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Polynomials which form the Sheffer sequence for(1)(2)which have generating function(3)The first few are(4)(5)(6)

The central factorials form an associated Sheffer sequence with(1)(2)giving the generating function(3)The first central factorials are(4)(5)(6)(7)(8)(9)(10)(11)(12)

Polynomials which form the Sheffer sequence for(1)and have exponential generating function(2)The first few are(3)(4)(5)(6)(7)(8)

Polynomials which form a Sheffer sequence with(1)(2)and have generating function(3)The first few are(4)(5)(6)Jordan (1965) considers the related polynomials which form a Sheffer sequence with(7)(8)These polynomials have generating function(9)The first few are(10)(11)(12)(13)The Peters polynomials are a generalizationof the Boole polynomials.

Roman (1984, p. 2) describes umbral calculus as the study of the class of Sheffer sequences. Umbral calculus provides a formalism for the systematic derivation and classification of almost all classical combinatorial identities for polynomial sequences, along with associated generating functions, expansions, duplication formulas, recurrence relations, inversions, Rodrigues representation, etc., (e.g., the Euler-Maclaurin integration formulas, Boole's summation formula, the Chu-Vandermonde identity, Newton's divided difference interpolation formula, Gregory's formula, Lagrange inversion).The term "umbral calculus" was coined by Sylvester from the word "umbra" (meaning "shadow" in Latin), and reflects the fact that for many types of identities involving sequences of polynomials with powers , "shadow" identities are obtained when the polynomials are changed..

The polynomials which form the Sheffer sequence for(1)(2)which have generating function(3)The first few are(4)(5)(6)

Roman (1984, p. 26) defines "the" binomial identity as the equation(1)Iff the sequence satisfies this identity for all in a field of field characteristic 0, then is an associated sequence known as a binomial-type sequence.In general, a binomial identity is a formula expressing products of factors as a sum over terms, each including a binomial coefficient . The prototypical example is the binomial theorem(2)for . Abel (1826) gave a host of such identities (Riordan 1979, Roman 1984), some of which include(3)(4)(Abel 1826, Riordan 1979, p. 18; Roman 1984, pp. 30 and 73), and(5)(Saslaw 1989).

Polynomials which form the Sheffer sequence for(1)(2)where is the inverse function of , and have generating function(3)The first few polynomials are(4)(5)(6)(7)The Stirling polynomials are related to the Stirling numbers of the first kind by(8)where is a binomial coefficient and is an integer with (Roman 1984, p. 129).

Polynomials which form the Sheffer sequence for(1)(2)and have generating function(3)The are given in terms of the hypergeometricseries by(4)where is the Pochhammer symbol (Koepf 1998, p. 115). The first few are(5)(6)(7)Koekoek and Swarttouw (1998) defined the Meixner polynomials without the Pochhammersymbol as(8)The Krawtchouk polynomials are a specialcase of the Meixner polynomials of the first kind.

A sequencewhere is a Sheffer sequence, is invertible, and ranges over the real numbers. If is an associated Sheffer sequence, then is called a cross sequence. If , thenis called an Appell cross sequence.An example is the Laguerre polynomial.

Polynomials which form the Sheffer sequence for(1)where is the inverse function of , and have generating function(2)The first few are(3)(4)(5)(6)(7)(8)

Polynomials which form a Sheffer sequence with(1)(2)giving generating function(3)Roman (1984) defines Bernoulli numbers of the second kind as . They are related to the Stirling numbers of the first kind by(4)(Roman 1984, p. 115), and obey the reflection formula(5)(Roman 1984, p. 119).The first few Bernoulli polynomials of the second kind are(6)(7)(8)(9)(10)

A sequence is called a Sheffer sequence iff its generating function has the form(1)where(2)(3)with . Sheffer sequences are sometimes also called poweroids (Steffensen 1941, Shiu 1982, Di Bucchianico and Loeb 2000).If is a delta series and is an invertible series, then there exists a unique sequence of Sheffer polynomials satisfying the orthogonality condition(4)where is the Kronecker delta (Roman 1984, p. 17). Examples of general Sheffer sequences include the actuarial polynomials, Bernoulli polynomials of the second kind, Boole polynomials, Laguerre polynomials, Meixner polynomials of the first and second kinds, Poisson-Charlier polynomials, and Stirling polynomials.The Sheffer sequence for is called the associated sequence for , and Roman (1984, pp. 53-86) summarizes properties of the associated Sheffer sequences and gives a number of specific examples (Abel polynomial, Bell polynomial, central factorial,..

The polynomials given by the associated Sheffer sequence with(1)where . The inverse function (and therefore generating function) cannot be computed algebraically, but the generating function(2)can be given in terms of the sum(3)This results in(4)where is a falling factorial. The first few are(5)(6)(7)(8)(9)The binomial identity obtained from the Sheffer sequencegives the generalized Chu-Vandermonde identity(10)(Roman 1984, p. 69; typo corrected).In the special case , the function simplifies to(11)which gives the generating function(12)giving the polynomials(13)(14)(15)(16)(17)

A polynomial sequence is called the basic polynomial sequence for a delta operator if 1. , 2. for all , 3. . If is a basic polynomial sequence for some delta operator , then it is a binomial-type sequence of polynomials. Furthermore, if is a binomial-type sequence of polynomials, then it is a basic polynomial sequence for some delta operator.

A Sheffer sequence for is called the associated sequence for , and a sequence of polynomials satisfying the orthogonality conditionswhere is the delta function, is said to be associated to .

A formula for the Bell polynomial and Bellnumbers. The general formula states that(1)where is a Bell polynomial (Roman 1984, p. 66). Setting gives the special case of the th Bell number,(2)It can be derived by dividing the generating function formula for a Stirling number of the second kind by , yielding(3)Then(4)and(5)Now setting gives the identity (Dobiński 1877; Rota 1964; Berge 1971, p. 44; Comtet 1974, p. 211; Roman 1984, p. 66; Lupas 1988; Wilf 1994, p. 106; Chen and Yeh 1994; Pitman 1997).Dobinski also published a curious infinite productsometimes also known as Dobiński's formula.

An Appell sequence is a Sheffer sequence for . Roman (1984, pp. 86-106) summarizes properties of Appell sequences and gives a number of specific examples.The sequence is Appell for iff(1)for all in the field of field characteristic 0, and iff(2)(Roman 1984, p. 27). The Appell identity states that the sequence is an Appell sequence iff(3)(Roman 1984, p. 27).The Bernoulli polynomials, Euler polynomials, and Hermite polynomials are Appell sequences (in fact, more specifically, they are Appell cross sequences).

A shift-invariant operator for which is a nonzero constant. 1. for every constant . 2. If is a polynomial of degree , is a polynomial of degree . 3. Every delta sequence has a unique basicpolynomial sequence.

A sequencewhere is a Sheffer sequence, is invertible, and ranges over the real numbers is called a Steffensen sequence. If is an associated Sheffer sequence, then is called a cross sequence. If , thenis called an Appell cross sequence.Examples include the Bernoulli polynomial, Euler polynomial, and Hermite polynomial.

Polynomials which form the Sheffer sequence for(1)(2)and have generating function(3)The first few are(4)(5)(6)(7)The Pidduck polynomials are related to the Mittag-Leffler polynomials by(8)(Roman 1984, p. 127).

A sequencewhere is a Sheffer sequence, is invertible, and ranges over the real numbers is called a Steffensen sequence. If is an associated Sheffer sequence, then is called a cross sequence.Examples include the actuarial polynomialand Poisson-Charlier polynomial.

The polynomials given by the Sheffer sequence with(1)(2)giving generating function(3)The Sheffer identity is(4)where is a Bell polynomial. The actuarial polynomials are given in terms of the Bell polynomials by(5)(6)They are related to the Stirling numbers of the second kind by(7)where is a binomial coefficient and is a falling factorial. The actuarial polynomials also satisfy the identity(8)(Roman 1984, p. 125; Whittaker and Watson 1990, p. 336).The first few polynomials are(9)(10)(11)(12)

Polynomials which are a generalization of the Boole polynomials, form the Sheffer sequence for(1)(2)and have generating function(3)The first few are(4)(5)and(6)

A polynomial given by the associated Sheffer sequence with(1)given by(2)The generating function is(3)where is the Lambert W-function. The associated binomial identity is(4)where is a binomial coefficient, a formula originally due to Abel (Riordan 1979, p. 18; Roman 1984, pp. 30 and 73).The first few Abel polynomials are(5)(6)(7)(8)(9)

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