Conjecturing and proving identities of the form A = B (where A is sum of “nice” terms (such as binomial)  and B is a closed form or a sum of “nice” terms)  is  among ancient and attractive mathematical problems. There are several types of proof techniques from various areas of mathematics that can be used to prove such identities. Among these techniques, we mention three here: enumerative combinatorics, generatingfunctionology and the Wilf-Zeilberger (WZ) method.  Enumerative combinatorics deals with counting the number of certain combinatorial objects. It gives meaning and understanding of such objects, and provides an elegant and creative way of verification. Many problems that arise in applications have a relatively simple combinatorial interpretation. For example the binomial coefficient nCk  counts the number of different ways of selecting k objects from a set of n objects.  Even though this method gives combinatorial interpretations, it is at times challenging to find combinatorial descriptions for identities that have multiple parameters or involve non-integral values. For an introduction on combinatorial argument techniques, see, among others, the books:

❶ Proofs that Really Count: The Art of Combinatorial Proof, by A. T. Benjamin and J. J. Quinn.

❷ A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory, by M. Bona.

Another classical method for proving identities is the generatingfunctionology technique which is  time-tested since Euler. It uses formal series expansions  and is flexible enough to be applied to many situations. Formal series expansions satisfy additive and multiplicative properties making them one of the most widely used methods for proving identities. However, this approach involves tedious algebraic manipulations and lacks to give meaning of the identities. A classical book that discusses the generatingfunctionology method is generatingfunctionology, by H. S. Wilf. The Wilf-Zeilberger (WZ)  method is one of the most recent, efficient, computer-assisted/automated revolutionary technique for proving identities. Among its powerful features are its instant generation of elegant and short proofs and its ability to validate identities for non- integral values of free parameters (such as, real, complex, even indeterminates) as well as broader generalizations. For a superb exposition of the WZ method see, among others, the book A = B which is devoted to this and other methods. For an abridged (10 minutes) introduction to the WZ proof style see What Is the WZ pairs?.

Participants will:
●explore, conjecture and formulate formulas for summation expressions,
●develop and master fundamental and useful skills for identity proving: combinatorial (counting) methods and computerized proof techniques (the Wilf-Zeilberger Method), and,
●apply various computer summation algorithms for discovering and proving challenging and interesting combinatorial-identities.

● familiarity in combinatorial concepts from a discrete mathematics course,
● positive attitude and a good work ethic,
● willingness and eagerness to work independently as well as collaboratively,
● familiarity in using  computer algebra systems or software (such as Maple or Mathematica)  are recommended but not required.

For application information and instructions, please visit the GVSU Summer Mathematics REU home page.