Binomial type

:For other concepts using the name "binomial", see binomial (disambiguation).

Characterization by delta operators

It can be shown that a polynomial sequence { pn(x) : n = 0, 1, 2, ... } is of binomial type if and only if all three of the following conditions hold:

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• The linear transformation on the space of polynomials in x that is characterized by

::p_n(x)mapsto np_{n-1}(x)

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:is shift-equivariant, and

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• p₀(x) = 1 for all x, and
• p_n(0) = 0 for n > 0.

(The statement that this operator is shift-equivariant is the same as saying that the polynomial sequence is a Sheffer sequence; the set of sequences of binomial type is properly included within the set of Sheffer sequences.)

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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

Related Topics:
Delta operator - Difference operator - Power series

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:Q=sum_{n=1}^infty c_n D^n

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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

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• $p\_0(x)=1,\; ,$
• $p\_n(0)=0quad\{\; m\; for\; \}ngeq\; 1,\{\; m\; and\}$
• $Qp\_n(x)=np\_\{n-1\}(x).\; ,$

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.

Related Topics:
1973 - Rota - Odlyzko

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