Sheffer sequence explained

In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence 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 (p_n). Define a linear operator Q on polynomials in x by$$Qp\_n(x)\; =\; np\_(x)\backslash ,\; .$$

This determines Q on all polynomials. The polynomial sequence p_n is a Sheffer sequence if the linear operator Q just defined is shift-equivariant; such a Q is then a delta operator. Here, we define a linear operator Q on polynomials to be shift-equivariant if, whenever f(x) = g(x + a) = T_a g(x) is a "shift" of g(x), then (Qf)(x) = (Qg)(x + a); i.e., Q commutes with every shift operator: T_aQ = QT_a.

Properties

The set of all Sheffer sequences is a group under the operation of umbral composition of polynomial sequences, defined as follows. Suppose ( p_n(x) : n = 0, 1, 2, 3, ... ) and ( q_n(x) : n = 0, 1, 2, 3, ... ) are polynomial sequences, given by$$p\_n(x)=\backslash sum\_^n\; a\_x^k\backslash \; \backslash mbox\backslash \; q\_n(x)=\backslash sum\_^n\; b\_x^k.$$

Then the umbral composition

p \circ q is the polynomial sequence whose nth term is$$(p\_n\backslash circ\; q)(x)\; =\; \backslash sum\_^n\; a\_q\_k(x)\; =\; \backslash sum\_\; a\_b\_x^\backslash ell$$(the subscript n appears in p_n, since this is the n 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$$e\_n(x)\; =\; x^n\; =\; \backslash sum\_^n\; \backslash delta\_\; x^k.$$

Two important subgroups are the group of Appell sequences, which are those sequences for which the operator Q is mere differentiation, and the group of sequences of binomial type, which are those that satisfy the identity$$p\_n(x+y)\; =\; \backslash sum\_^np\_k(x)p\_(y).$$A Sheffer sequence ( p_n(x) : n = 0, 1, 2, ... ) is of binomial type if and only if both$$p\_0(x)\; =\; 1\backslash ,$$and$$p\_n(0)\; =\; 0\backslash mbox\; n\; \backslash ge\; 1.\; \backslash ,$$

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 s_n(x) is a Sheffer sequence and p_n(x) is the one sequence of binomial type that shares the same delta operator, then$$s\_n(x+y)=\backslash sum\_^np\_k(x)s\_(y).$$

Sometimes the term Sheffer sequence is defined to mean a sequence that bears this relation to some sequence of binomial type. In particular, if ( s_n(x)) is an Appell sequence, then$$s\_n(x+y)=\backslash sum\_^nx^ks\_(y).$$

The sequence of Hermite polynomials, the sequence of Bernoulli polynomials, and the monomials (xⁿ : n = 0, 1, 2, ...) are examples of Appell sequences.

A Sheffer sequence p_n is characterised by its exponential generating function$$\backslash sum\_^\backslash infty\; \backslash frac\; t^n\; =\; A(t)\; \backslash exp(x\; B(t))\; \backslash ,$$where A and B are (formal) power series in t. 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 polynomial;
• The central factorial polynomials;
• The Hermite polynomials;
• The Laguerre polynomials;
• The monomials (xⁿ : n = 0, 1, 2, ...);
• The Mott polynomials;
• The Bernoulli polynomials of the second kind;
• The Falling and rising factorials;
• The Touchard polynomials;
• The Mittag-Leffler polynomials;

References

• Rota . G.-C. . Gian-Carlo Rota . Kahaner . D. . Odlyzko . A. . Andrew Odlyzko . On the Foundations of Combinatorial Theory VIII: Finite Operator Calculus . Journal of Mathematical Analysis and Applications . 42 . 3 . June 1973 . 684–750 . 10.1016/0022-247X(73)90172-8. free . Reprinted in the next reference.
• Book: Rota . G.-C. . Gian-Carlo Rota . Doubilet . P. . Greene . C. . Kahaner . D. . Odlyzko . A. . Stanley . R. . Finite Operator Calculus . Academic Press . 1975 . 0-12-596650-4.
• Sheffer . I. M. . Isador M. Sheffer . Some Properties of Polynomial Sets of Type Zero . . 5 . 3 . 590–622 . 1939 . 10.1215/S0012-7094-39-00549-1.
• Book: Roman, Steven . The Umbral Calculus . Academic Press Inc. [Harcourt Brace Jovanovich Publishers] . London . Pure and Applied Mathematics . 978-0-12-594380-2 . 741185 . 1984 . 111 . Reprinted by Dover, 2005.