Throughout, we have emphasized operator methods at the expense of generating functions, which were used almost exclusively in the past. Let Q be a delta operator and let F be the ring of formal power series in the variable t over the same Jield. FINITE OPERATOR CALCULUS 693 In the following we shall write P = p(Q), where P is a shift-invariant operator and p(t) is a formal power series, to indicate that the operator P corresponds to the formal power series p (t) under isomorphism of Theorem 3.
THE PINCHERLE DERIVATIVE
In the same way, from the isomorphism theorem and from the previous theorem we can easily deduce the following. The preceding corollaries show that a set of basic polynomials is completely determined by the coefficients of their first power
SHEFFER POLYNOMIALS
If s*(x) is a Sheffeer set with respect to the invertible shift-invariant operator S and the delta operator Q, then. Let T be an invertible shift-invariant operator, Zet Q be a delta operator, and let sn(x) be a polynomial sequence. By the preceding proposition, the polynomial set x-~&+~(x), n > 0, is the Sheffer set of the invertible shift-invariant operator Q' with respect to the delta operator Q, as desired.
RECURRENCE FORMULAS
END OPERATOR CALCULUS 703 Since xn is the basis set for the delta operator D, we also have after changing the variable. The isomorphism theorem allows us to proceed to indicators in the expansion for FE% obtained thereby. By the isomorphism theorem (see also Theorem 4 of Section 4) it is easily shown that there exists an invertible shift-invariant operator R with the property that Q = AR.
UMBRAL COMPOSITION
We develop the umbra1 device in a form that leads to a general result that embodies some of the more recondite identities satisfied by special polynomials. A simple example of the use of umbra1 notation is the definition of a polynomial sequence of binomial type, which can be denoted umbrally as . Then T-l exists and (a) the map S-+ TST-1 is an automorphism of the algebra 2 of shift-invariant operators;.
In light of the previous result, it follows that the umbra1 composition of two sequences of basis operators is again a basis sequence. We begin by establishing the special case where S and T are the identity operators, so we want to find the delta operator for the sequence u,(x) = s&W, w ic we know is a fundamental sequence by Proposition 1. the following result gives the solution to the so-called "connection constants problem".
The following result gives one of several closed-formula expressions for the coefficients of Sheffer polynomials. The proof does not provide an explicit method for computing the coefficients c(n, K, h), but see Proposition 4. From Corollary 1 to Proposition 2 of Section 5 it follows that every linear combination of polynomials of the form (* ) is again a Sheffer group in relation to A.
For cross Appell sequences, namely of the form pp(x) = P-Ax\x” 7 we have the composition umbra1.
EIGENFUNCTION EXPANSIONS
An operator F is thus the generator of a necessarily unique cross sequence of polynomials if and only if F( 1) = 0. a) If F and G are the generators of cross sequences prl(x) and 4$' ](x) with the same basic sequence, then F + G is the generator. For any Shefler sequence sn(x) with delta operator Q and operator S there exists a unique operator of the form a) A is essentially self-adjoint (and tightly defined) in the Hilbert space H obtained by completing the space P of polynomials in the associated inner product. FINITE OPERATOR CALCULUS 719 and comparing this to the right-hand side of (**), we see that the coefficients of the two expansions must agree.
We have shown that As,(x) = m,(x) for all rz > 0, so that the Sheffer set s,(x) is a set of eigenfunctions of A; since it includes the Hilbert space H, we conclude that A is an unbounded, essentially self-adjoint operator in H, with the non-negative integers as its simple spectrum, with eigenfunctions s,(x), as we wanted to show. The calculation of the coefficients ak and b is greatly simplified by using the corollary of Theorem 4 and by various umbra1 devices. Changes in variables allow these identities to be rearranged into a form suitable for computation in each particular case.
Another relevant issue in the present context is the representativeness of the inner product (*) of integral operators, evaluations of a function and its derivatives at specific points, etc. It would take us too far to address this issue here; suffice it to say that it can be answered completely. From the recurrence relations for orthogonal polynomials, it is easy to determine (by Sheffer) all Sheffer sets which are orthogonal polynomials over an interval of the real line.
It is interesting to speculate on possible generalizations of the notion of the classical orthogonal polynomial that are suggested by the "natural" inner product.
HERMITE POLYNOMIALS
Note that the definition of the Weierstrass operator with (*) is only valid for ZI > 0, but (***) always applies. However, we prefer to derive the spectral theory directly from the general results of Section 9. Operational identity (*) can also be used to provide a quick proof of the formulas for Burchndl-Feldheim-.
FINITE OPERATOR CALCULUS 725 Proposition 1 of Section 9 shows that the Hermite polynomials are orthogonal to the inner product. However, definition (37) is valid for arbitrary u and, combined with the results of Section 9, gives a formally valid eigenfunction expansion, whose inner product is non-degenerate but in general not positive definite. On the other hand, the positive-definite inner product (38), as defined in Section 9, gives a Hilbert space eigenfunction expansion for arbitrary V.
The interaction of the two bilinear forms for non-positive ZI leads to interesting analytical developments that we must necessarily leave for a later publication. It remains to find the operator of which the Hermite polynomials are the eigenfunctions, and this is immediately given by Theorem 9. We conclude that the Hermite polynomials are a complete set of eigenfunctions, with eigenvalues n, of the operator.
The present treatment shows that, apart from this one fact of analysis, the whole theory of Hermite expansions can be made purely algebraic.
LAGUERRE POLYNOMIALS One of the simplest cross-sequences is
The cross set ME](x) is related to Laguerre-type polynomials, the Sheffer sets with respect to the delta operator. We will deal with Laguerre type sets with respect to the operators (Laguerre operators of order a):. as can be easily verified by the first expansion theorem. The Sheffer sets with respect to these operators are polynomial sets Lo], classically known as Laguerre polynomials of order 01.
Note that our definition of Laguerre polynomials .. differs from that used by many authors by a factor of n!. Expanding the third formula on the right-hand side of the string of identities gives the coefficients of the Laguerre polynomials. Umbra1 composition of ME'(x) with ) L:)(x) gives, by an application of Theorem 7, the Sheffer set relative to-oh.
The so-called "doubling formulas" for Laguerre's polynomials (see, e.g., Rainville) are trivial consequences of Theorem 7; we will only extract one of them to show the method. The second inner product is, however, positive definite for all a; whereas, the first is symmetric for all 01 and gives From formula (2) of theorem 4 we find that the base polynomials JEmB)(~) for La.0 are given by. we see that L,., form a group in the convolution and that this group is in fact isomorphic to the matrix multiplication group.
Deeper properties can be obtained by developing Sheffer group theory with respect to these operators.
VANDERMONDE CONVOLUTION
The identity which states that these polynomials are of binomial type is sometimes known as the Van&~- monde convolution, although the name is also applied to other identities. This is done most simply by expanding the first set in terms of the second. Gould's summation formula 5.5 and Bateman's alternating convolution can also be obtained from the expansion theorem.
EXAMPLES AND APPLICATIONS Appell Polynomials
Extensions of the product p,(ax)g&) of Appell groups in terms of a third group were considered by Carlitz (1963); his results are special cases of those of Section 5. By specializing in suitable sets of Sheffer polynomials, many of the inverse relations in the literature can be explained. This makes some of the results in Carl& (1957) special cases of the present theory.
All identities in Riordan (pp. 18-23) can be obtained by either of the expansion theorems or by umbra1 composition (sometimes by both methods). Similarly, Riordan's inverse Abel relations (pp. 92-99) can be obtained by any of the previous methods or by recognizing a cross series. Several authors have considered basic polynomials with respect to the operator Q = Ea(l + 0)” D. The connection constants with the Abel polynomials are.
The theory of cross sequences expresses them at once in terms of the Hermitian polynomials HJx), i.e. The coupling constants with x” can be calculated using the summation formula, given that the inverse polynomials can be expressed in terms of the inverses of the Abelian polynomials. Remember also that the coupling constants with xn are the Stirling numbers of the second kind.
The connection constants are therefore given by the coefficients of the basis sequence of the backward difference operators V = I - E-1, namely the polynomials x(x + 1).
PROBLEMS AND HISTORY
Some other analogies with classical expansions of eigenfunctions can be noted, which suggest an extension of the theory to classes of special functions. There are at least three possibilities; interpretation as composite Poisson processes; interpretation through stationary stochastic processes, as in relating Hermite polynomials to Brownian motion of Poisson-Charlier polynomials from the Poisson process, and, finally, combinatorial interpretation through the enumeration of binomial-type structures, such as reluctance functions (see III). In any case, the relationship between the two sets of coefficients seems simple enough and should be worked out, especially given the mystery underlying cumulants.
CALCULUS OF THE LAST OPERATOR 7.53 (13) Laguerre polynomials are formally related to range distributions such as Hermite to normal, Poisson-Charlier to Poisson; however, a specific construction of the corresponding stochastic process or a set of transformations with respect to which they are "spherical harmonics" still seems to be lacking. We conjecture that analogous results exist for Laguerre and Hermite polynomials and are related to the position of the zeros of these polynomials. It is impossible to account for the detailed development of the Heaviside calculus from its beginnings; we will mention only the papers dealing with the current approach.
Other authors have used characterizations in terms of operators, missing one of the most important techniques. The recurrence formulas are due to Sheffer, as are the eigenfunction expansion formulas, except for the explicit inner products; however, his proofs use power series. An extensive bibliography has been added as a hunting ground for further applications and extensions of current methods.
Although some of the results are completely new, many have been proposed, inspired and partially proven by previous authors.
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