A generalized q-binomial Vandermonde convolution of Sulanke is proved using a generalization of the Durfee square of a partition.
Bender [4] proved the following generalization of the q-binomial Vandermonde convolution:
where for l~ O,
Bender's identity was generalized by Evans [5], who Obtained a formula for the difference between the sides of (1) when the restriction on A~Γ·I-A~ is removed. Sulanke [7] generalized Bender's identity in a different direction, using lattice paths.
We prove here a more symmetrical version of Sulanke's identity, using a generalization of the concept of the Duffee square of a partition. Although our proof is ultimately equivalent to Sulanke's, the connection with partitions suggests further generalization and applications.
A partition ~ of a nonnegative integer l is a (possibly empty) sequence )t1~>)t2>~"" "~>Xk Of positive integers with sum l=lxl. The Ferrets graph of )t is the set of all pairs (i, ]) of integers with 1 ~< i ~< k and 1 ~<] ~< ~. The points of the Ferrers graph are displayed like entries of a matrix; thus the Ferrers graph of 421 is The Durfee square of a partition ,k is the largest square which fits in the upper \* Research partially supported by NSF Grant MCS 8105188