Recursive matrices and umbral calculus

The Umbral Calculus of Symmetric Functions

“…By an expansion argument, we have only to prove that for every partition :Á , (L ; 9 | p :Á (X)) Ä 0. Assuming that l=l( ;9 )>l(:Á ), for any sequence of partitions (:Á (1) , :Á (2) , ..., :Á (l ) ), adding up to :Á , at least one of them is equal to zero. Let ({ 1 { 2 } } } { l ) be the partition ;9 in standard notation.…”

“…Consider now the sequence of infinite sets of variables X (1) , X (2) , ..., X (k) . Since 4(X (1) , X (2) , ..., X (k) ) is isomorphic to the tensor product…”

“…where the sum of the above ranges over the tuples of partitions (:Á (1) , :Á (2) , ..., :Á (k) ) satisfying …”

“…For three partitions :Á , ;9 , and #Á satisfying ;9 +#Á =:Á we define the binomial coefficient \ :Á :Á (1) , :Á (2) , ..., :Á…”

Representation of doubly infinite matrices as non-commutative Laurent series

“…Jabotinky [3] represented composition operators by means of doubly in nite matrices to study iteration of analytic functions. Barnabei, Brini, and Nicoletti studied recurrence properties and connections with the Umbral Calculus of doubly in nite matrices in [1]. More recently, several authors have extended the Riordan groups, which are generated by multiplication and composition operators, to the bi-in nite context.…”