4 Delta operators

That linear transformation is clearly a delta operatorIn mathematics, a delta operator is a shift-equivariant linear operator Q on the vector space of polynomials in a variable x that reduces degrees by one. To say that Q is shift-equivariant means that if f ''x g ''x + a , i. f is a "shift" of g then Qf x Q, i.e., a shift-equivariant linear transformation on the space of polynomials in x that reduces degrees of polynomials by 1. The most obvious examples of delta operators are difference operatorIn mathematics, a difference operator maps a function f ''x to another function f ''x + a − f ''x + b . The forward difference operator : occurs frequently in the calculus of finite differences, where it plays a role formally similar to that of thes and differentiation. It can be shown that every delta operator can be written as a power seriesIn mathematics, a power series (in one variable) is an infinite series of the form :: where the coefficients a the center a and the argument x are usually real or complex numbers. These series usually arise as the Taylor series of some known function; the of the form

where D is differentiation (note that the lower bound of summation is 1). Each delta operator Q has a unique sequence of "basic polynomials", i.e., a polynomial sequence satisfying

It was shown in 1973 by Rota, Kahaner, and Odlyzko, that a polynomial sequence is of binomial type if and only if it is the sequence of basic polynomials of some delta operator. Therefore, this paragraph amounts to a recipe for generating as many polynomial sequences of binomial type as one may wish.

5 Umbral composition of polynomial sequences

The set of all polynomial sequences of binomial type is a group in which the group operation is "umbral composition" of polynomial sequences. That operation is defined as follows. Suppose { p_n(x) : n = 0, 1, 2, 3, ... } and { q_n(x) : n = 0, 1, 2, 3, ... } are polynomial sequences, and

Then the umbral composition p o q is the polynomial sequence whose nth term is

With the delta operator defined by a power series in D as above, the natural bijection between delta operators and polynomial sequences of binomial type, also defined above, is a group isomorphism, in which the group operation on power series is (perhaps surprisingly) formal composition of formal power series.

6 Characterization by generating functions

Polynomial sequences of binomial type are precisely those whose generating functions are formal (not necessarily convergent) power series of the form

where f(t) is a formal power series whose constant term is zero and whose first-degree term is not zero. It can be shown by the use of the power-series version of Faà di Bruno's formula that

The delta operator of the sequence is f⁻¹(D), so that

6.1 A way to think about these generating functions

The coefficients in the product of two formal power series

and

are

(see also Cauchy product). If we think of x as a parameter indexing a family of such power series, then the binomial identity says in effect that the power series indexed by x + y is the product of those indexed by x and by y. Thus the x is the argument to a function that maps sums to products: an exponential function

where f(t) has the form given above.

7 Cumulants and moments

The sequence κ_n of coefficients of the first-degree terms in a polynomial sequence of binomial type may be termed the cumulants of the polynomial sequence. It can be shown that the whole polynomial sequence of binomial type is determined by its cumulants, in a way discussed in the article titled cumulant. Thus

the nth cumulant

and

the nth moment.

These are "formal" cumulants and "formal" moments, as opposed to cumulants of a probability distribution and moments of a probability distribution.

Let

be the (formal) cumulant-generating function. Then

is the delta operator associated with the polynomial sequence, i.e., we have

8 Applications

The concept of binomial type has applications in combinatorics, probability, statistics, and a variety of other fields.

9 References

• G.-C. Rota, D. Kahaner, and A. Odlyzko , "Finite Operator Calculus," Journal of Mathematical Analysis and its Applications, vol. 42, no. 3, June 1973. Reprinted in the book with the same title, Academic Press, New York, 1975.

• R. Mullin and G.-C. Rota, "On the Foundations of Combinatorial Theory III: Theory of Binomial Enumeration," in Graph Theory and Its Applications, edited by Bernard Harris, Academic Press, New York, 1970.

As the title suggests, the second of the above is explicit about applications to combinatorial enumeration.

Polynomials Combinatorics

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