Sheffer sequence
Template:Use American English Template:Short description In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence Template:Math of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics. They are named for Isador M. Sheffer.
Definition
Fix a polynomial sequence Template:Math Define a linear operator Template:Mvar on polynomials in Template:Mvar by <math display="block"> Q[ p_n(x) ] = n p_{n-1}(x) ~.</math>
This determines Template:Mvar on all polynomials. The polynomial sequence Template:Math is a Sheffer sequence if the linear operator Template:Mvar just defined is shift-equivariant; such a Template:Mvar is then a delta operator. Here, we define a linear operator Template:Mvar on polynomials to be shift-equivariant if, whenever Template:Math is a "shift" of Template:Math then Template:Math i.e., Template:Mvar commutes with every shift operator: Template:Math.
Properties
The set of all Sheffer sequences is a group under the operation of umbral composition of polynomial sequences, defined as follows. Suppose Template:Math and Template:Math are polynomial sequences, given by <math display="block"> p_n(x) = \sum_{k=0}^n a_{n,k}x^k\ \mbox{and}\ q_n(x) = \sum_{k=0}^n b_{n,k}x^k ~.</math>
Then the umbral composition <math>p \circ q</math> is the polynomial sequence whose Template:Mvarth term is <math display="block">(p_n\circ q)(x) = \sum_{k=0}^n a_{n,k}q_k(x) = \sum_{0\le \ell \le k \le n} a_{n,k}b_{k,\ell}x^\ell</math> (the subscript Template:Mvar appears in pn, since this is the Template:Mvar term of that sequence, but not in q, since this refers to the sequence as a whole rather than one of its terms).
The identity element of this group is the standard monomial basis <math display="block">e_n(x) = x^n = \sum_{k=0}^n \delta_{n,k} x^k.</math>
Two important subgroups are the group of Appell sequences, which are those sequences for which the operator Template:Mvar is mere differentiation, and the group of sequences of binomial type, which are those that satisfy the identity <math display="block"> p_n(x+y) = \sum_{k=0}^n\ {n \choose k}\ p_k(x)\ p_{n-k}(y) ~.</math> A Sheffer sequence Template:Mathis of binomial type if and only if both <math display="block"> p_0(x) = 1\ </math> and <math display="block"> p_n(0) = 0 \quad \mbox{ for } \quad n \ge 1 ~.</math>
The group of Appell sequences is abelian; the group of sequences of binomial type is not. The group of Appell sequences is a normal subgroup; the group of sequences of binomial type is not. The group of Sheffer sequences is a semidirect product of the group of Appell sequences and the group of sequences of binomial type. It follows that each coset of the group of Appell sequences contains exactly one sequence of binomial type. Two Sheffer sequences are in the same such coset if and only if the operator Q described above – called the "delta operator" of that sequence – is the same linear operator in both cases. (Generally, a delta operator is a shift-equivariant linear operator on polynomials that reduces degree by one. The term is due to F. Hildebrandt.)
If Template:Math is a Sheffer sequence and Template:Math is the one sequence of binomial type that shares the same delta operator, then <math display="block"> s_n(x+y) = \sum_{k=0}^n\ {n \choose k}\ p_k(x)\ s_{n-k}(y) ~.</math>
Sometimes the term Sheffer sequence is defined to mean a sequence that bears this relation to some sequence of binomial type. In particular, if Template:Math is an Appell sequence, then <math display="block"> s_n(x+y) = \sum_{k=0}^n\ {n \choose k}\ x^k\ s_{n-k}(y) ~.</math>
The sequence of Hermite polynomials, the sequence of Bernoulli polynomials, and the monomials Template:Math are examples of Appell sequences.
A Sheffer sequence Template:Math is characterised by its exponential generating function <math display="block"> \sum_{n=0}^\infty\ \frac{p_n(x)}{n!}\ t^n = A(t)\ \exp\!\bigl(\ x\ B(t)\ \bigr)\ </math> where Template:Mvar and Template:Mvar are (formal) power series in Template:Mvar. Sheffer sequences are thus examples of generalized Appell polynomials and hence have an associated recurrence relation.
Examples
Examples of polynomial sequences which are Sheffer sequences include:
- The Abel polynomials
- The Bernoulli polynomials
- The Euler polynomials
- The central factorial polynomials
- The Hermite polynomials
- The Laguerre polynomials
- The monomials Template:Math
- The Mott polynomials
- The Bernoulli polynomials of the second kind
- The Falling and rising factorials
- The Touchard polynomials
- The Mittag-Leffler polynomials
References
- Template:Cite journal Reprinted in the next reference.
- Template:Cite book
- Template:Cite journal
- Template:Cite book Reprinted by Dover, 2005.