Sheffer Polynomial
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Sheffer sequence or poweroid is a
polynomial sequence In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Polynomial sequences are a topic of interest in ...
, i.e., a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
s in which the index of each polynomial equals its degree, satisfying conditions related to the
umbral calculus The term umbral calculus has two related but distinct meanings. In mathematics, before the 1970s, umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to prove ...
in
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
. They are named for Isador M. Sheffer.


Definition

Fix a polynomial sequence (''p''''n''). Define a
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
''Q'' on polynomials in ''x'' by Qp_n(x) = np_(x)\, . 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 In mathematics, a delta operator is a shift-equivariant linear operator Q\colon\mathbb \longrightarrow \mathbb /math> on the vector space of polynomials in a variable x over a field \mathbb that reduces degrees by one. To say that Q is shift-equi ...
. 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 In mathematics, and in particular functional analysis, the shift operator, also known as the translation operator, is an operator that takes a function to its translation . In time series analysis, the shift operator is called the '' lag opera ...
: ''T''''a''''Q'' = ''QT''''a''.


Properties

The set of all Sheffer sequences is a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
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)=\sum_^n a_x^k\ \mbox\ q_n(x)=\sum_^n b_x^k. Then the umbral composition p \circ q is the polynomial sequence whose ''n''th term is (p_n\circ q)(x) = \sum_^n a_q_k(x) = \sum_ a_b_x^\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 = \sum_^n \delta_ x^k. Two important
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
s are the group of
Appell sequence In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence \_ satisfying the identity :\frac p_n(x) = np_(x), and in which p_0(x) is a non-zero constant. Among the most notable Appell sequences besides the ...
s, which are those sequences for which the operator ''Q'' is mere differentiation, and the group of sequences of
binomial type In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by non-negative integers \left\ in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities : ...
, which are those that satisfy the identity p_n(x+y) = \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\, and p_n(0) = 0\mbox n \ge 1. \, The group of Appell sequences is
abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a group ...
; the group of sequences of binomial type is not. The group of Appell sequences is a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
; the group of sequences of binomial type is not. The group of Sheffer sequences is a
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol . There are two closely related concepts of semidirect product: * an ''inner'' sem ...
of the group of Appell sequences and the group of sequences of binomial type. It follows that each
coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
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 In mathematics, a delta operator is a shift-equivariant linear operator Q\colon\mathbb \longrightarrow \mathbb /math> on the vector space of polynomials in a variable x over a field \mathbb that reduces degrees by one. To say that Q is shift-equi ...
" 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)=\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)=\sum_^nx^ks_(y). The sequence of
Hermite polynomials In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well a ...
, the sequence of
Bernoulli polynomials In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula. These polynomials occur ...
, and the
monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called a power product or primitive monomial, is a product of powers of variables with n ...
s ( ''xn'' : ''n'' = 0, 1, 2, ... ) are examples of Appell sequences. A Sheffer sequence ''p''''n'' is characterised by its
exponential generating function In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression invo ...
\sum_^\infty \frac t^n = A(t) \exp(x B(t)) \, where ''A'' and ''B'' are (
formal Formal, formality, informal or informality imply the complying with, or not complying with, some set of requirements ( forms, in Ancient Greek). They may refer to: Dress code and events * Formal wear, attire for formal events * Semi-formal atti ...
)
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
in ''t''. Sheffer sequences are thus examples of
generalized Appell polynomials In mathematics, a polynomial sequence \ has a generalized Appell representation if the generating function for the polynomials takes on a certain form: :K(z,w) = A(w)\Psi(zg(w)) = \sum_^\infty p_n(z) w^n where the generating function or kernel K( ...
and hence have an associated
recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
.


Examples

Examples of polynomial sequences which are Sheffer sequences include: * The
Abel polynomials The Abel polynomials are a sequence of polynomials named after Niels Henrik Abel, defined by the following equation: :p_n(x)=x(x-an)^ This polynomial sequence is of binomial type: conversely, every polynomial sequence of binomial type may be obtai ...
; * The
Bernoulli polynomials In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula. These polynomials occur ...
; * The
Euler polynomial In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula. These polynomials occur ...
; * The central factorial polynomials; * The
Hermite polynomials In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well a ...
; * The
Laguerre polynomials In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation: xy'' + (1 - x)y' + ny = 0,\ y = y(x) which is a second-order linear differential equation. Thi ...
; * The
monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called a power product or primitive monomial, is a product of powers of variables with n ...
s ( ''xn'' : ''n'' = 0, 1, 2, ... ); * The
Mott polynomials In mathematics the Mott polynomials ''s'n''(''x'') are polynomials given by the exponential generating function: : e^=\sum_n s_n(x) t^n/n!. They were introduced by Nevill Francis Mott who applied them to a problem in the theory of electrons. Be ...
; * The
Bernoulli polynomials of the second kind Bernoulli can refer to: People *Bernoulli family of 17th and 18th century Swiss mathematicians: **Daniel Bernoulli (1700–1782), developer of Bernoulli's principle **Jacob Bernoulli (1654–1705), also known as Jacques, after whom Bernoulli number ...
; * The
Falling and rising factorials In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial \begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) . \end ...
; * The
Touchard polynomials The Touchard polynomials, studied by , also called the exponential polynomials or Bell polynomials, comprise a polynomial sequence of binomial type defined by :T_n(x)=\sum_^n S(n,k)x^k=\sum_^n \left\x^k, where S(n,k)=\left\ is a Stirling ...
; * The Mittag-Leffler polynomials;


References

* Reprinted in the next reference. * * * Reprinted by Dover, 2005.


External links

*{{MathWorld, title=Sheffer Sequence, id=ShefferSequence Polynomials Factorial and binomial topics