mode)}.
In the case in point we set problem to formalize the main
operators of the method to make them consistent with the
grammatical structure of the SANTRA language.
The preliminary command set and its prototypical operations
are presented in subsections COMMANDS (I) -- COMMANDS(VI)
of Sec.2, Ref.\cite{prep2}.
A representative example illustrating the main
features of the both modes "emulating" one another will be
given in Sec.5 where operations presented in Sec.2,
Ref.\cite{prep2} are only used.\\[-5mm]
{\bf 3.} So far we used another computational mode. Considering
separate classes of formulas we used the factorization method
to obtain a limited number of formulas playing the role of
basis relations. Combining these relations (see MACRO -- COMMANDS
in Ref.\cite{prep2}, Sec.2)
one can obtain all relations belonging
to the class. Interactive and automated computer programs
have been developed for 5 classes of formulas inaccessible
in any other way ({\bf bounded universal computer mode}).
Some examples are given in Sec.~6.\\[-4mm]
\begin{center}
{\Large {\bf 5.\,
Formula derivation based on the use of the operator factorization
method}}
\end{center}
{\large {\bf 5.1. Passing to examples
}} \\[-5mm]
\noindent
Given hundreds and thousands of multiple hypergeometric series
and innumerable scores of formulas fulfilling for these series it
seems surprising that a {\bf moderate set of operations} listed
in COMMANDS and MACRO -- COMMANDS sections {\bf is sufficient} to
derive any property of an arbitrarily complicated series.
{\bf Formal proof} of derivation potential of the method {\bf can
hardly be given} at present. One can make certain of the merits
of the method in a practical manner by deriving sufficiently
large number of formulas. We further give derivation of {\bf a
formula illustrating four step analytical scheme} (see 4.7 (I)).
\\[-5mm]
\noindent
{\large {\bf 5.2. Transformation of $F_4$ }} \\[-5mm]
\noindent
{\bf 1. Factorization allows the function $F_4$ to be
expressed in terms of simpler functions $U$}
\noindent
Introduce the $F_4$ function \\[-2mm]
$$
F_4 \equiv F\left[\begin{array}{ccccc}a_1, a_2 & :\,*\,; & * & ;
{\displaystyle
\frac{z_1}{(1-v)(1-u)}},{\displaystyle\frac{z_2}{(1-v)(1-u)}} \\
{\,}*&:\,b_1;&b_2&&\end{array}\right].
$$
\noindent
The most interesting factorization of the $F_4$ gives
$$
F_4 =
U\left[a_1,b_1;\,\,\frac{d(s_2)}{1-v},\,\,\frac{d(s_1)}{1-v}\right]\left.
U\left[a_2,b_2;\frac{z_1s_1}{1-u},\frac{z_2s_2}{1-u}\right]
\right|_{s_1=s_2=0}\,,
$$
$$
U[a,b;x_1,x_2] = F\left[\begin{array}{ccccc} a: & *; & *; & x_1, & x_2 \\
{\,}*: & *; & b && \end{array}\right]\,.
$$\\[-5mm]
\noindent
{\bf 2. Function U, in contrast to $F_4$, exhibits
obvious simple properties}\\[-4mm]
\noindent
The function $U$\\[-3mm]
$$
U[a,b;x_1,x_2] = F\left[\begin{array}{ccccc} a: & *; & *; & x_1, & x_2 \\
{\,}*: & *; & b && \end{array}\right]
$$
has {\bf
binomial type} $(1/\!/0)$ with respect to $x_1$ and {\bf Kummer type}
$(1/\!/1)$ with respect to $x_2$.
Specialization of {\bf general} binomial and Kummer transformations gives
$$
U[a,b;\,x_1+u,x_2]
= (1-u)^{-a}\,U_1\left[a,b;\frac{x_1}{1-u},
\frac{x_2}{1-u}\right]\,,
$$
\noindent
$$
U[a,b;x_1,x_2]
= e^{x_2}\,F\left[\begin{array}{ccccc}\langle b-a|1,
\overline{1}\rangle & :\,*; & a,\,1+a-b; & -x_2, & -x_1 \\
{\,} * & :\,b; & * && \end{array}\right]\,.
$$
\\[-5mm]
\noindent
{\bf 3. Using binomial properties of $U$-functions
}\\[-4mm]
\noindent
Binomial transformation of both
$\Omega$-multipliers gives
\noindent
\begin{multline}
F_4 = (1-v)^{a_1}(1-u)^{a_2}\,\\
\times
U[a_1,\,b_1;d(s_2)+v, d(s_1)] \nonumber
{\,}\left. U[a_2,\,b_2;z_1s_1+u, z_2s_2]\right|_{s_1=s_2=0}.
\end{multline}
Applying the {\bf operator displacement formula}
$$
F[d(s)+v] = \exp(-vs) F[d(s)] \exp(vs)
$$
to the first multiplier and the {\bf shift operator formula}
$$
F[s+u] = \exp(u{\,}d/ds) F[s]
$$
to the second multiplier ensures {\bf elimination of interfering
constant terms} $u$ and $v$ from the arguments.
\noindent
{\bf 4. Binomial and Kummer transformations give us
a complicated $\Omega$-product} \\[-4mm]
After "uniformization of arguments" in $U$ functions the
result of binomial transformations in $F_4 = U*U$ is\\[-4mm]
$
F_4 = (1-v)^{a_1}(1-u)^{a_2} U[a_1,b_1;\,d(s_2),\,d(s_1)]
$
$$
\times
\exp[uz_1^{-1}{\,}d(s_1)] \exp\,(s_2v)
\,U\left.[a_2,\,b_2;\,z_1s_1,\,z_2s_2]
\right|_{s_1=s_2=0}.
$$
\noindent
{\bf Kummer transformation} of the both $U$'s gives us\\[-4mm]
$
F_4 = (1-v)^{a_1}(1-u)^{a_2}
$
$$
\times
F\left[\begin{array}{cccccc}\langle b_1-a_1|1,
\overline{1}\rangle&:&*&;&a_1,\,1+a_1-b_1;&-d(s_1),-d(s_2) \\
{\,}*&:&b_1&;&*&\end{array}\right]
$$
$
\times{\,}\exp\left[\left((z_1+u)/(z_1)\right)d(s_1)\right]{\,}
\exp\left[(z_2+v)s_2\right]
$
$$
\times F\left.\left[\begin{array}{cccccc}\langle b_2-a_2|1,
\overline{1}\rangle&:&*&;&a_2,\,1+a_2-b_2;&-z_2s_2,z_1s_1) \\
{\,}*&:&b_2&;&*&\end{array}\right]\right|_{s_1=s_2=0}.
$$
\\[-5mm]
\noindent
{\bf 5. Applying again a factorization formula we obtain
the desired result} \\[-6mm]
To simplify the $\Omega$-product representation for $F_4$ we
eliminate the exponential terms by putting $u=-z_1,\,v=-z_2$.
Using factorization formula we finally obtain\\[-8mm]
$$
F\left[\p{a_1,a_2}{*}\dv\p{*}{b_1}\tz\p{*}{b_2}\p{;
\displaystyle{\frac{z_1}{(1+z_1)(1+z_2)},
\frac{z_2}{(1+z_1)(1+z_2)}}}{}\right] =
$$
$$
=(1+z_2)^{a_1}(1+z_1)^{a_2}
F\left[\p{\an{b_1-a_1}{1,\bar{1}},\an{b_2-a_2}{\bar{1},1}}{*}\dv
\right.
$$
$$
\qquad\qquad\qquad\left.\dv
\p{a_2,1+a_2-b_2}{b_1}\tz\p{a_1,1+a_1-b_1}{b_2}\p{;\,z_1,z_2}{}
\right]\,.
$$ \\[-4mm]
This formula transforms the complete series of the second order
(of the type $[2/\!/1,2/\!/1]$) into the complete series of the
third order (of the type $[3/\!/2,3/\!/2]$).
\\[-2mm]
\noindent
{\bf 5.3. Analytical corollaries } \\[-5mm]
Factorizing the r.h.s. of the above formula we can express
the general $F_4$ as an $\Omega$-product of a Kummer
function $F^1_1$ and two Gauss functions $F^2_1$. This
implies that {\bf many new formulas} for $F_4$ can be deduced
from the known properties of $F^1_1$ and $F^2_1$.
Many known hard-hitting results follow immediately from the
above formula. If $b_1+b_2 = a_1+a_2+1$ then
$F_4\sim F^2_1F^2_1$ ({\bf Watson formula}). If $b_2 = a_2$ then
using the "indefinite" transformations and an appropriate
linear transformation we obtain $F_4\sim F_1$ ({\bf Bailey
formula}). If $a_2 = b_1+b_2-1$ then using a couple of linear
transformations we can see that $F_4\sim F_2$ ({\bf another
Bailey formula}).\\[-2mm]
\begin{center}
{\Large {\bf 6.\,
Computer generation of formulas using bounded
universal programs}}
\end{center}
{\large {\bf 6.1. Programs
}}
A complex of programs have been developed with the help of
macro -- commands realizing Kummer-type and Gauss-type {\bf
linear} transformations, {\bf quadratic} transformations,
analytic {\bf continuation} formulas and {\bf reduction} formulas
turning multiple series into series depending on lesser number of
variables. Provision was made for both {\bf interactive} and
{\bf automated} modes of processing. \\[-5mm]
{\large {\bf 6.2.~Example of computer generation: the case of
the Appell function $F_4$
}}
In the following we confine ourselves to the only
example of special Appell function
$F_4[a_1,a_2,a_1,b_2;x_1\,,x_2]$ having one
constraint on parameters.
Using the special transformation of general $F_4$ and utilizing,
for the case of restricted $F_4[a_1,a_2,a_1,b_2;x_1\,,x_2]$,
either contraction reduction or the auxiliary "indefinite"
transformation we represent the $F_4$ as non-Hornian functions:
$$
K_{gb}=F\left[\p{\alpha}{\beta}\dv\p{a_1,a_1'}{b_1}\tz\p{a_2}{*}
\p{;\,x_1,x_2}{}\right]\,,
$$
$$
\Gamma_{bg}=F\left[\p{\an{\alpha_1}{1,\bar{1}}\an{\alpha_2}%
{\bar{1},1}}{}\dv\p{a_1}{*}\tz\p{a_2,a_2'}{b_2}
\p{;\,x_1,x_2}{}\right].
$$
The processing of these functions consisted in using all possible
linear commands {\bf lin(G)} along with an auxiliary bilinear
transformation (for the functions with parameter
$\an{0}{1,{\bar1}}$). The process performed in automatic mode gave
us functions symbolically presented by double black circles on
Fig.~1. \\
\noindent
\unitlength=1.5pt
\begin{picture}(120,220)(-85,10)
%\put(0,0){\line(1,0){145}}
%\put(0,230){\line(1,0){145}}
%\put(0,0){\line(0,1){230}}
%\put(145,0){\line(0,1){230}}
\put(15,15){\circle*{2}}
\put(45,15){\circle*{2}}
\put(27.5,20){\circle*{2}}
\put(57.5,20){\circle*{2}}
\put(10,35){\circle*{2}}
\put(40,35){\circle*{2}}
\put(70,35){\circle*{2}}
\put(100,35){\circle*{2}}
\put(22.5,40){\circle*{2}}
\put(52.5,40){\circle*{2}}
\put(82.5,40){\circle*{2}}
\put(112.5,40){\circle*{2}}
\put(35,55){\circle*{2}}
\put(65,55){\circle*{2}}
\put(95,55){\circle*{2}}
\put(125,55){\circle*{2}}
\put(47.5,60){\circle*{2}}
\put(77.5,60){\circle*{2}}
\put(107.5,60){\circle*{2}}
\put(137.5,60){\circle*{2}}
\put(90,75){\circle*{2}}
\put(120,75){\circle*{2}}
\put(102.5,80){\circle*{2}}
\put(132.5,80){\circle*{2}}
\put(27.5,20){\line(5,4){25}}
\put(57.5,20){\line(5,4){25}}
\put(10,35){\line(5,4){25}}
\put(40,35){\line(5,4){25}}
\put(70,35){\line(5,4){25}}
\put(100,35){\line(5,4){25}}
\put(77.5,60){\line(5,4){25}}
\put(107.5,60){\line(5,4){25}}
\put(27.5,20){\line(1,0){30}}
\put(10,35){\line(1,0){30}}
\put(70,35){\line(1,0){30}}
\put(52.5,40){\line(1,0){30}}
\put(35,55){\line(1,0){30}}
\put(95,55){\line(1,0){30}}
\put(77.5,60){\line(1,0){30}}
\put(102.5,80){\line(1,0){30}}
\put(15,15){\line(5,2){12.5}}
\put(45,15){\line(5,2){12.5}}
\put(10,35){\line(5,2){12.5}}
\put(40,35){\line(5,2){12.5}}
\put(70,35){\line(5,2){12.5}}
\put(100,35){\line(5,2){12.5}}
\put(35,55){\line(5,2){12.5}}
\put(65,55){\line(5,2){12.5}}
\put(95,55){\line(5,2){12.5}}
\put(125,55){\line(5,2){12.5}}
\put(90,75){\line(5,2){12.5}}
\put(120,75){\line(5,2){12.5}}
%теперь скопируем низ этажерки на верх
\put(0,140){\begin{picture}(145,90)%
\put(15,15){\circle*{2}}
\put(45,15){\circle*{2}}
\put(27.5,20){\circle*{2}}
\put(57.5,20){\circle*{2}}
\put(10,35){\circle*{2}}
\put(40,35){\circle*{2}}
\put(70,35){\circle*{2}}
\put(100,35){\circle*{2}}
\put(22.5,40){\circle*{2}}
\put(52.5,40){\circle*{2}}
\put(82.5,40){\circle*{2}}
\put(112.5,40){\circle*{2}}
\put(35,55){\circle*{2}}
\put(65,55){\circle*{2}}
\put(95,55){\circle*{2}}
\put(125,55){\circle*{2}}
\put(47.5,60){\circle*{2}}
\put(77.5,60){\circle*{2}}
\put(107.5,60){\circle*{2}}
\put(137.5,60){\circle*{2}}
\put(90,75){\circle*{2}}
\put(120,75){\circle*{2}}
\put(102.5,80){\circle*{2}}
\put(132.5,80){\circle*{2}}
\put(27.5,20){\line(5,4){25}}
\put(57.5,20){\line(5,4){25}}
\put(10,35){\line(5,4){25}}
\put(40,35){\line(5,4){25}}
\put(70,35){\line(5,4){25}}
\put(100,35){\line(5,4){25}}
\put(77.5,60){\line(5,4){25}}
\put(107.5,60){\line(5,4){25}}
\put(27.5,20){\line(1,0){30}}
\put(10,35){\line(1,0){30}}
\put(70,35){\line(1,0){30}}
\put(52.5,40){\line(1,0){30}}
\put(35,55){\line(1,0){30}}
\put(95,55){\line(1,0){30}}
\put(77.5,60){\line(1,0){30}}
\put(102.5,80){\line(1,0){30}}
\put(15,15){\line(5,2){12.5}}
\put(45,15){\line(5,2){12.5}}
\put(10,35){\line(5,2){12.5}}
\put(40,35){\line(5,2){12.5}}
\put(70,35){\line(5,2){12.5}}
\put(100,35){\line(5,2){12.5}}
\put(35,55){\line(5,2){12.5}}
\put(65,55){\line(5,2){12.5}}
\put(95,55){\line(5,2){12.5}}
\put(125,55){\line(5,2){12.5}}
\put(90,75){\line(5,2){12.5}}
\put(120,75){\line(5,2){12.5}}
\end{picture} }
\put(15,92.5){\circle*{2}}
\put(27.5,87.5){\circle*{2}}
\put(70,82.5){\circle*{2}}
\put(82.5,87.5){\circle*{2}}
\put(132.5,107.5){\circle*{2}}
\put(120,112.5){\circle*{2}}
\put(77.5,107.5){\circle*{2}}
\put(65,112.5){\circle*{2}}
\put(15,125){\circle*{2}}
\put(27.5,120){\circle*{2}}
\put(70,125){\circle*{2}}
\put(82.5,120){\circle*{2}}
\put(132.5,140){\circle*{2}}
\put(120,145){\circle*{2}}
\put(77.5,140){\circle*{2}}
\put(65,145){\circle*{2}}
\put(35,142.5){\circle*{2}}
\put(47.5,147.5){\circle*{2}}
\put(100,142.5){\circle*{2}}
\put(112.5,147.5){\circle*{2}}
\put(100,85){\circle*{2}}
\put(112.5,90){\circle*{2}}
\put(35,95){\circle*{2}}
\put(47.5,100){\circle*{2}}
%ga1lum2l
\put(35,95){\line(3,5){30}}
%um2lt5l
\put(65,145){\vector(0,1){48}}
%ga1lb4l
\put(35,95){\vector(0,-1){38}}
\put(27.5,87.5){\line(1,0){55}}
\put(77.5,107.5){\line(1,0){55}}
\put(27.5,120){\line(1,0){55}}
\put(77.5,140){\line(1,0){55}}
\put(27.5,87.5){\line(5,2){50}}
\put(82.5,87.5){\line(5,2){50}}
\put(27.5,120){\line(5,2){50}}
\put(82.5,120){\line(5,2){50}}
\put(27.5,87.5){\line(-5,2){12.5}}
\put(82.5,87.5){\line(-5,-2){12.5}}
\put(132.5,107.5){\line(-5,2){12.5}}
\put(77.5,107.5){\line(-5,2){12.5}}
\put(27.5,120){\line(-5,2){12.5}}
\put(82.5,120){\line(-5,2){12.5}}
\put(132.5,140){\line(-5,2){12.5}}
\put(77.5,140){\line(-5,2){12.5}}
%lm4lga3l
\put(70,82.5){\line(1,2){30}}
%ga3lt10l
\put(100,142.5){\vector(0,1){30.5}}
%lm4lb11l
\put(70,82.5){\vector(0,-1){45.5}}
%ga4lum4l
\put(100,85){\line(-3,4){30}}
%um4lt11l
\put(70,125){\vector(0,1){48}}
%ga4lb10l
\put(100,85){\vector(0,-1){48}}
%lm2lga2l
\put(65,112.5){\line(-1,1){30}}
%ga2lt4l
\put(35,142.5){\vector(0,1){50.5}}
%lm2lb5l
\put(65,112.5){\vector(0,-1){55.5}}
\put(15,92.5){\line(0,1){32.5}}
\put(120,112.5){\line(0,1){33}}
\put(35,95){\line(5,2){12.5}}
\put(35,142.5){\line(5,2){12.5}}
\put(100,142.5){\line(5,2){12.5}}
\put(100,85){\line(5,2){12.5}}
\put(15,125){\vector(0,1){28}}
\put(15,92.5){\vector(0,-1){75}}
\put(120,145){\vector(0,1){68}}
\put(120,112.5){\vector(0,-1){35}}
%
{ \normalsize {\bf
\put(11,180){\llap {$K_{bg}^{\de} $}}
\put(12,150){\llap {$ G_{ke}^{\de}$}}
\put(10,120){\llap {$F_2^{\ga} $}}
\put(10,90){\llap {$ F_2^{\be}$}}
\put(11,40){\llap {$ K_{bg}^{\be}$}}
\put(12,10){\llap {$ G_{ke}^{\be}$}}
\put(87.5,212.5){\llap {$K_{gb}^{\ga} $}}
\put(40,200){\llap {$ G_{ek}^{\ga}$}}
\put(70,200){\llap {$F_1^{\ga} $}}
\put(105,202.5){\llap {$ F_3^{\ga}$}}
\put(130,190) {$ K_{bg}^{\ga}$}
\put(130,212.5) {$ G_{ke}^{\ga}$}
\put(130,145) {$ F_2^{\de}$}
\put(130,112.5) {$ F_2^{\al}$}
\put(130,50) {$ K_{bg}^{\al}$}
\put(130,83.75) {$ G_{ke}^{\al}$}
\put(78.5,91.5){$\widetilde H_2^\be$}
\put(75,125){$\widetilde H_2^\de$}
\put(131,92.5){\llap{$\Gamma_{bg}^{\de}$}}
\put(50,172.5) {$ F_3^{\de}$}
\put(85,178.5) {$ F_1^{\de}$}
\put(52.5,152.5){$K_{gb}^{\de} $}
\put(107.5,170){$ G_{ek}^{\de}$}
\put(72.5,145){$\widetilde H_2^{\ga} $}
\put(32.5,141.5){\llap {$ \Gamma_{bg}^{\al}$}}
\put(107.5,152.5) {$ \Gamma_{bg}^{\be}$}
\put(62.5,107.5){\llap{$\widetilde H_2^{\al} $}}
\put(32.5,97.5){\llap {$ \Gamma_{bg}^{\ga}$}}
\put(86.5,72.5){\llap {$K_{gb}^{\al} $}}
\put(32.5,55.5){\llap {$ G_{ek}^{\be}$}}
\put(80,51){\llap {$F_1^{\be} $}}
\put(75,26.5){$ F_1^{\al}$}
\put(55,10) {$ K_{gb}^{\be}$}
\put(100,26.5) {$ G_{ek}^{\al}$}
\put(99.5,46){\llap {$F_3^{\al} $}}
\put(50,32){$F_3^{\be} $}
}}
\end{picture}\\ [0.5mm]
\begin{quote}
Fig. 1.
The diagram representing 36 series generated by
the function $F_4[a_1,a_2,a_1,b_2]$. The arrows
correspond to additional auxiliary transformations
connected with "indefinite" parameters (see Eqs.~(37) and (38)
in \cite{prep2}).
\end{quote}
\noindent
Long lines symbolize general linear transformations (see
Sec.~2.7.3 in \cite{prep2}). Each series is represented by short
inclined
segments with two black nodes corresponding to the arguments
of the series. The arrows indicate additional transformations
connected with "indefinite" parameters.
Along with functions $K_{gb}$ and $\Gamma_{gb}$
we obtained, in an automatic mode, the following 5 functions:
$$
G_{ek}=F\left[\p{\alpha_1,\alpha_2}{\beta}\dv\p{*}{*}\tz\p{a_2}{b_2}
\p{;\,x_1,x_2}{}\right] \qquad \;
$$ \\[-7mm]
$$
F_1=F\left[\p{\alpha}{\beta}\dv\p{a_1}{*}\tz\p{a_2}{*}
\p{;\,x_1,x_2}{}\right]\,, \quad \; \;
$$
$$
F_3=F\left[\p{*}{\beta}\dv\p{a_1,a_1'}{*}\tz\p{a_2,a_2'}{*}
\p{;\,x_1,x_2}{}\right]\,, \quad \;
$$
$$
F_2=F\left[\p{\alpha}{*}\dv\p{a_1}{b_1}\tz\p{a_2}{b_2}
\p{;\,x_1,x_2}{}\right] \,, \; \; \;\:
$$
$$
\widetilde{H}_2=F\left[\p{\an{\alpha}{\bar{1},1}}{}\dv\p{a_1,%
a_1'}{*}\tz\p{a_2}{b_2}\p{;\,x_1,x_2}{}\right]\,.
$$ \\[-8mm]
\noindent
Transformation
properties of the series $G_{ke}$ and $G_{ek}$ make it
possible to obtain a toroidal construction made of three etageres
jointed by their upper and lower facets with one another. Of
course it would be difficult to establish such connection
between 108 double series without using computer--aided symbolic
approach based upon the new computational principle. Note that
the use of new transformations permits our program to make start
from an arbitrary point of the diagram. \\
\begin{center}
{\Large {\bf 7.\, Prospects of the globally-universal approach
within the system of analytical transformations Santra~3 based on
Refal language}}
\end{center}
\noindent
We discuss in brief the problem of implementation of a globally
universal approach (see Sec. 4.8.2) using the domestic
algorithmic language Refal. Envision the 4-layer structure
\begin{align}
{\mbox{\sf Hypertrans graphic shell}} \qquad& (4)\notag \\
{ \mbox{\sf Hypertrans functions}} \qquad& \notag (3) \\
{ \mbox{\sf Santra 3}} \qquad& (2) \notag \\
{ \mbox{\sf Refal 6 }} \qquad& (1) \notag
\end{align}
which is graphical representation of a conceived
system of analytical transformations of hypergeometric series.
The system will be called, for brevity, the Hypertrans system.
A computer satisfying common general conditions plays the role
of the "bottom" zero level. All peculiar features of system are
inherent in the level 1 which plays the role of the "specialized
basement" for all other upper levels. This leads to the machine --
independent system. Any change in the level 1, for example,
transfer to other Refal version may cause serious re-building of
other levels. Having presently only the level 1 facilities with
Refal~6 (http://www.refal.net/\~{}arklimov/refal6/)
functioning in DOS window with tight graphic interface we
aim, for the time being, at development of a model Hypertrans
system having complete set of program facilities with but a
limited set of graphic conveniences.
Level 2 is exhausted at present by tools of the system Santra~2
\cite{1}-\cite{6}. These tools are not convenient for
implementation of Hypertrans functions. However Santra~2 provides
a reliable basis for development of more convenient system
Santra~3 because Santra~2 was conceived as a universal
system of algebraic transformations using symbolic manipulations
similar to those of Refal. The experience gained in course of
implementing different algebraic transformations with the help
of Refal tools \cite{7}-\cite{10} served us as a basis for
development of Santra~2. The universality feature inhereted by
Sanra~2 from Refal allows us to build up various
problem-oriented superstructures. The aim of the improvements is
development of more convenient tools without change of
already written code. Really, the universality of Santra~2
implies that any algebraic transformations can be implemented,
in principle, with the help of basic system tools. Practically,
we need more convenient tools for a specific applied problem. It
is important that any "applied subsystem" can be readily
modified without any change in the basic system.
Specifically, it is Refal~2 \cite{11}-\cite{13} that plays the
role of paradigm for Santra~2 language. One of the additional
capabilities is that the left parts of statements can be formed
algorithmically. This allows algorithmic recognition of objects
having fixed structures, for example, the typed expressions.
In turn, this allows to differentiate operations depending on the
type of object on the same principle as in the case of abstract
data types. Besides, the Santra~2 language admits dynamical
program formation that allows the use of macrocommands and
subroutine relocation depending on a specific data set. However
the processing of the normal forms which are used in Santra~2
for representation of algebraic expressions is not sufficient
for operations required by operator factorization method.
For example, it is necessary to process functions
whose arguments, exponents, coefficients and free terms play the
role of parameters in contextual definition. In Santra~2, there
are no convenient tools for extraction such functions from normal
forms. One more deficiency is that Santra~2 is closely related to
Refal~2 which is characterized by less convenient form of program
record compare to Refal~5 and Refal~6
(http://www.refal.net/\~{}arklimov/refal6/) which gives
an extension of Refal~5. Santra~3 is being developed on the basis
of Refal~5 supplemented by ideas of Refal~6 and some additional
options.
All extensions introduced in Santra~3 persue the goal to simplify
programming by development of more simple and illustrative tools
close to the typical manipulations of researcher engaged in
derivation of a mathematical formula. The work on the project of
Santra~3 is still in progress now.
The approach to construction of the level 3 functions will be
illustrated by analysis of several typical transformations of the
factorization method \cite{prep2}. The level 3 as a whole is
constituted by the set of commands corresponding to the complete set of
operations used in the factorization method. It is just what we
call the command set of the globally universal approach. The
program implementation of the commands is illustrated by
model constructions of Santra~3 \cite{prep2}. \\
\newpage
\begin{center}
{\Large{\bf
Summary }}
\end{center}
He who has to deal with hypergeometric series systematically and
even he who confronts with them occasionally should familiarize
himself with the operator factorization method.
The reasons giving advantage to the method over innumerable
traditional approaches to the theory of hypergeometric series
and special functions are as follows:
\begin{itemize}
\item
Factorization principle is a {\bf key -- stone}
of the new theory
\item
A set of auxiliary identities together with the factorization
formulas play the role of a {\bf meta -- language} giving us the
shortest way to write the derivation story for any formula
\item
Hypergeometric series are expressed only through
hypergeometric series ({\bf closure property}).
On the one hand the hypergeometric series are subjects of
investigation, on the other hand they are investigatory tools.
Such {\bf an ideal correspondence} is the main reason of
extraordinary efficiency of the method
\item
New theoretical concepts ($\Omega$-equivalent relations,
$\Omega$-equivalent identities, $\Omega$-identical
transformations, etc.) have a significant heuristic value and
give a well-structured form to the theory
\item
Bounded universal approach based on the use of
{\bf mac\-ro -- commands} allowed extensive classes of formulas
to be obtained with the help of computer-aided symbolic
transformations
\item
Instead of {\bf separate tools} inherent in other computer
algebra approaches the factorization method gives us an
{\bf integrated system of commands}. Prospective development may
give us a computer algebra superstructure allowing us to work with
any types of hypergeometric series without need to tear ourselves
from keyboard
\end{itemize}
\newpage
\begin{thebibliography}{99}
\bibitem{fpm1}
Niukkanen A.W.\, New theory of hypergeometric series
and new prospects for computer algebra systems.
Fundamental and applied mathematics (in Russian)
v.~5, no~3, pp.~716-745, 1999.
\bibitem{cpc1}
Niukkanen A.W.\, A transformation of $F_4$ suggestive
of a new symbolic software. Computer Physics
Communications v.~126, pp.~137--140, 2000.
\bibitem{cpc2}
Niukkanen A.W., Paramonova O.S.\, Computer generation
of complicated transformations and reduction
formulas for multiple hypergeometric series.
Computer Physics Communications v.~126,
pp.~141 -- 148, 2000.
\bibitem{mz1}
Niukkanen A.W.\, Extention of factorization principle
on general hypergeometric series. Mat. Zametki
(in Russian) v.~67, no~4, pp.~573-581, 2000.
\bibitem{mz2}
Niukkanen A.W.\, General linear transformations of
hypergeometric functions. Mat.Zametki (in Russian)
v.~70, no~5, pp.~769-779, 2001.
\bibitem{fpm2}
Niukkanen A.W.\, Analytical continuation formulas
for multiple hypergeometric series.
Fundamental and applied mathematics (in Russian)
v.~7, no~1, pp.~1-16, 2001.
\bibitem{fpm3}
Niukkanen A.W.\, Quadratic transformations of
multiple hypergeometric series.
Fundamental and applied mathematics (in Russian)
v.~8, no~2, pp.~517-531, 2002.
\bibitem{pro}
Paramonova O.S., Niukkanen A.W.\, Computer--aided
analysis of transformation formulas of the Appell
and the Horn functions. Programmirovanie
(in Russian) no~2, pp.~1-6, 2002.
\bibitem{nima}
Niukkanen A.W.\, On the way to computerizable scientific
knowledge (by the example of the operator factorization
method). Nucl. Instr. and Methods A v.~502,
pp.~639-642, 2003.
\bibitem{1}
I.B. Shchenkov. System of symbolic analytical transformations
SANTRA-2. Description of formal part of command language.
Keldysh Inst. of Appl. Math. Academy of Science USSR, Preprint
No~1, 1989 (32p.).
\bibitem{2}
I.B. Shchenkov. System of symbolic analytical transformations
SANTRA-2. Description of dynamical functions.
Keldysh Inst. of Appl. Math. Academy of Science USSR, Preprint
No~7, 1989 (24p.).
\bibitem{3}
I.B. Shchenkov. System of symbolic analytical transformations
SANTRA-2. Description of functions supporting non-algebraic
operations. Keldysh Inst. of Appl. Math. Academy of Science USSR,
Preprint No~21, 1989 (31p.).
\bibitem{4}
I.B. Shchenkov. System of symbolic analytical transformations
SANTRA-2. Operations over expression belonging to the main
classes. Keldysh Inst. of Appl. Math. Academy of Science USSR,
Preprint No~14, 1991 (38p.).
\bibitem{5}
I.B. Shchenkov. System of symbolic analytical transformations
SANTRA-2. Operations over matrices. Keldysh Inst. of Appl. Math.
Academy of Science USSR, Preprint No~15, 1991 (29p.).
\bibitem{6}
I.B. Shchenkov. System of symbolic analytical transformations
SANTRA-2. Program design and program checkout tools. Keldysh
Inst. of Appl. Math. Academy of Science USSR, Preprint No~1,
1993 (28p.).
\bibitem{7}
A.P.Budnik, E.V.Gai, N.S. Rabotnov, V.F. Turchin, S.V. Popov,
I.B. Shchenkov. Machine performance of analytical transformations
in mathematical physics with the help of REFAL language. Theses.
Symposium on processing of symbolic information. Computer Center
GSSR, Tbilisi, 1970.
\bibitem{8}
I.B. Shchenkov. Program
in mathematical physics with the help of REFAL language. Theses.
Symposium on processing of symbolic information. Computer Center
GSSR, Tbilisi, 1970.
\bibitem{9}
A.P.Budnik, E.V.Gai, N.S. Rabotnov, N.S.Klimov, V.F. Turchin,
I.B. Shchenkov. Basis wave functions and operator matrices
in the nuclei model. Yadernaya fizika (in Russian), V.~14,
No~2, pp.304-314, 1971.
\bibitem{10}
V.N. Vinogradov, F.V. Gai, S.V. Popov, N.C. Rabotnov,
I.B. Shchenkov. Construction of physical bases of O(5) and
SV(3) groups with automated execution of symbolic
transformations. Yadernaya fizika (in Russian), V.~16,
No~6, pp.1178-1187, 1972.
\bibitem{11}
And. V. Klimov, S.A. Romanenko. Programming system Refal-2 for
Unified Series of Electronoc Compating Machines. Description of
command language. IAM, Academy of Science USSR, Moscow: 1987.
\bibitem{12}
And. V. Klimov, S.A. Romanenko. Programming system Refal-2 for
Unified Series of Electronoc Compating Machines. Description of
functions library. IAM, Academy of Science USSR, Preprint No 200,
1996.
\bibitem{13}
S.A. Romanenko. Implementation of Refal-2. IAM, Academy of
Science USSR, Moscow: 1987.
\bibitem{prep2}
A.W. Niukkanen, I.B. Shchenkov, G.B. Efimov. A project of a
globally universal interactive program of formula derivation
based on the operator factorization technique.
Keldysh Inst. of Appl. Math. RAS, Preprint No 82, 2003,
(24 p.).
\bibitem{loz}
D.W. Lozier. The DLMF Project: A new initiative in classical
special functions, in C. Dunkl, M. Ismail and R. Wong, eds.,
Special Functions: Proceedings of the International Workshop,
World Scientific (Singapore), pp. 207-220, 2000.
\end{thebibliography}
\end{document}