Pochhammer Symbol - Relation To Umbral Calculus

Relation To Umbral Calculus

The falling factorial occurs in a formula which represents polynomials using the forward difference operator Δ and which is formally similar to Taylor's theorem of calculus. In this formula and in many other places, the falling factorial (x)_k in the calculus of finite differences plays the role of xk in differential calculus. Note for instance the similarity of

to

A similar result holds for the rising factorial.

The study of analogies of this type is known as umbral calculus. A general theory covering such relations, including the falling and rising factorial functions, is given by the theory of polynomial sequences of binomial type and Sheffer sequences. Rising and falling factorials are Sheffer sequences of binomial type:

where the coefficients are the same as the ones in the expansion of a power of a binomial (Chu-Vandermonde identity).

Similarly, the generating function of Pochhammer polynomials then amounts to the umbral exponential,

as Δ(1+t )x = t (1+t )x.