Binomial type 

Examples

• In consequence of this definition the binomial theorem can be stated by saying that the sequence { xⁿ : n = 0, 1, 2, ... } is of binomial type.

• The sequence of "lower factorials" is defined by

<math>(x)_n=x(x-1)(x-2)\cdot\cdots\cdot(x-n+1).</math>

(In the theory of special functions, this same notation denotes upper factorials, but this present usage is universal among combinatorialists.) The product is understood to be 1 if n = 0, since it is in that case an empty product. This polynomial sequence is of binomial type.

• Similarly the "upper factorials"

<math>x^{(n)}=x(x+1)(x+2)\cdot\cdots\cdot(x+n-1)</math>

are a polynomial sequence of binomial type.

• The Abel polynomials

<math>p_n(x)=x(x-an)^{n-1}</math>

are a polynomial sequence of binomial type.

• The Touchard polynomials[?]

<math>p_n(x)=\sum_{k=1}^n S(n,k)x^k</math>

where S(n, k) is the number of partitions of a set of size n into k disjoint non-empty subsets, is a polynomial sequence of binomial type. Eric Temple Bell called these the "exponential polynomials" and that term is also sometimes seen in the literature. The coefficients S(n, k ) are "Stirling numbers of the second kind". This sequence has a curious connection with the Poisson distribution: If X is a random variable with a Poisson distribution with expected value λ then E(Xⁿ) = p_n(λ). In particular, when λ = 1, we see that the nth moment of the Poisson distribution with expected value 1 is the number of partitions of a set of size n, called the nth Bell number. This fact about the nth moment of that particular Poisson distribution is "Dobinski's formula".

A simple characterization

It can be shown that a polynomial sequence { p_n(x) : n = 0, 1, 2, ... } is of binomial type if and only if the linear transformation on the space of polynomials in x that is characterized by

<math>p_n(x)\mapsto np_{n-1}(x)</math>

Delta operators

That linear transformation is clearly a delta operator, 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 operators and differentiation. It can be shown that every delta operator can be written as a power series of the form

<math>Q=\sum_{n=1}^\infty c_n D^n</math>

1. <math>p_0(x)=1,</math>
2. <math>p_n(0)=0\quad{\rm for\ }n\geq 1,{\rm\ and}</math>
3. <math>Qp_n(x)=np_{n-1}(x).</math>

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

<math>p_n(x)=\sum_{k=0}^n a_{n,k}x^k.</math>

<math>(p_n\circ q)(x)=\sum_{k=0}^n a_{n,k}q_k(x).</math>

Applications

The concept of binomial type has applications in combinatorics.

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.

instinctive conservatism.html">conservatism of mankind is sure to make all processes of to this by the avoidable conservatism which is forced upon organized does not spur them on. I think it is only a small percentage of whom might easily become industrious if they were given more congenial work by any such methods are probably to be regarded as pathological cases, residue must be set the very much larger number who are now ruined in and the great irregularity of their employment.html">employment. To very many, health. The most dangerous aspect of the tyranny of the employer is the power working hours. A man may.html">may be dismissed because the employer dislikes immoral. He may be dismissed because he tries to produce a spirit.html">spirit of find employment merely on the ground that he is better educated than present. This evil.html">evil would not be remedied, but rather intensified, there is no refuge from its prejudices such as may now accidentally would be able to enforce any system of beliefs it happened to like, would be penalized, and all independence of spirit would die out. Any rigid system would involve this evil. It is very necessary that Minorities must be able to live and develop their opinions freely. If force all men into one mold and make all vital progress impossible. For these reasons, no one ought to be allowed to suffer destitution so to be made into opinion or private life. It is only on this basis tyranny and terror. productivity of labor. So long as it was necessary to the bare hours for a pittance, so long no civilization was possible except an .

 On wordlookup.net  

All is still licensed under the GNU FDL.
It uses material from the wikipedia.