No. 19 (00099)

Family name : Niukkanen
Given name : Arthur
Affiliation : Vernadsky Institute, RAS 
Abbreviation : GEOKHI RAS
E-mail address : NIUKKANEN@tula.net; elkor@geokhi.msk.su
Title : OPERATOR FACTORIZATION METHOD AS A BASIS FOR ADVANCED ANALYSIS
        AND ADVANCED SYMBOLIC COMPUTING OF MULTIPLE HYPERGEOMETRIC
        SERIES (theses)
Authors : A.W.Niukkanen
Abstract : 
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{\bf
OPERATOR FACTORIZATION METHOD AS A BASIS FOR
ADVANCED ANALYSIS AND ADVANCED SYMBOLIC COMPUTING OF MULTIPLE
HYPERGEOMETRIC SERIES (theses)
}

\vspace{5mm}
{\large\bf A.W. Niukkanen}\\

Vernadsky Institute for Geo and Analytical Chemistry (GEOKHI RAS),\\
Kossygin St. 19, Laboratory of Molecular Modelling and Spectroscopy, \\
%Mathematical and Computational Group,
119991 Moscow, Russia \\
Phone: (095)1376371, (0872)347383 \quad Fax: (095)9382054 \\
E-mail: elkor@geokhi.msk.su, NIUKKANEN@tula.net
\end{center}


\vspace{5mm}
{\bf 1. $\Omega$-multiplication of functions} $u(x),v(x)$
is defined  by
$w(x)=\,<u * v\,|\,x>\,=u[\,d(s)\,]\,v(xs)\,|_{s=0}$ where
$d(s)$ is differentiation operator.
$\Omega$-mul\-ti\-pli\-ca\-tion is commutative and associative;
$exp(x)$ plays the role of $\Omega$-unit.

{\bf 2. On the products of power series and the products of
their coefficients.}
Let $F[A;x],F[B;x]$ and $F[A,B;x]\equiv F[A\times B;x]$ be the
formal series of functions $x^i/i!\,\,(i=0,1,\ldots)$ with
coefficients $A(i), B(i)$ and $A(i)B(i)$, respectively. It is
known that multiplication of power series corresponds to
convolution of their coefficients. {\it Vice versa},
multiplication of coefficients of two series corresponds to
$\Omega$-multiplication of the series, that is
$F[A\times B;x]\linebreak = \,<F[A]\,*\,F[B]\,|x>\,\equiv
F[A;\,d(s)] F[B;xs]\,|_{s=0}\,$. This formula can serve us as a
simplest example of factorization formula.

{\bf 3. Power series form a commutative group with respect to
$\Omega$-mul\-ti\-pli\-ca\-tion}. This statement follows from
Sec.~1 because the series $F[*/\!/A;\,x]$ with coefficients
$[A(i)]^{-1}$ is $\Omega$-invertion of the $F[A;\,x]$.

{\bf 4. Operator factorization principle.}
Operator factorization formula \\
$
F[A,B,\ldots\,/\!/D,E,\ldots\,;\,x] =\, <F[A] * F[B]\,*\,\ldots
\,*\,F[*/\!/D]\,*\,F[*/\!/E]\,*\ldots\,|\,x\,>\,,
$
\\
reduces any series to simple series (analysis stage). Using the
properties of the simple series and some auxiliary identities
(see Sec.8) one can obtain then a transformed algebraic
expression for the initial series using an appropriate
factorization formula "in reverse order" (synthesis stage).

{\bf 5. For the practical applicability of the factorization
method} it is necessary that the
series under consideration should satisfy
some relations following from their structure rather than being
imposed at will of researcher.
Due to a wide variety of different types of hypergeometric series
arising in applications including combinatorics and probability
theory and great number of their properties these series give us a
natural "touchstone" for the factorization method.

{\bf 6. Hypergeometric series}. Any hypergeometric series is a
particular case of a general hypergeometric series $^N\!F$ in
functions $x_1^{i_1}\,\cdots\,x_N^{i_N}\,/\,i_1!\,\cdots\,i_N!$
\,with coefficients
$(a,\, m_1i_1\,+\cdots+\,m_Ni_N)\,
(b,\,l_1i_1\,+\cdots+\,l_Ni_N) \cdots\,  (\,i_n
=0,1,2,\,\ldots\,;\, \linebreak n=1,\,\ldots\,,N)$, where
$(a,\,I) = a(a+1)\ldots(a+I-1) = \Gamma(a+I)\,/\,\Gamma(a)$ is
the Pochhammer symbol. Denoting by ${\bf m}, {\bf l}$ the
sets of "spectral numbers" $m_n, l_n\, (n=1,\,\ldots\,,N)$, the
general hypergeometric series  $G$ can be written as
$G\equiv ^N\!F[<a\,|\,{\bf m}>,
<b\,|\,{\bf l}>,\,\ldots\,;\, x_1,\,\ldots\,,x_N]$.

{\bf 7. General and special factorization formulas}. General
factorization formula of the series $G$ is analogous to formulas
given in Secs.~2,4: \\
$
G =\! ^N\!F[<\!\!a\,|\,{\bf m}\!>,\,\ldots\,;\,d(s_1),\,\ldots\,,d(s_N)]
^N\!F[<\!\!b\,|\,{\bf l}>,\,\ldots\,;\,x_1s_1,\,\ldots\,,x_Ns_N]|_{{\bf
s}={\bf 0}}.
$
\\
Special dependence of the multipliers
$(a,m_1i_1+\cdots+m_Ni_N)$ on $i_1,\,\ldots\,,i_N$ gives rise to
the special factorization formula  \\
\hspace*{15mm}
$
 G = \,^1F[a;\,d(s)]\,^N\!F[<b\,|\,{\bf
l}>,\,\ldots\,;x_1s^{m_!}\,,\ldots\,,x_Ns^{m_N}]|_{s=0}\,.
$
\\
These two formulas allow any series to be reduced to the
binomial series $F^1_0[a;\,x]$ and Bessel-type series
$F^0_1[*/\!/b;\,z]$.

{\bf 8. Auxiliary identities}. Along with formulas given in
Sec.~7 we use the following "derivation rules":  operator
identities (for example, the shift operator formula
$\exp[u\,d(x)]\,f(x)=f(x+u)$,
operator argument displacement formula
$\exp(-vx)F[d(x)]\exp(vx)=F[d(x)+v]$,
the relations
$F[d(x)]\,\exp(ux)\linebreak =F(u)\,\exp(ux)$,
$d^n(x)\,F[{\bf d};\,ux] =
u^n({\bf d},n)\,F[{\bf d}+n;\,ux]$,
etc.),  operator $\Omega$-eq\-u\-i\-va\-len\-ce formulas
(for example,
$F[d(s)]s^n \Psi(s)|_{s=0} =
F^{(n)}[d(s)] \Psi(s)|_{s=0}$;
$F[d(s)]\Psi(s)|_{s=0} =
f(s)F[d(s)]\Psi(s)|_{s=0}$, if
$f(0)=1$;
$F^1_0[\,a;d(s)\,] \exp(xs)\, \Psi(s)|_{s=0}\,\linebreak
=\, (1-x)^{-a}\, F^1_0[\,a;\,(1-x)^{-1}\,d(s)\,]
\Psi(s)|_{s=0}$\,)  as well as simple
algebraic identities for $F^1_0[a;\,x]$ and
$F^0_0[*;\,z]$, permutations of parameters, etc.

{\bf 9. Within the new mathematical paradigm} (see Sec.~4)
the above formulas together with new theoretical concepts (of
$\Omega$-equivalent operators -- see Sec.~8,
$\Omega$-identical transformations, $\Omega$-equivalent
relations) play the role of "meta-language" making possible
derivation of any formula for any hypergeometric series. These
"derivation rules" can be also looked upon as prototypes
of instructions for {\bf a new algorithmic language} allowing a
universal interactive program of computer analysis of
hypergeometric series to be developed in the near future.

{\bf 10. The fundamentals of a new theory of hypergeometric
functions have been developed}. The main results have been
obtained in the following domains: tranformation properties of
hypergeometric series; finding of analytical coefficients of
functional expansions; reducibility conditions for multiple
series; duality relationship between coefficients occuring in
addition and linearization formulas; recurrence relations;
diagram technique for representation of multiple series; special
properties of particular types of hypergeometric series.

{\bf 11. Analytical merits of the new theory} will be
demonstrated by derivation of 4 universal generating functions
for classical polynomials and by analysis of conditions assuring
positivity of linearization coefficients for the Laguerre
polynomials. {\bf Physical applictions} will be illustrated by
analysis of Davydychev multiple hypergeometric series connected
with Feynman diagrams. {\bf Computer advantages} of the developed
bounded-universal symbolic manipulation programs will be
confirmed by derivation of reduction formulas for multiple
Gel'fand series and automated analysis of transformation
properties of an Appell function. None of the existent computer
algebra systems, nor any of the more specialized symbolic
software can cope with thus much complicated problems.
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