In honor of Gian-Carlo Rota's 64th birthday, the
*RotaFest* conference was organized by Richard Stanley, Neil White
and Richard Ehrenborg at the *Massachusetts Institute of Technology*,
Cambridge, Massachusetts, USA from April 17 to April 20, 1996.

A
workshop on Umbral Calculus was held on
April 22--23 as an addition to the* RotaFest*. The workshop was
organized by Daniel Loeb, Nigel Ray and Alessandro Di Bucchianico with
much support from Rotafest organizer Richard Stanley and secretarial staff
of M.I.T.

The goal of this workshop was to give an overview of the
development since Rota's seminal papers on the subject in the early
seventies and to present new developments. Although Umbral Calculus is
classified under combinatorics in the AMS classification, it has
applications to several fields of mathematics, including special
functions. We refer to
Dynamic Survey 3 of the
* Electronic
Journal of Combinatorics* for more details.

Talks were delivered (in chronological order) by Brian Taylor, Marilena Barnabei, Heinrich Niederhausen, Luis Verde-Star, Jet Wimp, George Andrews, Mourad Ismail, Miguel Méndez, William Chen, Daniel Loeb, Ottavio D'Antona, Nigel Ray, Philip Feinsilver, Henryk Gzyl, Alessandro Di Bucchianico, and Jack Freeman. Below we will briefly summarize those talks related to orthogonal polynomials or special functions.

- Brian Taylor presented joint work with Gian-Carlo Rota on a rigorous foundation of the classical umbral calculus of the previous century. Brian illustrated the high computational power of this approach by giving one-line proofs of properties of the Bernoulli polynomials.
- Luis Verde-Star presented a very general Hopf algebra structure
that he
showed to be the underlying structure of many analytic methods such as
Laplace
transforms etc. As a nice application he gave a one-line calculation
without
induction of the integral
\int_{0}^{\infty}\, x^n\, e^{-x}\, dx.

- Jet Wimp showed how cut operators on Laurent series form a powerful tool for proving hypergeometric identities.
- Mourad Ismail discussed various properties of the Askey-Wilson operators.
- William Chen showed how elementary identities and a clever use of
the
operator
\vartheta: p(x) \mapsto \frac{p(qx)-p(x)}{x}

yields many basic hypergeometric identities and transformations.

- Daniel Loeb presented ongoing joint research with Gian-Carlo Rota
and
Alessandro Di Bucchianico on a basis-free infinite-dimensional umbral
calculus.
As a first application, he showed how to calculate integrals of the form
\int_{\mid\mid x\mid\mid=1}\, x_1^{d_1}\, \ldots x_n^{d_n}\, d\mu(x),

where \mu is an orthogonally invariant measure.

- Philip Feinsilver showed results from joint research with René Schott on linearization and abelianization of Lie algebras with various applications to hypergeometric functions.
- Alessandro Di Bucchianico gave an umbral approach to variance functions of exponential families (joint work with Daniel Loeb). He gave a new proof of the following theorem of Philip Feinsilver: The variance function of a natural exponential family is a polynomial of degree at most 2 if and only if the associated Sheffer polynomials are orthogonal.
- Jack Freeman showed an algebraic transform method that can be used to find orthogonal polynomials (with respect to a linear functional) within a certain class. He applied his method to the classes of Sheffer polynomials and Chebyshev-like polynomials.

The workshop was attended by approximately 40 to 60 people.

Alessandro DiBucchianico

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