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 $F^{\left(y\right)}:K\left[x\right]\to K[x,y]$ .

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 $\phi :K\left[x\right]\to K\left[x\right]$ . Then $\phi$ has a unique $K\left[y\right]$ -linear extension $\phi \prime$ to $K[x,y]$ . By an abuse of notation, we will denote $\phi$ and $\phi \prime$ by the same symbol $\phi$ . Note that if $\phi$ is a linear map on $K\left[x\right]$ and $\theta$ is a linear map on $K\left[y\right]$ , then $\phi$ and $\theta$ commute when considered as maps on $K[x,y]$ . Nevertheless, $\phi$ and $\theta$ do not necessarily commute with maps $\psi :K\left[x\right]\to K[x,y]$ such as the shift operator.

Now, consider the natural isomorphism $\pi :K\left[x\right]\to K\left[y\right]$ . There is a unique map which we will denote $T_x^y\phi$ such that $T_x^y\phi \circ \pi =\pi \circ \phi$ . Often a linear map $K\left[x\right]\to K\left[x\right]$ will be denoted ${\phi }_x$ with $x$ as a subscript. In that case, $T_x^y{\phi }_x$ will be denoted ${\phi }_y$ . For example, if $D_x$ is the derivative with respect to $x$ , then $D_y$ is the derivative with respect to $y$ . Similarly, if ${\epsilon }_x$ is the evaluation map at $x=0$ , ${\epsilon }_{x}p\left(x\right)=p\left(0\right)$ , then ${\epsilon }_y$ is the evaluation map at $y=0$ . Essentially, ${\phi }_y$ behaves with respect to $y$ in the same way as ${\phi }_x$ does with respect to $x$ .

For $n$ a nonnegative integer, let $p_n\left(x\right)$ be a polynomial of degree $n$ with coefficients in $K$ and let $F^{\left(y\right)}$ be a $K$ -linear operator $K\left[x\right]\to K[x,y]$ which we propose as a possible solution to equation 3.

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

Finally, we note that in the above notation there are really two kinds of `shifts' $E^y:p\left(x\right)\mapsto p(x+y)$ . If $y$ is taken as a constant, then $E^y:K\left[x\right]\to K\left[x\right]$ . Whereas, if $y$ is taken as a variable, then $E^y:K\left[x\right]\to K[x,y]$ . 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 $p\left(x\right),$ the expressions $E^y\theta p\left(x\right)$ and $\theta E^{y}p\left(x\right)$ agree infinitely often when thought of as polynomials taking values in $K\left[x\right]$ . Thus, they must be the same polynomial, and $\theta E^y=E^y\theta .\square$

2.2 Sheffer Theorem

In this section, we make the additional assumption that $F^{\left(y\right)}$ is shift-invariant for all $y.$

Proof: (2 implies 1): Define $q_n\left(x\right)=P_x^{-1}{p}_n\left(x\right)$ or equivalently $q_n\left(y\right)=P_y^{-1}{p}_n\left(y\right)$ . By hypothesis, ${\left(q_n\left(x\right)\right)}_{n=0}^{\infty }$ is a divided power sequence. That is to say,

We may now let $q_n\left(x\right)=P_x^{-1}{p}_n\left(x\right)$ and $G^{\left(y\right)}=P_y^{-1}{F}^{\left(y\right)}$ . It remains now to show that $G^{\left(y\right)}=E^y$ .

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 ${\left(p_n\left(x\right)\right)}_{n=0}^{\infty }$ be a Sheffer sequence relative to ${\left(q_n\left(x\right)\right)}_{n=0}^{\infty }$ . By the First Expansion Theorem ([17, Theorem 2]), the operator $P_y$ which maps $q_n\left(y\right)$ to $p_n\left(y\right)$ has expansion

2.2.1 Hermite

Let ${\left(H_n^{\nu }\left(x\right)\right)}_{n=0}^{\infty }$ be the sequence of Hermite polynomials of variance $\nu$ where $\nu$ is a real number (see [17, sect. 10]). Its generating function is

2.2.2 Laguerre

Let ${\left(L_n^{\alpha }\left(x\right)\right)}_{n=0}^{\infty }$ be the sequence of Laguerre polynomials of order $\alpha$ where $\alpha$ is a real number (see [17, sect. 11]). Its generating function is

2.2.3 Bernoulli

Let ${\left(b_n^{\alpha }\left(x\right)\right)}_{n=0}^{\infty }$ 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 $F^{\left(y\right)}$ 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 ${\left(p_n\left(x\right)\right)}_{n=0}^{\infty }$ (with $deg{p}_n\left(x\right)=n$ ) gives rise to a unique operator of $F^{\left(y\right)}$ which verifies equation 3.

Proof: Let us first check that all the objects mentioned above are well-defined. Since ${\left(p_n\left(x\right)\right)}_{n=0}^{\infty }$ is a basis for $K\left[x\right]$ , $Q$ 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 $q_n\left(x\right)$ is given, $q_n\left(x\right)$ is well defined. By induction, $Q^n$ lowers the degree of any polynomial by $n$ ; thus, the sum giving $G^{\left(y\right)}$ is in fact convergent. $P_x$ is of course well defined and invertible since ${\left(q_n\left(x\right)\right)}_{n=0}^{\infty }$ and ${\left(p_n\left(x\right)\right)}_{n=0}^{\infty }$ are both sequences of polynomials. Thus, $F^{\left(y\right)}$ is well defined.

$P_x$ is $Q$ -invariant because

Two explicit examples that illustrate Theorem 2.2 are:

• $p_n\left(x\right)=\frac{{(x-1)}^n}{{\left(n!\right)}^2}{P}_n\left(\frac{x+1}{x-1}\right)$ where $P_n\left(x\right)$ is the $nth$ Legendre polynomial. Here, $Q=DxD=D+x{D}^2$ . [16, Chapter 13, Exercise 11]
• $p_n\left(x\right)=H_n(x/2)/{\left(n!\right)}^2$ where $H_n\left(x\right)=H_n^1\left(x\right)$ is the $nth$ Hermite polynomial as defined in Section 2.2.1. Here, $Q=\frac{1}{2}D+\frac{1}{2}x{D}^2-\frac{1}{4}{D}^3$ . [16, Chapter 13, p. 220]

In these examples, operators of the form ${\sum }_{n=0}^{\infty }{a}_n\left(x\right)D^k$ appear where the $a_n\left(x\right)$ 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 $a_n\left(x\right)$ explicitly.

We see that equation 3 imposes no conditions on the sequence ${\left(p_n\left(x\right)\right)}_{n=0}^{\infty }$ . 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 ${\left(q_n\left(x\right)\right)}_{n=0}^{\infty }$ with $q_n\left(0\right)={\delta }_{n0}$ obeys equation 3. These sequences are to $F^{\left(y\right)}$ as divided power sequences are to the shift operator. The theorem above says not only that ${\left(p_n\left(x\right)\right)}_{n=0}^{\infty }$ is generalized Sheffer, but also how it is so. That is to say, given ${\left(p_n\left(x\right)\right)}_{n=0}^{\infty }$ there is a unique operator $G^{\left(y\right)}$ with a unique $G^{\left(y\right)}$ -invariant operator $Q$ with a unique basic sequence ${\left(q_n\left(x\right)\right)}_{n=0}^{\infty }$ . It is this sequence that ${\left(p_n\left(x\right)\right)}_{n=0}^{\infty }$ 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 ${\left(p_n\left(x\right)\right)}_{n=0}^{\infty }$ , it does impose certain conditions on the operators $F^{\left(y\right)}$ . As seen above, we can multiply $F^{\left(y\right)}$ by any invertible operator $K\left[x\right]\to K\left[x\right]$. Moreover, ${\epsilon }_y\circ F^{\left(y\right)}$ is clearly invertible. Thus, without a real loss of generality we can assume that ${\epsilon }_y\circ F^{\left(y\right)}$ 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 $F^{\left(y\right)}$ such that ${\epsilon }_y\circ F^{\left(y\right)}$ is the identity.

In particular, the class of shift-invariant operators contains only such solutions of the form $F^{\left(y\right)}=E^y$ as we saw above.

We now want to point out some connections with generalized translation operators. The operators $G^{\left(y\right)}$ 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 $G^{\left(y\right)}$ . In our case it is easy to show that

Proof: Apply the left hand side to the basis ${\left(q_n\left(x\right)\right)}_{x=0}^{\infty }$ . 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 $u(x,0)=p\left(x\right)$ because $G^{\left(0\right)}=I$ . Since ${\left(q_n\left(x\right)\right)}_{n=0}^{\infty }$ is a basis, it suffices to show that $Q_x{G}^{\left(y\right)}{q}_n\left(x\right)=Q_y{G}^{\left(y\right)}{q}_n\left(x\right)$ . This follows directly from equation 5. $\square$

If $Q=D$ in Proposition 2.2, then we can easily compute $u$ as follows: Define new variables $\xi =x$ and $\eta =x+y$ . Since $D_x=D_{\xi }+D_{\eta }$ and $D_y=D_{\eta }$ , the differential equation transforms into $D_{\xi }u=0$ with solution $u(x,y)=f\left(\eta \right)=f(x+y)$ . 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 $u(x,y)=e^{y{D}_x}p\left(x\right)$ which equals $p(x+y)$ by the First Expansion Theorem [17, Theorem 2].

If $Q=cD+x{D}^2$ (cf. the second example below Theorem 2.2 where $c=1$ ) and $c\ge \frac{1}{2}$ , 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 $\Delta :V\to V\otimes V$ and a counitary map $\epsilon :V\to K$ . These maps must be coassociative

Now, $K[x,y]$ is isomorphic to the tensor product $K\left[x\right]\otimes K\left[x\right]$ , so any $F=F^{\left(y\right)}$ (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 $F^{\left(y\right)}$ is an algebra map, then $F^{\left(y\right)}$ 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 $F^{\left(y\right)}$ 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, $F^{\left(y\right)}=E^{y-c}$ . The remaining results are easily verified. $\square$

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 ${\left(p_n(x1,x2,\dots )\right)}_{n=0}^{\infty }$ is a sequence of homogeneous symmetric functions—one of each degree—obeying the following convolution identity

3.2 Notation

A partition $\lambda$ is an eventually zero, decreasing sequence of natural numbers ${\lambda }_1\ge {\lambda }_2\ge \cdots =0$ . Its conjugate, denoted $\lambda \prime$ , is defined by the rule

• First, they can be compared coordinate wise: $\alpha \le \beta$ if and only if ${\alpha }_i\le {\beta }_i$ 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). $\alpha \ll \beta$ if and only if there is an $i$ such that ${\alpha }_i<{\beta }_i$ and ${\alpha }_j={\beta }_j$ 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 $\ll$ is a total ordering of $P$ . In fact, $\ll$ is a strengthening of the $<$ relation which itself is so weak as to be equality when restricted to $P_n$ .

The monomial symmetric functions $m_{\lambda }(x1,x2,\dots )$ for $\lambda \in P_n$ form a basis for the vector space of homogeneous symmetric functions of degree $n$ . In fact, ${\left(m_{\lambda \prime }(x1,x2,\dots )\right)}_{\lambda \in P}$ will be our canonical example of a full sequence (just as ${\left(x^n\right)}_{n=0}^{\infty }$ is the typical sequence of polynomials).

In general, in a full sequence ${\left(p_{\lambda }(x1,x2,\dots )\right)}_{\lambda \in P}$ , the symmetric functions $p_{\lambda }(x1,x2,\dots )$ must be homogeneous of degree $n$ (for $\lambda \in P_n$ ). Moreover, they must have expansions in terms of the monomial symmetric functions whose index follows $\lambda \prime$ in reverse lexicographical order

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

Even though $p_{\lambda }(x1,x2,\dots )$ is only defined for $\lambda$ a partition, it will be convenient to extend its definition to all vectors of integers with finite support. If ${\alpha }_i$ is always nonnegative, then there is a unique partition $\lambda$ which is a permutation of $\alpha$ . We then write

3.3 Sheffer Theorem

What linear operators $F^y$ and full sequences $p_{\lambda }(x1,x2,\dots )$ 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 $\epsilon$ is the counitary map, and the antipode is the classical involution of symmetric functions

References