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{\Large Stephen C. Milne}\\

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The Ohio State University, U.S.A.

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{\bf Lecture 1\\
Transformations of U(n+1) multiple basic hypergeometric series\\}

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{\bf Abstract}
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The purpose of this talk is to survey the transformation theory of $U(n+1)$
multiple basic hypergeometric series---starting with the $U(n+1)$
terminating very-well-poised $_6\phi_5$ summation theorems.
These series were strongly motivated by L. C. Biedenharn and J. D. Louck
and coworkers mathematical physics research involving angular momentum theory
and the unitary groups $U(n+1)$, or equivalently $A_n$.  They are directly
related to the corresponding Macdonald identities.  This $U(n+1)$ or $A_n$
theory has also been extended to the root systems $C_n$ and $D_n$.
There are now many applications of $A_n$ and/or subsequently $C_n$ and $D_n$
multiple basic hypergeometric series. These include the following topics:
A. N. Kirillov--quantum groups; R. Gustafson--multidimensional beta and/or
Barnes integral evaluations; C. Krattenthaler and I. Gessel--plane partition
enumeration; S. Milne--analytic number theory-sums of  squares; S. Milne, G. Lilly,
G. Bhatnagar, C. Krattenthaler, and M. Schlosser--multidimensional matrix inversions;
S. Milne and V. Leininger--new infinite families of eta function identities;
Y. Kajihara and M. Noumi--applications to raising operators for Macdonald polynomials
(these operators were first introduced by A. N. Kirillov and M. Noumi,
then by L. Lapointe and L. Vinet, and later by Y. Kajihara and M. Noumi.)
As an introduction to this area we discuss some of the main results and techniques
from the following outline of the development of $U(n+1)$ basic hypergeometric series.

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The $U(n+1)$ terminating very-well-poised $_6\phi_5$ summation theorems extend
Rogers' classical one-variable work and are central to our theory. They may be
proved using $q$-difference equations---arising from the Lagrange interpolation
formula---and partial fraction expansions. The $U(n+1)$ $_6\phi_5$ summation
theorems may in turn be specialized to obtain the $U(n+1)$ extension of
Andrews' matrix formulation of the Bailey Transform. The $U(n+1)$ Bailey transform
is then applied to the $U(n+1)$ $_6\phi_5$ summation theorems to derive
the $U(n+1)$ terminating, balanced $_3\phi_2$ summation theorems, whose special cases
include $U(n+1)$ $q$-Gauss summation theorems, $q$-Chu-Vandermonde theorems
and $U(n+1)$ $q$-binomial theorems. An analytic continuation argument
applied to a $U(n+1)$ $q$-binomial theorem yields the $U(n+1)$ extension of
Ramanujan's $_1\psi_1$ sum, and Gustafson's $U(n+1)$ $_6\psi_6$ summation
turns out to be the next higher dimensional version of the $U(n+1)$ $_1\psi_1$ sum.
For example, the two-dimensional $U(3)$ $_1\psi_1$ summation is equivalent
to Bailey's classical one-dimensional $_6\psi_6$ summation.
We also obtain $U(n+1)$ extensions of the Jacobi triple product identity.
The Bailey transform coupled with the $U(n+1)$ balanced $_3\phi _2$ summation
theorems yields the $U(n+1)$ extension of Andrews' explicit formulation of
the Bailey Lemma, which---upon iteration---gives several $U(n+1)$ generalizations
of Watson's $q$-analogue of Whipple's transformation formula.
Special and limiting cases include the non-terminating $U(n+1)$ $_6\phi_5$ summation,
the $U(n+1)$ extension of the terminating balanced $_8\phi_ 7$ summation theorem,
and the $U(n+1)$ Rogers--Ramanujan--Schur identities. A classical interchange
of summation argument leads to the $U(n+1)$ $_{10}\phi_9$ transformation formulas.
Important limiting cases include the $U(n+1)$ generalization of Bailey's
non-terminating extension of Watson's transformation. This in turn leads to
the non-terminating $U(n+1)$ extension of Bailey's balanced $_8\phi_ 7$
summation theorem.  The classical case of all this work, corresponding
to $A_1$ or equivalently  $U(2)$, contains a substantial amount of
the theory and application of one-variable basic hypergeometric series.

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{\bf Lecture 2\\
Infinite families of exact sums of squares formulas,\\
Jacobi elliptic Functions, continued fractions, and Schur functions}

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{\bf Abstract}
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In this talk we give several infinite families of explicit exact formulas
involving either squares or triangular numbers, two of which generalize Jacobi's (1829)
$4$ and $8$ squares identities to $4n^2$ or $4n(n+1)$ squares, respectively,
without using cusp forms.  (In fact, we similarly have generalized to infinite
families all of Jacobi's explicitly stated degree $2$, $4$, $6$, $8$ Lambert series
expansions of classical theta functions. In addition, we have extended
Jacobi's special analysis of $2$ squares, $2$ triangles, $6$ squares, $6$ triangles
to $12$ squares, $12$ triangles, $20$ squares, $20$ triangles, respectively.)
These results, depending on new expansions for powers of various products of
classical theta functions, arise in the setting of Jacobi elliptic functions,
associated continued fractions, regular C-fractions, Hankel or Tur\'anian determinants,
Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula
for determinants, and Schur functions.  The Schur function form of
these infinite families of identities are analogous to the $\eta$-function identities
of  Macdonald. Moreover, the powers $4n(n+1)$, $2n^2+n$, $2n^2-n$ that appear in
Macdonald's work also arise at appropriate places in our analysis.
We also utilize a special case of our methods to give a proof of
the two Kac--Wakimoto conjectured identities involving representing
a positive integer by sums of $4n^2$ or $4n(n+1)$ triangular numbers, respectively.
Our $16$ and $24$ squares identities were originally obtained via
multiple basic hypergeometric series, Gustafson's $C_{\ell}$ nonterminating
${}_6\phi_5$ summation theorem,  and Andrews' basic hypergeometric series
proof of Jacobi's $2$, $4$, $6$, and $8$ squares identities.
We have (elsewhere) applied symmetry and Schur function techniques to
this original approach to prove the existence of similar infinite families
of sums of squares identities for $n^2$ or $n(n+1)$ squares, respectively.
Our sums of more than $8$ squares identities are not the same as the formulas
of Mathews (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, 1919),
Hardy (1918, 1920), Kac and Wakimoto (1994), and many others.


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