An Extension of Ito’s Differentiation Formula

An extension of the Cayley-Sylvester formula

We extend the Cayley-Sylvester formula for symmetric powers of SL2(C)modules, to plethysms defined by rectangle partitions. Ordinary partitions are replaced by plane partitions, and an extension of the Hermite reciprocity law follows. 1. Let Sd denote the irreducible SL2(C)-module of dimension d + 1, which can be identified with the space of homogeneous polynomials of degree d in two variables....

An extension of the Clark-Ocone formula

ON AN EXTENSION OF A QUADRATIC TRANSFORMATION FORMULA DUE TO GAUSS

The aim of this research note is to prove the following new transformation formula begin{equation*} (1-x)^{-2a},_{3}F_{2}left[begin{array}{ccccc} a, & a+frac{1}{2}, & d+1 & & \ & & & ; & -frac{4x}{(1-x)^{2}} \ & c+1, & d & & end{array}right] \ =,_{4}F_{3}left[begin{array}{cccccc} 2a, & 2a-c, & a-A+1, & a+A+1 & & \ & & & & ; & -x \ & c+1, & a-A, & a+A & & end{array} right], end{equation*} wher...

An Extension of Reduction Formula for Littlewood-richardson Coefficients

There is a well-known classical reduction formula by Griffiths and Harris for Littlewood-Richardson coefficients, which reduces one part from each partition. In this article, we consider an extension of the reduction formula reducing two parts from each partition. This extension is a special case of the factorization theorem of Littlewood-Richardson coefficients by King, Tollu, and Toumazet (th...

An extension of the exponential formula in enumerative combinatorics

Let α be a formal variable and Fw be a weighted species of structures (class of structures closed under weight-preserving isomorphisms) of the form Fw = E(F c w), where E and F c w respectively denote the species of sets and of connected Fw-structures. Multiplying by α the weight of each F c wstructure yields the species Fw(α) = E(F c αw). We introduce a “universal” virtual weighted species, Λ,...