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