1 Introduction

The basis of the Umbral Calculus (see [15] and [17]) is the convolution identity

At the Franco-Québecois Workshop in May 1991, we asked what sort of `shiftless' Umbral Calculus would arise if the operator E y was replaced by some other shift-invariant operator F ( y ) : K [ x ] → K [ x , y ] .

In section 3, we seek parallel results for divided power sequences of symmetric functions. Surprisingly, up to a constant, the only `Sheffer' sequences of symmetric functions are the divided power sequences themselves.

We end sections 2 and 3 with applications to coalgebra theory. These results may be safely skipped by any non-specialist. Solutions to equation 3 are interpreted as cocommutative coalgebras, and classified according to their coalgebraic properties.

2 Polynomials

2.1 Notation

Let K be a field of characteristic zero, and let x , y be indeterminates. Consider a K -linear map φ : K [ x ] → K [ x ] . Then φ has a unique K [ y ] -linear extension φ ′ to K [ x , y ] . By an abuse of notation, we will denote φ and φ ′ by the same symbol φ . Note that if φ is a linear map on K [ x ] and θ is a linear map on K [ y ] , then φ and θ commute when considered as maps on K [ x , y ] . Nevertheless, φ and θ do not necessarily commute with maps ψ : K [ x ] → K [ x , y ] such as the shift operator.

Now, consider the natural isomorphism π : K [ x ] → K [ y ] . There is a unique map which we will denote T x y φ such that T x y φ ^ π = π ^ φ . Often a linear map K [ x ] → K [ x ] will be denoted φ x with x as a subscript. In that case, T x y φ x will be denoted φ y . For example, if D x is the derivative with respect to x , then D y is the derivative with respect to y . Similarly, if ε x is the evaluation map at x = 0 , ε x p ( x ) = p ( 0 ) , then ε y is the evaluation map at y = 0 . Essentially, φ y behaves with respect to y in the same way as φ x does with respect to x .

For n a nonnegative integer, let p n ( x ) be a polynomial of degree n with coefficients in K and let F ( y ) be a K -linear operator K [ x ] → K [ x , y ] which we propose as a possible solution to equation 3.

Note that we must assume that F ( y ) is linear in order to characterize such operators satisfying equation 3, since equation 3 specifies the value of F ( y ) only on a basis of K [ x ] .

Finally, we note that in the above notation there are really two kinds of `shifts' E y : p ( x ) ↦ p ( x + y ) . If y is taken as a constant, then E y : K [ x ] → K [ x ] . Whereas, if y is taken as a variable, then E y : K [ x ] → K [ x , y ] . However, commutation with one shift guarantees commutation with the other as the next lemma shows.

Proof: (If) Trivial, evaluate at y = c .

(Only if) For any polynomial p ( x ) , the expressions E y θ p ( x ) and θ E y p ( x ) agree infinitely often when thought of as polynomials taking values in K [ x ] . Thus, they must be the same polynomial, and θ E y = E y θ . □

2.2 Sheffer Theorem

In this section, we make the additional assumption that F ( y ) is shift-invariant for all y .

Proof: (2 implies 1): Define q n ( x ) = P x - 1 p n ( x ) or equivalently q n ( y ) = P y - 1 p n ( y ) . By hypothesis, ( q n ( x ) ) n = 0 ∞ is a divided power sequence. That is to say,

We may now let q n ( x ) = P x - 1 p n ( x ) and G ( y ) = P y - 1 F ( y ) . It remains now to show that G ( y ) = E y .

By the above reasoning,

The above theorem yields interesting Sheffer sequences identities. We illustrate this with three examples: the Hermite polynomials, the Laguerre polynomials and the Bernoulli polynomials of the second kind. Let ( p n ( x ) ) n = 0 ∞ be a Sheffer sequence relative to ( q n ( x ) ) n = 0 ∞ . By the First Expansion Theorem ([17, Theorem 2]), the operator P y which maps q n ( y ) to p n ( y ) has expansion

2.2.1 Hermite

Let ( H n ν ( x ) ) n = 0 ∞ be the sequence of Hermite polynomials of variance ν where ν is a real number (see [17, sect. 10]). Its generating function is

2.2.2 Laguerre

Let ( L n α ( x ) ) n = 0 ∞ be the sequence of Laguerre polynomials of order α where α is a real number (see [17, sect. 11]). Its generating function is

2.2.3 Bernoulli

Let ( b n α ( x ) ) n = 0 ∞ be the sequence of Bernoulli polynomials of the second kind. Its generating function is

2.3 Generalized Sheffer

Let us now remove the condition that F ( y ) be shift-invariant which was so crucial to Theorem 2.1. Immediately, we have new solutions to equation 3. In fact, any sequence of polynomials ( p n ( x ) ) n = 0 ∞ (with deg p n ( x ) = n ) gives rise to a unique operator of F ( y ) which verifies equation 3.

Proof: Let us first check that all the objects mentioned above are well-defined. Since ( p n ( x ) ) n = 0 ∞ is a basis for K [ x ] , Q is well defined and it lowers the degree of any polynomial by one. Thus, Q - 1 is well defined up to a constant. Since the constant term of q n ( x ) is given, q n ( x ) is well defined. By induction, Q n lowers the degree of any polynomial by n ; thus, the sum giving G ( y ) is in fact convergent. P x is of course well defined and invertible since ( q n ( x ) ) n = 0 ∞ and ( p n ( x ) ) n = 0 ∞ are both sequences of polynomials. Thus, F ( y ) is well defined.

P x is Q -invariant because

Two explicit examples that illustrate Theorem 2.2 are:

• p n ( x ) = ( x - 1 ) n ( n ! ) 2 P n x + 1 x - 1 where P n ( x ) is the n t h Legendre polynomial. Here, Q = D x D = D + x D 2 . [16, Chapter 13, Exercise 11]
• p n ( x ) = H n ( x / 2 ) / ( n ! ) 2 where H n ( x ) = H n 1 ( x ) is the n t h Hermite polynomial as defined in Section 2.2.1. Here, Q = 1 2 D + 1 2 x D 2 - 1 4 D 3 . [16, Chapter 13, p. 220]

In these examples, operators of the form ∑ n = 0 ∞ a n ( x ) D k appear where the a n ( x ) are polynomials. In fact, any linear operator on the vector space of polynomials can be represented in this way (see [7, Proposition 1], cf. [16, Theorems 70 and 77]). The paper [7] shows an efficient way to calculate the polynomials a n ( x ) explicitly.

We see that equation 3 imposes no conditions on the sequence ( p n ( x ) ) n = 0 ∞ . So what does it mean to be a generalized Sheffer sequence if every sequence is a generalized Sheffer sequence? We can answer this question as follows. In [12], it is shown that any sequence ( q n ( x ) ) n = 0 ∞ with q n ( 0 ) = δ n 0 obeys equation 3. These sequences are to F ( y ) as divided power sequences are to the shift operator. The theorem above says not only that ( p n ( x ) ) n = 0 ∞ is generalized Sheffer, but also how it is so. That is to say, given ( p n ( x ) ) n = 0 ∞ there is a unique operator G ( y ) with a unique G ( y ) -invariant operator Q with a unique basic sequence ( q n ( x ) ) n = 0 ∞ . It is this sequence that ( p n ( x ) ) n = 0 ∞ is Sheffer with respect to, and P x is the Sheffer operator relating the two sequences (cf. [18] or [16, Chapter 13]). The Hermite example above shows that a suitable choice of norming constants may change the associated basic sequence. For more information, see [3].

Finally, we note that although equation 3 does not impose any conditions on ( p n ( x ) ) n = 0 ∞ , it does impose certain conditions on the operators F ( y ) . As seen above, we can multiply F ( y ) by any invertible operator K [ x ] → K [ x ] . Moreover, ε y ^ F ( y ) is clearly invertible. Thus, without a real loss of generality we can assume that ε y ^ F ( y ) is the identity.

Let us partition the set of all linear operators (other than constant multiples of the identity) according to which operators commute with which operators (cf. [12], [19] and [16, Chapter 13]). Then using the methods of [12] it can be shown that each equivalence class contains exactly one possible value of F ( y ) such that ε y ^ F ( y ) is the identity.

In particular, the class of shift-invariant operators contains only such solutions of the form F ( y ) = E y as we saw above.

We now want to point out some connections with generalized translation operators. The operators G ( y ) of equation 4 are generalized translation operators in the sense of Levitan (see [8]). The series on the right-hand side of equation 4 is called a Taylor-Delsarte series since they were studied in [5]. Levitan stresses the importance of the infinitesimal generator of the operators G ( y ) . In our case it is easy to show that

Proof: Apply the left hand side to the basis ( q n ( x ) ) x = 0 ∞ . Then equation 3 yields

In [8], Levitan also gives a systematic exposition of the relation between generalized translation operators and Cauchy problems (i.e, partial differential equations with initial data). In our case, we have the following Cauchy problem (cf. [2, Theorem 5]):

Proof: First, note that u ( x , 0 ) = p ( x ) because G ( 0 ) = I . Since ( q n ( x ) ) n = 0 ∞ is a basis, it suffices to show that Q x G ( y ) q n ( x ) = Q y G ( y ) q n ( x ) . This follows directly from equation 5. □

If Q = D in Proposition 2.2, then we can easily compute u as follows: Define new variables ξ = x and η = x + y . Since D x = D ξ + D η and D y = D η , the differential equation transforms into D ξ u = 0 with solution u ( x , y ) = f ( η ) = f ( x + y ) . Now, set y = 0 which yields p = f . Hence, u ( x , y ) = p ( x + y ) as expected.

Another way to solve this Cauchy problem is to proceed as Heaviside did in the previous century: Fix x and treat D x as a formal constant. Then the Cauchy problem becomes an ordinary differential equation whose solution is readily seen to be u ( x , y ) = e y D x p ( x ) which equals p ( x + y ) by the First Expansion Theorem [17, Theorem 2].

If Q = c D + x D 2 (cf. the second example below Theorem 2.2 where c = 1 ) and c ≥ 1 2 , then it follows from [6, Theorem 2.4.2.6] that

2.4 Coalgebra

The above can be profitably recast in the terminology of coalgebras (see [14] for the relation between Umbral Calculus and coalgebras). A coalgebra is a vector space V equipped with a comultiplication Δ : V → V ⊗ V and a counitary map ε : V → K . These maps must be coassociative

Now, K [ x , y ] is isomorphic to the tensor product K [ x ] ⊗ K [ x ] , so any F = F ( y ) (satisfying equation 3) would be a potential candidate for a comultiplication map. Equation 6 is automatically satisfied:

By equation 7,

We have thus proven the following proposition.

Proof: If F ( y ) is an algebra map, then F ( y ) is the substitution for x of some polynomial r ( x , y ) . By degree considerations in equation 3, r ( x , y ) must be of degree one. Moreover, since F ( y ) is cocommutative, r ( x , y ) must be symmetric in x and y . Thus, r ( x , y ) = a ( x + y ) - c . Consideration of the leading coefficients in equation 3 indicates that a must be zero. Thus, F ( y ) = E y - c . The remaining results are easily verified. □

3 Symmetric Functions

3.1 Introduction

In [9], the notion (and combinatorial interpretation) of divided power sequences is extended to the domain of symmetric functions. A linear divided powers sequence of symmetric functions ( p n ( x 1 , x 2 , … ) ) n = 0 ∞ is a sequence of homogeneous symmetric functions—one of each degree—obeying the following convolution identity

3.2 Notation

A partition λ is an eventually zero, decreasing sequence of natural numbers λ 1 ≥ λ 2 ≥ ⋯ = 0 . Its conjugate, denoted λ ′ , is defined by the rule

• First, they can be compared coordinate wise: α ≤ β if and only if α i ≤ β i for all i .
• Second, they can be compared using the reverse lexicographical order. That is to say, they are ordered as if they were words written in Hebrew or Arabic (from right to left). α ≪ β if and only if there is an i such that α i < β i and α j = β j for all j > i .

Let P be the set of all partitions and P n be the set of all partitions summing to n . Clearly, only ≪ is a total ordering of P . In fact, ≪ is a strengthening of the < relation which itself is so weak as to be equality when restricted to P n .

The monomial symmetric functions m λ ( x 1 , x 2 , … ) for λ ∈ P n form a basis for the vector space of homogeneous symmetric functions of degree n . In fact, ( m λ ′ ( x 1 , x 2 , … ) ) λ ∈ P will be our canonical example of a full sequence (just as ( x n ) n = 0 ∞ is the typical sequence of polynomials).

In general, in a full sequence ( p λ ( x 1 , x 2 , … ) ) λ ∈ P , the symmetric functions p λ ( x 1 , x 2 , … ) must be homogeneous of degree n (for λ ∈ P n ). Moreover, they must have expansions in terms of the monomial symmetric functions whose index follows λ ′ in reverse lexicographical order

A full sequence is thus a basis for the space of symmetric functions.

Even though p λ ( x 1 , x 2 , … ) is only defined for λ a partition, it will be convenient to extend its definition to all vectors of integers with finite support. If α i is always nonnegative, then there is a unique partition λ which is a permutation of α . We then write

3.3 Sheffer Theorem

What linear operators F y and full sequences p λ ( x 1 , x 2 , … ) obey

Proof: First, consider F 0 .

Since F y and E y are both shift-invariant, so is their difference which we can then expand in the form

3.4 Coalgebra

As seen in [9], all operators of the form F y obeying 9 serve as the comultiplication of a (stronly) cohomogeneous cocommutative Hopf algebra over the symmetric functions, and conversely. For the symmetric shift operator, for example, the augmentation ε is the counitary map, and the antipode is the classical involution of symmetric functions

References