Summer Conference

Mini-conference on q-Series
Combinatorics, Classical Number Theory, Special Functions

For three brief days of summer, some of us followed Euler's advice and confugiamus ergo ad series.
The dates were June 20, 21 and 22, and the place was the Department of Mathematics, at The Ohio State University. Several interesting people agreed to visit, partly supported by the Ohio State University Mathematical Institute. The titles of their talks and abstracts are below. The conference is over, but feel free to look at the information below. Click on Euler's picture for the credits.

WHO TALKED ABOUT WHAT

Graduate Student Seminar
Richard Askey
John Bascom Professor of Mathematics
(The University of Wisconsin at Madison)
The binomial theorem and extensions through 25 centuries
ABSTRACT

The origin of the binomial theorem is lost, but we know something about the development of aspects of it. Serious counting as reflected in the binomial coefficients starts in India by 500 BC, and both the Pascal triangle and an explicit formula for the binomial coefficients were found there more than 1000 years ago. The first extension of the binomial theorem was found in China about 700 years ago. The q-extensions of the binomial theorem start with special cases found by Euler, and then a missed opportunity of Rothe. The modern viewpoint which is reflected in quantum groups starts with work of Rodrigues about 1840. The history of this and other work related to the binomial theorem will be the focus of this talk.

Richard Askey
How do special functions arise?
ABSTRACT

Some examples of problems which lead to special functions and questions about them will be given. These include such seemingly simple questions as trying to show that 1/[(1-r)(1-s)+(1-r)(1-t)+(1-s)(1-t)] has positive power series coefficients.
Bruce Berndt
(University of Illinois at Urbana-Champaign)
The Rogers-Ramanujan Continued Fraction
ABSTRACT

Beginning with a paper of L. J. Rogers and Ramanujan's first and second letters to G. H. Hardy, we present a survey of what is known about this famous continued fraction. In particular, focus shall be given to exact evaluations at particular values of the argument, and to modular equations, which are equations relating the continued fraction at different powers of the argument. Almost all of the results are due to Ramanujan and are found in his three notebooks and lost notebook. Some of the results were first proved only in the past few months.

Douglas Bowman
(University of Illinois at Urbana-Champaign)
Analytic continuation of basic hypergeometric series and the inversive closure.
ABSTRACT

In this talk I discuss the analytic continuation of basic hypergeometric functions which works analogous to the proof of the existence of the inversive closure of a difference ring. As applications I give a couple of identities which can be written down by analytically continuing a q-series outside of its circle of convergence; happily the continuation is again easily expressed as convergent q-series. Generalizations of this idea will be briefly discussed.

Heng Huat Chan
(IAS, Princeton)
On Ramanujan's Cubic Transformation for $_2F_1(1/3,2/3;1;z)$
ABSTRACT

Let $$_2F_1(a,b;c;z) = \sum_{n=0}^\infty \dfrac{(a)_n(b)_n}{(c)_n}\dfrac{z^n}{n!} $$ where $(u)_n = (u)(u+1)...(u+n-1)$ and $|z|<1$. In this talk, we present a new proof of the following transformation formula of S. Ramanujan : $$_2F_1\left(\dfrac{1}{3},\dfrac{2}{3};1;1-\left\{\dfrac{1-x}{1+2x}\right\} \right) = (1+2x)\, _2F_1\left(\dfrac{1}{3},\dfrac{2}{3};1;x^3\right),$$ where $|x|<1$.

The TeX version of this abstract.

Sheldon Degenhardt
(Ohio State University)
Weighted Inversion Statistics and their Symmetry Groups
ABSTRACT

A statistic w on the symmetric group S(n) is a weighted-inversion statistic if there is an upper-triangular matrix W = (w[i,j]) such that w(s) is the sum of the elements w[i,j] where s[i]>s[j] for each s in S(n). The two most famous examples are major index (w[i,i+1] = i; w[i,j] = 0 otherwise) and inversion count (w[i,j] = 1). It is well known that these two statistics share the same distribution over S(n), and many bijections on S(n) have been described to prove this. Our question: Which bijections on S(n) have the property that, given any weighted-inversion statistic w there exists another w-i statistic w_f such that w(f(s))=w_f(s) for all s in S(n)?

The LaTeX version of this abstract.

James Haglund
(University of Illinois at Urbana-Champaign)
Rook Theory and Hypergeometric Series
ABSTRACT

We show how to express the number of ways to place k non-attacking rooks on a Ferrers board as a hypergeometric series of Karlsson-Minton type, and discuss how identities for rook polynomials look when translated into hypergeometric notation. We also show how q-rook polynomials count the number of matrices over a finite field satisfying certain constraints.

Christian Krattenthaler
(Institut für Mathematik der Universität Wien)
A new bijective proof of Stanley's hook-content formula for semistandard tableaux
ABSTRACT

A bijective proof for Stanley's hook-content formula for the generating function for semistandard tableaux of a given shape is given that does not involve the involution-principle of Garsia and Milne. The bijection is based on the Hillman-Grassl algorithm and Schützenberger's jeu de taquin.

Christian Krattenthaler
Advanced determinant calculus
ABSTRACT

A complicated binomial determinant is evaluated, thus proving a conjecture of Robbins and Zeilberger. As a special case we obtain a new proof of the enumeration of totally symmetric self-complementary plane partitions. Main purpose of the talk will be to outline the simple ideas that lead to the evaluation of this determinant.

Verne Leininger
(Ohio State University)
Expansions for $(q)_{\infty}^{n^2-1}$ and basic hypergeometric series in U(n).
ABSTRACT

The main theorem is a general eta-function identity. Several special cases have been worked out including $(q)_{\infty}^{24}$ which leads to a formula for the tau function. This identity is equivalent to Lie Theory versions in Lepowsky, Macdonald, and Kac after considerable transformation and rewriting. One advantage is that the proof generalizes the proof of Jacobi of $(q)_ \infty^3$ which Andrews cleaned up using derivatives.

Steve Milne
(Ohio State University)
New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan's tau function
ABSTRACT

In this talk we give two infinite families of explicit exact formulas that generalize Jacobi's (1829) 4 and 8 squares identities to 4n^2 or 4n(n+1) squares, respectively, without using cusp forms. Our new 24 squares identity leads to a new formula for Ramanujan's tau function tau(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Turánian determinants, Fourier series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. We have also obtained many additional infinite families of identities in this same setting that are analogous to the eta-function identities in Appendix I of Macdonald (1972). A special case of our methods contains a proof of the two Kac-Wakimoto (1994) 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_l nonterminating _6phi_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), Bulygin (1914), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Estermann (1936), Lomadze (1948), van der Pol (1954), Krätzel (1961, 1962), Rankin (1962), Gundlach (1978), and, Kac and Wakimoto (1994).

The TeX version of this abstract.

Michael Schlosser
(Institut für Mathematik der Universität Wien)
Multidimensional matrix inversions and multiple basic hypergeometric series.
ABSTRACT

We compute the inverse of a specific infinite r-dimensional matrix, thus unifying multidimensional matrix inversions recently found by Milne, Lilly and Milne, and Bhatnagar and Milne. Our inversion is an r-dimensional extension of a matrix inversion previously found by Krattenthaler. We also compute the inverse of another infinite r-dimensional matrix. As applications of our matrix inversions, we derive new summation and transformation formulas for A_r, C_r, and D_r basic hypergeometric series.

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