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Basic Polynomial Sequence

A polynomial sequence p_n(x) is called the basic polynomial sequence for a delta operator Q if

1. p_0(x)=1,

2. p_n(0)=0 for all n>0,

3. Qp_n(x)=np_(n-1)(x).

If p_n(x) is a basic polynomial sequence for some delta operator Q, then it is a binomial-type sequence of polynomials. Furthermore, if p_n(x) is a binomial-type sequence of polynomials, then it is a basic polynomial sequence for some delta operator.

See also

Binomial-Type Sequence, Delta Operator, Polynomial Sequence, Umbral Operator

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References

Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684-760, 1973.

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Basic Polynomial Sequence

Cite this as:

Weisstein, Eric W. "Basic Polynomial Sequence." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BasicPolynomialSequence.html

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