Triangular expansions

Journal of Approximation Theory 162 (2010) 2106–2128 www.elsevier.com/locate/jat

Triangular expansions Kamel Belbahri University of Montreal, Canada Received 26 February 2010; received in revised form 15 May 2010; accepted 26 June 2010 Available online 31 July 2010 Communicated by Francisco Marcellan Dedicated to R.S. Pinkham

Abstract We use umbral methods to obtain several general expansion theorems for linear operators and linear functionals. We show in particular that every linear operator admits Newton-type expansions. These expansions are prototypes of numerous classical ones as well as new ones, some of which are powerful enough for use in numerical approximation. c 2010 Elsevier Inc. All rights reserved. ⃝ Keywords: Umbral calculus; Numerical approximation; Approximations and expansions

1. Introduction and terminology Let P = K[x] be the ring of polynomials in the variable x over a field K of characteristic zero (in practice, K = R or C). Let P∗ be the algebraic dual space of P. Every polynomial sequence qn (x) (deg(qn ) = n) with mild norming conditions (qn (0) = δn,0 ) is called a basic set [26] (or, [3, chapter 3] and [18] with other naming conventions) and defines an algebra K[[Q]] of linear operators (K[[Q]] = {P : P Q = Q P}) over K generated by the delta operator Q (Qqn (x) = qn−1 (x) and Q1 = 0). Conversely, every delta operator Q (Q1 = 0 and deg(Qx n ) = n − 1, n ≥ 1) admits a unique basic set qn (x). A typical member P of K[[Q]] can uniquely be expanded as  − P= Sc0 Pqn (x) Q n . (1.1) n≥0

E-mail address: [email protected]. c 2010 Elsevier Inc. All rights reserved. 0021-9045/$ - see front matter ⃝ doi:10.1016/j.jat.2010.06.007

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(Sch is the scale operator: Sch p(x) = p(hx); in particular Sc0 is the annihilation functional: Sc0 p(x) = p(0)). Of particular importance is the convolution operator Th defined by Th qn (x) = qn (h) ∗ qn (x) =

n −

qk (h)qn−k (x).

(1.2)

k=0

The operator Th can in turn be used to define K[[Q]] as the algebra of Th -invariant operators: K[[Q]] = {P : Th P = P Th for all h in K}. We thus have − Th = qn (h)Q n = q(h, Q), (1.3) n≥0

where q(x, t) =

∑

n≥0 qn (x)t

n

is the generating function of qn (x).

Remark 1. Introduce the umbral notation an = a n [19, p. 199], let P be any Q-invariant operator and put Sc0 P x n = an . Then Sc0 Pqn (x) = qn (a) (qn (a) is the umbral composition of the polynomial qn (x) with the sequence an ) and −   P= qn a Q n = Ta . (1.4) n≥0

The algebras K[[Q]] are isomorphic to the algebra K[[t]] of formal power series in the variable t over the field K and are essentially distinct in the following sense: if R and Q are two delta operators, then we have either R Q = Q R (that is K[[Q]] = K[[R]]) or R Q ̸= Q R, in which case K[[Q]] ∩ K[[R]] = K (the kernel of every delta operator). See [18,29]. ∑ n If R and Q commute, then obviously Th = n≥0 rn (h)R , where rn (x) is the basic set relative to R. Let R(t) be the Q-indicator of R: Rq(x, t) = R(t)q(x, t). Then we have − − rn (h)R(t)n = qn (h)t n , n≥0

(1.5)

n≥0

so that the generating function r (x, t) of rn (x) is given by r (x, t) = q(x, R(t)), where R(t) is the compositional inverse of R(t). This result can be written in a more suggestive way. Putting ∑ R(t) = n≥0 αn t n (α0 = 0 and α1 ̸= 0), we get rn (x) =

n −

  αnk∗ qk (x) = q x, αn∗ ,

(1.6)

k=0

where αnk∗ is the k-fold convolution of the sequence αn . The norming condition qn (0) = δn,0 excludes many interesting polynomial sequences, such as orthogonal polynomials. However, every polynomial sequence sn (x) is a (generalized) Sheffer set [8], [23, chapter 2] relative to some delta operator Q. Indeed, sn (x) can trivially and uniquely be written as the convolution sn (x) = sn (0) ∗ qn (x) where qn (x) is automatically a basic set. Conversely, given a basic set qn (x) and a scalar sequence cn , c0 ̸= 0, then sn (x) = cn ∗ qn (x) is Sheffer relative to qn (x) (and to its associated delta operator Q). Let C be the (Q-invariant)

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operator that sends qn (x) to sn (x) (C =

∑∞

n=0 sn (0)Q

n ).

Then we have

C Th sn (x) = C Th (sn (0) ∗ qn (x)) = sn (0) ∗ C Th qn (x) = sn (0) ∗ qn (h) ∗ Cqn (x) = sn (h) ∗ sn (x), ∑ n and therefore C Th = ∞ n=0 sn (h)Q is the convolution operator with respect to the sequence sn (x). Any Q-invariant operator P can be uniquely expanded as [26], [29, proposition 1]: P=

∞ −

(Sc0 C −1 Psn (x))Q n .

(1.7)

n=0

Now, what can we say about an arbitrary linear operator A (that is, not necessarily a member of some K[[Q]]) acting on P? It is known [17] that it can be expanded as − A= an (x)Q n , (1.8) n≥0

where an (x) are polynomials, or, alternatively, as − A= qn (x)Pn ,

(1.9)

n≥0

where Pn is a sequence in K[[Q]], as we shall see. These expansions are unique up to the choice of the algebra K[[Q]] for the second one and of the delta operator Q for the first. They are both related by their common Q-indicator [11, lemma 1.1] − − a(x, t) = an (x)t n = qn (x)Pn (t) n≥0

n≥0

1 = Aq(x, t). q(x, t)

(1.10)

(Pn (t) is the Q-indicator of Pn ). Another way to achieve this dual expansion is via the concept of adjoint operator [23, chapter 3]. It is to be noted that the delta operator used in expansion (1.8) is completely arbitrary. In the context of K[[Q]], a particularly studied umbral calculus is the cn -umbral calculus 1 ) where Th = E h is the shift operator, and [23,30], notably the shift invariant case (cn = n! the divided differences case (cn = 1: Th p(x) = x p(x)−hp(h) ). We may also mention the recent x−h 1 hyperbolic case (cn = (2n)! ) [9,28]. One question that comes immediately to mind is: is it true that every algebra K[[Q]] is an instance of a cn -umbral calculus? In other words, does there exist a sequence of non-zero scalars cn such that cn x n is the basic set associated to a delta operator R in K[[Q]]? While the converse is obviously true, it is easy to see that the answer to the question is negative. The above expansions are of the “geometric” type (involve powers of Q) [23–25]. We intend to extend them to a triangular, or Newton-like more general form (the classical divided differences scheme being a case in point). The idea of using triangular expansions is not new [7, p. 41], [13]. In fact, some of our results may be considered to be umbral versions of the Lagrange Representation Formula [7, p. 35] and the Generalized Newton Representation [7, p. 41]. Let us conclude this introduction with a word on linear functionals. To each linear functional L in P∗ , we associate a unique linear operator P in K[[Q]] as follows: L = Sc0 P and conversely. The product of two linear functionals L 1 = Sc0 P1 and L 2 = Sc0 P2 is defined as

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L 1 ×L 2 = Sc0 P1 P2 (this is induced by the underlying discrete convolution) so that the dual space P∗ is an algebra in which there is defined a concept of convergence (“the umbral algebra” [23]), isomorphic to each K[[Q]]. Finally, let A be any linear operator acting on P, and put Aqn (x) = pn (x) (qn (x) is the basic set relative to the delta operator Q). Then Sc0 Th Aqn (x) = pn (h), and therefore the formal polynomials pn (x) are uniquely associated to the polynomial functions pn (h) via the companion linear functional Ah = Sc0 Th A. Further, Ah does not depend on the particular choice of the convolution operator Th . In particular, Sc0 Th is the “evaluation functional” [23]. This observation means that the study of linear operators may to some extent be reduced to that of linear functionals (as well known to algebraists, this is not true if K is an arbitrary field or ring). 2. General expansion theorems Let Q k be a sequence of linear operators in the algebra A of all linear operators acting on P = K[x] (K = R or C). The sequence Q k is a delta (or triangular) sequence if: (1) Q k x n = 0 for k > n ≥ 0, and (2) deg (Q k x n ) = n − k for n ≥ k ≥ 0. The second condition implies in particular that Q k x k must be a non-zero scalar for each k ≥ 0. Example 1. The obvious example of such a sequence is Q k = Q k , where Q is an arbitrary delta operator. For instance: n

(1) Q k = D k x k D k = (Dx D)k . The basic set of Q = Dx D is qn (x) = x 2 . Note that q(x, t) = n! √ I0 (2 xt) (I0 (x) is the modified Bessel function of the first kind of order 0). (2) Q k = x −k D −k x −k = (x −1 D −1 x −1 )k (basic set of Q = x −1 D −1 x −1 : qn (x) = n!x n ). Here, p(0) x −1 is the division operator: x −1 p(x) = p(x)− . x x k (3) Other trivial examples are: Q k = E k D , Q k = E xk 1k , Q k = E xk x −k , etc. Example 2. Lanczos’ generalized derivative Dh is defined by Groetsch [14] ∫ h 3 Dh f (x) = 3 u f (u + x)du. 2h −h It is readily seen that Dh is a shift invariant delta operator with indicator 3 d eht − e−ht t 2h 3 dt 3 ht cosh (ht) − sinh (ht) = 3 h t2 1 2 3 1 4 5 1 =t+ h t + h t + h6t 7 + · · · . 10 280 15 120 This expression being even with respect to the parameter h, an application of Richardson’s extrapolation technique gives an interesting approximation of the derivative of a function. Moreover, a reversion of the above series (using a computer algebra system), gives the following expansion of D: dh (t) =

D = Dh −

h 2 3 37h 4 5 3481h 6 7 284453h 8 9 D + D − D + D + ···. 10 h 1400 h 378000 h 77616000 h

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Starting from Dh , one may construct a delta sequence as follows: Q k = Dh k Dh k−1 . . . Dh 1 , Q 0 = I , where h 1 , h 2 , . . . is an arbitrary sequence of non-zero parameters. Proposition 1. Let Q k be a delta sequence and let Q be a delta operator. Then there exists a unique sequence Pk of invertible operators such that Q k = Pk Q k . Proof. Let qn (x) be the basic set of Q. Fix k and let n ≥ k. Then Q k qn (x) = pm (x) is a polynomial of degree m = n − k. Now, Q k qn (x) = qn−k (x) = qm (x). Since both qm (x) and pm (x) are bases of the vector space of polynomials K[x], there exists a unique invertible operator Pk that sends qm (x) to pm (x).  It should be stressed that, in general, Pk and Q do not commute (for example, if Q k = E xk x −k , then Pk = E xk and Q = x −1 do not commute). Furthermore, the unicity of the sequence Pk is up to the choice of Q. In what follows, we shall make use of the following convergence concept. Definition 1. A sequence of linear operators An is said to converge to the operator A if, given any polynomial p(x), there exists a positive integer N such that An p(x) = Ap(x) for all n ≥ N . We now state the following general result. Theorem 1 (General Triangular Expansion Theorem). Every linear operator A can be uniquely expanded as − A= cn (x)Q n , (2.1) n≥0

where cn (x) are polynomials and Q n is a delta sequence. The series in (2.1) converges in the sense of Definition 1. Proof. Let m k (x) = Ax k be the moments of A. For every fixed n ≥ 0, the determinant of the triangular system − m r (x) = ck (x)Q k x r , r = 0, 1, . . . , n, k≥0

is (Q 0 1) × (Q 1 x) × · · · × (Q n x n ), a non-zero scalar. The corresponding linear system has a unique solution c0 (x), c1 (x), . . . , cn (x). This solution can be expressed recursively as c0 (x) =

m 0 (x) , Q01

(2.2)

r −1 m r (x) 1 − − ck (x)Q k x r , r = 1, . . . , n. (2.3) Qr x r Q r x r k=0 ∑ The sequence An = nk=0 ck (x)Q k being convergent to A (since (A − An ) p(x) = 0 for all polynomials p(x) of degree < n), the result follows. 

cr (x) =

Corollary 1. Let Ah = Sc0 Th A be the companion linear functional of A. Then − Ah = cn (h)Sc0 Th Q n . n≥0

(2.4)

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Example 3. Consider the Laplace transform Ls

xn = s n+1 , n!

n = 0, 1, . . . .

Take Q k = 1k , where 1p(x) = p(x + 1) − p(x) is the difference operator. Then we have       1 1 1 x 1 x2 1 1 x Ls = + 2 − Sc0 1 + 3 − + 1) Sc0 12 + · · · . − − (2x s s 2 s 2 s2 s s s The following result may prove to be useful. Proposition 2. Let A be an arbitrary operator and Q a delta operator with basic set qn (x). Then there exists a unique P ∈ K[[Q]] such that Sc0 A = Sc0 P. Proof. The operator P is given by P=

∞  −

 Sc0 Aqn (x) Q n .



(2.5)

n=0

3. Delta sequences in K[[ Q]] The main difficulty with an arbitrary delta sequence Q k is that the members of the sequence may not commute with each other. Such sequences are too general to be useful for our purposes. We cannot, for example, associate a basic set of polynomials to these delta sequences in a way that extends the usual notion. We do, however, have the following result. Proposition 3. Let Rk be an arbitrary delta sequence. Then there exists a delta sequence Q k such that: (1) Q i Q j = Q j Q i for all i, j ≥ 0, and (2) Sc0 Rk = Sc0 Q k for all k ≥ 0. Proof. Let Q be a delta operator with basic set qn (x). Then  − Qk = Sc0 Rk qn (x) Q k

(3.1)

n≥k

is the unique delta sequence (in K[[Q]]) verifying both properties.



In what follows, we suppose that Q k is a sequence in ∑ K[[Q]] where Q is a fixed delta operator with basic set qn (x). The order of P ∈ K[[Q]] (P = n≥0 an Q n ) is the smallest index n for which an ̸= 0 (in other words: the order of P is the degree of the lowest degree polynomial p(x) for which P p(x) ̸= 0). It is denoted by ω(P). By convention, ω(0) = +∞. We have the following simple facts. Proposition 4. The following statements are equivalent: (1) Q k is a delta sequence. (2) Q k x k is a non-zero scalar for each k. (3) ω(Q k ) = k for all k.

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Proposition 5. Let Q k be a delta sequence in K[[Q]]. Then, for each k, there exists a unique invertible operator Pk in K[[Q]] such that Q k = Pk Q k . Proof. Trivial. This is a special case of Proposition 1.



Example 4 (Divided Differences). Let h be a parameter and define the delta operator δ(h) by p(h) (in particular, δ(0) = x −1 ) so that K[[Q]] = K[[x −1 ]]. In this case, δ(h) p(x) = p(x)− x−h x p(x)−hp(h) . Basic properties x−h δ(a)δ(b) = δ(a)−δ(b) = δ (a, b). a−b

Th p(x) =

of δ(h) and Th are:

• 0 • Th = T0 + hδ(h), or, equivalently, δ(h) = Th −T h . Note that T0 = I is the identity operator. a b • Tb Ta = a−b Ta + b−a Tb . We shall make use of this Lagrangian-type equality. Now, let x0 , x1 , . . . be a sequence of scalars. Put  T δ (x1 , x2 , . . . , xk ) if k > 0 Q k = x0 Tx0 if k = 0,

(3.2)

where the operators δ (x1 , x2 , . . . , xk ) are defined inductively the usual way: δ (x1 , x2 , . . . , xk−1 ) − δ (x2 , x3 , . . . , xk ) , k ≥ 2. x1 − xk The familiar divided differences expansion uses the sequence (3.2) (see Example 11). δ (x1 , x2 , . . . , xk ) =

(3.3)

Example 5. Now, consider the algebra K[[D]] of shift invariant operators. For any pair of parameters h 1 and h 2 , define the difference operator 1 (h 1 , h 2 ): 1 (h 1 , h 2 ) p(x) =

E h1 − E h2 p (x + h 1 ) − p (x + h 2 ) p(x) = h1 − h2 h1 − h2

(3.4)

if h 1 ̸= h 2 and 1 (h, h) p(x) = E h Dp(x) = p ′ (x + h) . Given sequences of scalars x0 , x1 , . . . and y0 , y1 , . . . put  1 (x1 , y1 ) 1 (x2 , y2 ) . . . 1 (xk , yk ) if k > 0 Qk = E x0 if k = 0.

(3.5)

(3.6)

Now, we associate a basic set to the sequence Q k . Definition 2. Let Q k be a delta sequence in K[[Q]]. A polynomial sequence ϕn (x) is the basic set relative to the sequence Q k if Sc0 Q k ϕn (x) = δn,k , n, k ≥ 0. We say that the sequences Q k and ϕn (x) are biorthonormal [7, p. 34]. Remark 2. As noted in the introduction, every polynomial sequence ϕn (x) is a Sheffer set relative to some delta operator R and its basic set rn (x) (ϕn (x) = rn (x) ∗ ϕn (0)). Let U be the umbral operator that sends rn (x) to qn (x). We have the string of equalities: QUrn (x) = Qqn (x) = qn−1 (x) = Urn−1 (x) = U Rrn (x). Sc0 Q

Sc0 R k ϕ

(3.7)

Hence R = so that k ϕn (x) = n (x). Then, why bother introducing delta sequences since in effect they may be replaced by delta operators? A possible answer is that U −1 QU

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in general R and Q do not commute (K [[Q]] ∩ K[[R]] = K) and we may lose the benefit of working within the algebra K[[Q]]. This distinction does not hold for linear functionals, however (see the last paragraph of the introduction). See also [8] or [29]. Proposition 6. Every delta sequence Q k admits a unique basic set ϕn (x). Conversely, every polynomial sequence ϕn (x) is the basic set relative to a unique delta sequence in K[[Q]]. Proof. Along the lines of [26] with minor modifications (the polynomials ϕn (x) are shown to exist through an induction argument). We can also use the orthonormalization method of [7, p. 42].  Example 6. The basic set associated to Q k = Tx0 δ (x1 ) δ (x2 ) . . . δ (xk ) is ϕn (x) = (x − x0 )(x − x1 ) . . . (x − xn−1 ), n ≥ 1 (ϕ0 (x) = 1). The x −1 -indicator of Q k is Q k (t) =

tk . (1 − x0 t) (1 − x1 t) . . . (1 − xk t)

(3.8)

Example 7. Let P be an invertible operator in K[[Q]] and put Q k = P Q k . The basic set associated to Q k is the Sheffer set associated to the pair (P, Q) [23]. Example 8 (Orthogonal Polynomials). Let ϕn (x) be a sequence of orthogonal polynomials [1] b relative to some weight function w(x): a w(x)ϕn (x)ϕm (x)dx = δn,m . Use a norming criterion to make the sequence unique (for example, the leading coefficient of ϕn (x), when expanded in terms of a basic set qn (x), is set to be equal to 1). Define the Fourier coefficients of a b function f (x) by L k f (x) = a w(x)ϕk (x) f (x)dx. We obviously have L k x n = 0 for n < k and L k x k ̸= 0. On the other hand, since the algebra of linear functionals and the algebra K[[Q]] are isomorphic, there exists a unique Q k in K[[Q]] such that L k = Sc0 Q k , and the sequence Q k is the delta sequence associated with ϕn (x). Furthermore, the operators Q k are given explicitly by the expansion Qk =

∞ −

(L k qn (x)) Q n .

(3.9)

n=0

This example can be extended to the more general case of formal orthogonal polynomials [4] (or polynomials orthogonal relative to a scalar sequence [31, p. 192]). Let L be a linear functional and put Lqn (x) = cn . A polynomial sequence ϕn (x) is said to be a sequence of formal orthogonal polynomials with respect to L (or with respect to the sequence cn , and relative to qn (x)) if, for each fixed n, L (qk (x)ϕn (x)) = δn,k ,

k = 0, 1, . . . , n.

(3.10)

If we put L k f (x) = L (qk (x) f (x)), then we are in the above configuration. It can be shown [31, p. 192] that the polynomial sequence ϕn (x) exists if and only if, for each fixed n, the determinant of the Hankel matrix (ck )nk=0 is non-zero. Proposition 7. Put ϕn (x) =

n − k=0

ank qk (x)

(3.11)

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and Qk =

−

bnk Q n ,

(3.12)

n≥k

where qn (x) is the basic set relative to Q. Then the infinite lower triangular matrices A = (ank ) and B = (bnk ) are inverses of each other. Proof. It is a straightforward verification. Let ϕ(x, t) be the generating function of the sequence ϕn (x). We have the string of equalities: ∞ ∞ − n − − ϕ(x, t) = ϕn (x)t n = ank qk (x)t n n=0

=

∞ −

n=0 k=0

qk (x)

−

ank t n =

n≥k

k=0

∞ −

qk (x)Ak (t),

(3.13)

k=0

where the formal series Ak (t) is of order exactly k for each k. Now, apply the linear functional Sc0 Q k to ϕ(x, t): Sc0 Q k ϕ(x, t) = Sc0 Q k

∞ −

ϕn (x)t n =

n=0

∞ −

δn,k t n = t k .

(3.14)

n=0

On the other hand, Sc0 Q k ϕ(x, t) = Sc0

−

bnk Q n

n≥k

=

=

−

bnk

∞ − i=n

−

−

n≥k

qi (x)Ai (t)

i=0

n≥k

bnk

∞ −

i≥n

δn,i Ai (t) = i

ain t =

−

bnk An (t)

n≥k  i − − i≥0

 ain bnk t i .

(3.15)

n=0

Identifying coefficients of like powers in the last members of (3.14) and (3.15), we get i −

ain bnk = δi,k .

(3.16)

n=0

The sum on the left of (3.16) is the product of row i of A with column k of B and it is equal to the corresponding term in the infinite dimensional identity matrix.  Remark 3. A word of caution is necessary here. Since we are dealing with lower triangular infinite matrices, a right inverse is automatically a left inverse, and it is unique. This may not be the case in general. For a detailed discussion of these aspects, see [20, pp. 37–39]. For an interesting supply of examples, see [32]. Theorem 2. Let Q k (t) be the Q-indicator of Q k . Then the generating function q(x, t) of qn (x), the basic set of Q, can be uniquely expanded as: q(x, t) =

∞ − n=0

qn (x)t n =

∞ − n=0

where ϕn (x) is the basic set of Q k .

ϕn (x)Q n (t),

(3.17)

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Proof. This is a direct consequence of the previous proposition. Using the same notations, we have    ∞ n − − − − r br n t ϕn (x)Q n (t) = ank qk (x) n=0

n≥0

=

k,r ≥0

−

=

r ≥n

k=0

 − −

 br n ank qk (x) t r

n≥0

δr,k qk (x)t r =

k,r ≥0

−

qk (x)t k = q(x, t).

k≥0

Note that we made use of the preceding remark.



Corollary 2. The convolution operator Th can uniquely expanded as: Th =

∞ −

ϕn (h)Q n .

(3.18)

n=0

Example 9 (An Extension of Euler’s Transformation). Let xn be an arbitrary sequence of scalars. Put ϕ0 (x) = 1 and ϕn (x) = (x − x0 ) . . . (x − xn−1 ) (n ≥ 1). Then, with Q = x −1 (see Example 6), we have the expansion − 1 tn = ϕn (x) . 1 − xt (1 − x0 t) (1 − x1 t) . . . (1 − xn t) n≥0

(3.19)

Let L be the linear functional that sends x n to an and apply it to both members of this expansion (see Remark 1): −

an t n =

n≥0

−

  ϕn a

n≥0

tn . (1 − x0 t) (1 − x1 t) . . . (1 − xn t)

(3.20)

∑ If the series n≥0 an converges, we may replace t by 1 (Abel’s lemma for infinite series [10, p. 92]), and an appropriate choice of the xn ’s gives the identity − n≥0

an =

− n≥0

  ϕn a

1 . (1 − x0 ) (1 − x1 ) . . . (1 − xn )

(3.21)

This result generalizes Euler’s h-transformation [16] (x0 = x1 = · · · = h): − n≥0

 − n an = a−h n≥0

1 1−h

n+1

,

(3.22)

widely used to accelerate the convergence of alternating series (usually with h = −1). The generalized Euler’s transformation (3.21) can be put to very effective use to accelerate the convergence of alternating series. Variants were obtained using classical analysis [5]. As this example indicates, series transformations by means of infinite row finite matrices (e.g. Borel, Cesaro, etc. See [10,16,20,27]) can be considered from the umbral calculus prism.

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4. Triangular expansions in K[[ Q]] The following result, while generalizing many classical operator expansions, may be viewed as the umbral calculus version of the Generalized Newton Representation Theorem [7, p. 41]. The formal series version is also a classical result. Theorem 3. Every P ∈ K[[Q]] can be uniquely expanded as  − P= Sc0 Pϕn (x) Q n .

(4.1)

n≥0

Proof. Apply to ϕr (x) and extend by linearity.



Corollary 3. Every linear functional L can be expanded as − L= (Lϕn (x)) Sc0 Q n .

(4.2)

n≥0

Corollary 4. Let P ∈ K[[Q]] and put Pϕn (x) = pn (x). Then − Sc0 Ta P = pn (a)Sc0 Q n .

(4.3)

n≥0

Note that Theorem 2 and its corollary are consequences of this result. For polynomials, we have the interpolation formula: Theorem 4. Let p(x) be a polynomial (or the generating function of a sequence of polynomials). Then we have the expansion  − p(x) = Sc0 Q k p(x) ϕk (x). (4.4) k≥0

Proof. Expand p(x) as a linear combination of the ϕn (x) and apply Sc0 Q k to both members of the resulting equality. This is also a direct consequence of the last corollary: Sc0 Ta p (x) = ∑ p(a) = n≥0 ϕn (a)Sc0 Q n p(x).  This theorem can be extended as follows. Theorem 5. Let p(x) be a polynomial (or a generating function of polynomials). Then we have the expansion  − p(x) = Th−1 ϕk (x) Sc0 Th Q k p(x). (4.5) k≥0

In particular, if Q k are shift invariant, − p(x) = ϕk (x − h)Sc0 Q k p(x + h). k≥0

(4.6)

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Proof. Since Th qn (x) = qn (x) ∗ qn (h), it is clear that Th p(x) = f (x; h) depends on x and h in a symmetric way: f (x; h) = f (h; x). Now consider the operator Th . It can be expanded as Th =

∞ −

ϕk (h)Q k .

n=0

Apply both sides of this expansion to p(x): Th p(x) =

∞ −

ϕk (h)Q k p(x).

n=0

Using the preliminary observation at the beginning of this proof, we exchange x and h on the right (taking into account that Q k acts on the variable x): Th p(x) =

∞ −

ϕk (x)Sc0 Th Q k p(x).

n=0

Apply Th−1 to both sides of this equality to get the desired expansion. In the particular case, use the fact that Th = E h is the shift operator.  Example 10 (An Abel-type Interpolation). Let Q k = E xk Q k where xk is an arbitrary sequence of scalars. Let ϕn (x) be the basic set relative to the delta sequence Q k . Then the sequences ϕn (x) and qn (x) are related by the identities: qn (x) =

n −

qn−k (xk )ϕk (x).

(4.7)

k=0

To see this, it suffices to apply Sc0 Q r to both members, for r = 0, 1, . . . , n. If we take Q = D, we have the beautiful n − xkn−k xn = ϕk (x). n! (n − k)! k=0

(4.8)

If xk = ka, where a is a constant, this identity reduces to the celebrated Abel identity [6,22] and ϕk (x) are the Abel polynomials. Now, if Q = 1, we have  n  x  − xk = ϕk (x). (4.9) n n−k k=0 If xk = ka, where a is a constant, the polynomials ϕk (x) are the Gould polynomials. Replacing qn (x) by an arbitrary polynomial p(x) in (4.7), we get  − p(x) = Sc0 E xk Q k p(x) ϕk (x). (4.10) k≥0

Specializing to Q = D, we obtain a generalization of Taylor’s expansion (the Abel–Gontscharoff interpolation formula [7]): − p(x) = ϕk (x) p (k) (xk ) (4.11) k≥0

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while the choice Q = 1 gives a generalization of Newton’s expansion: − ϕk (x)1k p(xk ). p(x) =

(4.12)

k≥0

Now, it is worth noting that the above expansions contain many classical formulas of numerical analysis (see the “lozenge diagram” [15, p. 304]). Let us give  a sample: Take Q = 1 and use (4.12). If x0 = x1 = · · · = 0, then ϕn (x) = nx and (4.12) is Newton’s forward formula. If x0 = 1, x1 = −1, x2 = −2, . . . , then ϕn (x) = x(x+1)...(x+n−1) and (4.12) is n! Newton’s backward formula. The choice x0 = x1 = 0, x2 = x3 = −1, x4 = x5 = −2, . . . gives x x 2 −1

ϕ0 (x) = 1, ϕ1 (x) = 1!x , ϕ2 (x) = x(x−1) , and (4.12) is Gauss’ formula. 2! , ϕ3 (x) = 3! E xk +E yk k Now, take Q k = 1 . Then the choice x0 = y0 = 0, x1 = −1, y1 = 0, x2 = y2 = −1, 2 x3 = −2, y3 = −1, x4 = y4 = −2, . . . gives Stirling’s formula, while the choice x0 = 0, y0 = 1, x1 = y1 = 0, x2 = −1, y2 = 0, x3 = y3 = −1, x4 = −2, y4 = −1, . . . gives Bessel’s formula. Example 11 (A General Divided Differences Formula). We describe a numerical scheme of general validity. Consider the sequence of linear functionals L n defined by L n p(x) =

n −

ank p (xnk ) ,

(4.13)

ank Txnk

(4.14)

k=0

i.e. L n = Sc0

n − k=0

(the particular choice of the convolution operators depends on the problem at hand), where the coefficients ank are chosen in such a way that  0 if k < n Ln xk = (4.15) 1 if k = n with L 0 1 = 1. The associated determinant of the above linear system is the Vandermonde determinant and the problem has thus a unique solution ∑ if and only if the xnk ’s are distinct for each fixed n. Hence, Corollary 3 applies (here Q n = nk=0 ank Txnk ) and every linear functional L can be expanded as L = λ0 L 0 + λ1 L 1 + λ2 L 2 + · · · .

(4.16)

Now, we can say more about the structure of the L n ’s. The coefficients ank can be expressed via the Lagrange polynomials: ank =

1 , πnk (xnk )

n ≥ 0, k = 0, 1, . . . , n,

(4.17)

where πnk (x) =

ωn (x) , x − xnk

k = 0, 1, . . . , n,

(4.18)

and ωn (x) = (x − xn0 ) (x − xn1 ) . . . (x − xnn ) ,

n = 0, 1, . . . .

(4.19)

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The coefficients λn of the expansion (4.16) are given by λn = Lϕn (x), where ϕn (x) is the biorthonormal sequence of polynomials associated with the sequence L n . The following recursive scheme gives the λn ’s: λ0 = m 0 , λn = m n −

n−1 −

λk L k x n ,

n ≥ 1,

(4.20)

k=0

where m n = L x n are the moments of L (compare with the proof of Theorem 1). A natural extension of this process is to replace x n by ϕn (x) in the definition of L n . Example 12. Another version of the previous example is to consider the sequence Q 0 = Tx00 , Q k = 1xk1 1xk2 . . . 1xkk , k = 1, 2, . . . . An effective integration scheme can be developed using a general formula of the form: ∫ 1 f (x)dx = a0 f (h 0 ) + a1 1x11 f (h 1 ) + a2 1x21 1x22 f (h 2 ) + · · · . (4.21) 0

Indeed, one can for instance characterize the parameters h k recursively using some criterion (exactness for a class of polynomials being the usual rule). Many definite integration schemes are described by (4.21) or a variant (using central differences instead). Consider now the vector subspace K0 [[Q]] of K[[Q]] defined by   n − K0 [[Q]] = ak Txk , n ∈ N, ak , xk ∈ K .

(4.22)

k=1

A typical member P0 of K0 [[Q]] is thus written as P0 =

n −

(4.23)

ak Txk

k=1

where the xk ’s are distinct. The coefficients ak can be expressed in terms of the Lagrange polynomials n ∏

lk (x) =

(x − xi )

i=1 i̸=k

n ∏

(4.24) (xk − xi )

i=1 i̸=k

as follows: Sc0 P0li (x) = Sc0

n −

ak Txk li (x) = ai .

(4.25)

k=1

Now, since P0 ∈ K[[Q]], it can uniquely be expanded as − P0 = bn Q n n≥0

(4.26)

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where bn = Sc0 P0 qn (x) = Sc0

n −

ak Txk qn (x).

(4.27)

k=1

Hence n −

bn =

ak qn (xk ) .

(4.28)

k=1

This is an inversion of ai = Sc0 P0li (x) = Sc0

−

bn Q n li (x).

(4.29)

n≥0

We relate the Lagrange representation to the Newton representation. Theorem 6. K0 [[Q]] is dense in K[[Q]]. In other words, given P ∈ K[[Q]] and an arbitrary polynomial p(x), we shall show there exists P0 ∈ K0 [[Q]] such that (P − P0 ) p(x) = 0. We will construct such an operator. Proof. Let x0 , x1 , . . . be sequence of distinct parameters and define the following sequence of operators in K0 [[Q]]: T (x0 ) = Tx0

(4.30)

T (x0 ) − T (x1 ) , T (x0 , x1 ) = x0 − x1 T (x0 , x1 ) − T (x1 , x2 ) , T (x0 , x1 , x2 ) = x0 − x2 .. . It is readily seen that we have the following properties: (1) T (x0 ), T (x0 , x1 ), T (x0 , x1 , x2 ) , . . . is a delta sequence. (2) T (x0 , x1 , . . . , xn ) = c0 T (x0 ) + c1 T (x1 ) + · · · + cn T (xn ) where ci = π(x) = (x − x0 ) . . . (x − xn )) and c0 + c1 + · · · + cn = 1.

1 πi (xi )

(πi (x) =

π(x) x−xi ,

Using the first property, we can expand any operator P ∈ K[[Q]] as follows: P=

∞ −

ak T (x0 , . . . , xk ) .

(4.31)

k=0

Now, given any polynomial p(x) of degree r , we get P0 by truncating the above expansion: P0 =

r −

ak T (x0 , . . . , xk ) .

(4.32)

k=0

Note that if we use the second property we get an alternative expression of P0 (Lagrange form).  Remark 4. We may instead use the operators of Example 11.

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Remark 5. If K[[Q]] = K[[D]] or K[[Q]] = K[[x −1 ]], the subspace K0 [[Q]] is actually a subalgebra since the product of two convolution operators Ta and Tb can be expressed as a linear combination of convolution operators. More precisely, in the first case, E a E b = E a+b and, in a b the second, Ta Tb = a−b Ta + b−a Tb . It would be interesting to characterize all subspaces that are closed under multiplication. Remark 6. We can also consider subspaces of the form   n r − − aik Q i Txik , r ∈ N, n ∈ N, aik , xik ∈ K .

(4.33)

i=0 k=1

Prototypes of such schemes are the Hermite interpolation formulas. 5. Expansions of arbitrary operators In this section, we consider arbitrary operators in A and give special versions of Theorem 1. Let Q be a delta operator with basic set qn (x). First, let us state the Kurbanov–Maksimov expansion formula [17]. Theorem 7. Let A ∈ A. Then − A= an (x)Q n

(5.1)

n≥0

where the polynomials an (x) are given by an (x) = [Aqn (x)] ∗ qn−1∗ (x).

(5.2)

The sequence of polynomials qn−1∗ (x) is the inverse of the polynomial sequence qn (x) for convolution (qn−1∗ (x) ∗ qn (x) = δn,0 ). Proof. First, qn−1∗ (x) exists because q0 (x) = 1 is a non-zero scalar. This condition is necessary and sufficient. To prove the theorem, it suffices to apply both members of (5.1) to qr (x) and then to extend by linearity. ∑ n Let a(x, t) = function of the sequence an (x) (a (x, t) is n≥0 an (x)t be the generating ∑ called the Q-indicator of A) and q(x, t) = n≥0 qn (x)t n the generating function of qn (x). Then we have the operational formula a(x, t) =

1 Aq(x, t). q(x, t)

(5.3)

In the shift invariant case (Q commutes with D), qn (x) is of the binomial type [26] and qn−1∗ (x) = qn (−x), q(x, t) = exp (xq (t)), where q(t) is the indicator of Q, and we have the Corollary 5. Let A ∈ A and let Q be shift invariant. Then A can be expanded as in (5.1), and the coefficients an (x) are given by an (x) = [Aqn (x)] ∗ qn (−x)

(5.4)

a(x, t) = q (−x, t) Aq(x, t).

(5.5)

and

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Example 13 (Scale Invariant Operators). Let P be an operator and put P x n = pn (x). We say that P is scale invariant if P Sch = Sch P for all h in K. It is easy to see that P is scale invariant if and only if P x n = an x n , where an = pn(1)[2]. Using the umbral notation introduced earlier, we may alternatively write P p(x) = p ax for any polynomial p(x). Hence the more suggestive P = Sca . The set of all scale invariant operators is readily seen to be an algebra to the is replaced by ∑ isomorphic ∑ ∑algebran of formal series where the usual Cauchy product a n t n × bn t n = an bn t . Now, it is a simple matter to expand Sca in terms of the shift invariant delta operator Q. The Q-indicator of Sca is       q (−x, t) Sca q(x, t) = q (−x, t) q ax, t = q x a − 1 , t , and therefore Sca =

∞ −

   qn x a − 1 Q n = E x (a−1) = E x1 a0 .

(5.6)

n=0

Taking Q = D and using (the binomial transform [21]) n    n − n a−1 = ak (−1)n−k = 1n a0 , k k=0

(5.7)

we get Sca =

∞  −

1n a 0

n=0

 xn n D . n!

(5.8)

An immediate consequence of this result is the case when an is a polynomial in n. The expansion is finite because 1n a0 is eventually ∑ equal to 0 (for all n > deg a). Apply both sides of (5.8) to 1 the generating function 1−xt = n xntn: −

an x n t n =

n≥0

∞  −

1n a 0

n=0

xntn



(1 − xt)n+1

.

(5.9)

Putting t = 1, we have the formal well-known identity −

an x n =

n≥0

∞  − n=0

1n a 0

xn



(1 − x)n+1

.

(5.10)

If an are polynomials in n, the right-hand side of (5.10) is related to the Eulerian polynomials. One also recognizes a form of Euler’s transformation (see Example 9). Finally, note in passing that the functional Sc0 has the expansion Sc0 =

∞ −

qn (−x) Q n = E −x ,

(5.11)

n=0

and that the umbral composition of a polynomial p(x) with a sequence an is obtained by setting x = 1. For instance, using (5.8),   − 1n a0 (n) p a = p (1) . n! n≥0

(5.12)

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Interesting as well as classical results are obtained if we expand Sca in terms of other classes of delta operators (or delta sequences). Example 14. It is well known (and easy to see [2]) that every automorphism A of K [x] is of the form Ap(x) = p (ax + b), where a and b are scalars (a ̸= 0). In other words, A = Sca E b . From q (−x, t) Aq(x, t) = q (x (a − 1) + b, t), we get the expansion (Q is shift invariant) Sca E b =

∞ −

qn (x (a − 1) + b) Q n = E x(a−1)+b .

(5.13)

n=0 n

Take Q = D in the previous corollary. Then qn (x) = xn! and [ n] n − x x n−k (−x)n (−x)k an (x) = A ∗ = A n! n! k! (n − k)! k=0 =

n   1 − n (−x)k Ax n−k . n! k=0 k

(5.14)

This is just the nth Pincherle derivative (see the last section of this paper) A(n) of the operator A, 1 up to the factor n! [26]. Corollary 6. For any A ∈ A, we have the expansion − A= an (x)D n

(5.15)

n≥0

with an (x) =

1 (n) A 1. n!

(5.16)

In the case of linear functionals, Theorem 7 specializes to Theorem 8. Every linear functional L can be expanded as  − L= an ∗ qn−1∗ (x) Q n n≥0

where an = Lqn (x). Example 15. Let Q be shift invariant. Then q(x, t) = exp(xq(t)), where q(t) is the indicator of Q. We see then that (see Corollary 5) a(x, t) = exp(−xq(t))A exp (xq(t)) .

(5.17)

Note that A is shift invariant if and only if a(x, t) does not depend on x, that is a(x, t) = a(t). Example 16. Now take Q in K[[x −1 ]]. Then q(x, t) = a(x, t) = (1 − xq(t))A(1 − xq(t))−1 .

1 1−xq(t)

and (5.18)

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Remark 7. Some of the above considerations and results ∑ naturally lead us to define the concept of an associate operator. Let Q be shift invariant and A = n an (x)Q n an arbitrary operator. Put Aq (x, t) = p(x, t) = a(x, t)q(x, t). The associate operator of A relative to Q is the operator AĎ whose Q-indicator is p (−x, t). In other words, AĎ sends qn (x) to an (−x). We list some of the properties of this operator. (1) (α A + β B)Ď = α AĎ + β B Ď . (2) (AĎ )Ď∑ = A. (3) A = n≥0 [Sc−1 AĎ qn (x)]Q n . (4) P is shift invariant if and only if P Ď is a linear functional. In this case P Ď = Sc0 P. (5) p(x)Ď = p(−x)Sc0 for any polynomial p(x). (6) ( p(x)A)Ď = p(−x)AĎ . In particular, if P is shift invariant, ( p(x)P)Ď = p(−x)Sc0 P. (7) (A P)Ď = AĎ P for any shift invariant operator P. In particular, if P and R are shift invariant, (P R)Ď = Sc0 P R. (8) (Sca )Ď = Sc1−a . In particular, (Sc0 )Ď = Sc1 = I (the identity operator) and (Sc1/2 )Ď = Sc1/2 . (9) The mapping P → P Ď is an algebra isomorphism from K[[Q]] onto P∗ . ∑ Let p(x, t) = n≥0 pn (x)t n be the generating function of a sequence pn (x) of polynomials. Then p(x, t) can alternatively be written as [11] − p(x, t) = qn (x) f n (t) (5.19) n≥0

where f n (t) is a sequence of formal series converging to 0 (i.e. ω( f n ) → ∞ as n → ∞). The sequence f n (t) is called the conjugate (or transpose) sequence (relative to the basic set qn (x)) of the sequence pn (x). It is easy to see that f n (t) converges if and only if it is the conjugate of a sequence of polynomials (here pn (x)). Another way to express this is as follows. Let λ be the linear operator acting on K [[t]]: λt n = f n (t),

(5.20)

or, equivalently, using generating functions, λq(x, t) =

∞ −

qn (x)(λt n ) =

n=0

∞ −

qn (x) f n (t).

(5.21)

n=0

The sequence f n (t) is the conjugate of a sequence gn (x) in K [[x]] via their common generating function (gn (x) is not necessarily a sequence of polynomials). This sequence defines the operator Aqn (x) = gn (x), ∞ ∞ − − Aq(x, t) = gn (x)t n . (Aqn (x)) t n = n=0

(5.22) (5.23)

n=0

Put more succinctly, we have the defining equality Aq(x, t) = λq(x, t).

(5.24)

The operator λ is the conjugate of A relative to the basic set qn (x) (if we think of formal series as linear functionals, the conjugate is just the adjoint [23, chapter 3]). We write λ = A∗ . The operator λ is continuous if and only if gn (x) is a sequence of polynomials (A ∈ A). The following properties of the conjugate operator (relative to qn (x)) are straightforward:

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(A + B)∗ = A∗ + B ∗ . (c A)∗ = c A∗ , for any scalar c. (AB)∗ = B ∗ A∗ . ABq(x, t) = AB ∗ q(x, t) = B ∗ Aq(x, t) = B ∗ A∗ q(x, t). (A−1 )∗ = (A∗ )−1 . If P ∈ K[[Q]] with Q-indicator P(t), then P ∗ = P(t). ∑ If L is a linear functional with Lqn (x) = an , then L ∗ t n = δn,0 A(t) (A(t) = n≥0 an t n ). ∑ If A is an arbitrary invertible operator, then q(x, t) = n (Aqn (x))((A−1 )∗ t n ) (see Theorem 2 and its corollary, and also Proposition 7). ∑ (9) If (ank ) is the matrix representation of A (Aqn (x) = k≥0 ank qk (x)), then (ank )t (the trans∑ pose matrix) is the matrix representation of A∗ (A∗ t k = n≥0 ank t n ) and conversely.

(1) (2) (3) (4) (5) (6) (7) (8)

In view of the last property, we define (A∗ )∗ = A. Example 17. Let qn (x) = cn x n . Then the scale invariant operators Sca are self-conjugate relative to this set.   1 t Example 18. The set of continuous operators λh f (t) = 1−ht f 1−ht (h ∈ K) is a group n under composition (λh λk = λh+k , λ−1 h = λ−h ). The conjugate of λh (relative to qn (x) = x , 1 ∗ h n q(x, t) = 1−xt ) is the shift operator λh = E . For h = 1 (and qn (x) = x ), this transformation n is known as Euler’s transformation (see Example 9). Now, if we take qn (x) = xn! (q(x, t) = ext ), the conjugate sequence of t n is given by

λ∗h

n   k − xn n x n−k = h = L n (x, h) . n! k k! k=0

(5.25)

We recognize in L n (−x, 1) the (simple, α = 0) Laguerre polynomials [6]. For further developments, see [12]. The continuous operator λ of this last example is of course not an isolated case. It is of the form (which gives an alternative definition of the generalized Sheffer sets) λ f (t) = P(t) f (R(t)), where P(0) ̸= 0 and R(0) = 0, R ′ (0) ̸= 0 (that is, they are respectively of order 0 and 1). For n a detailed study, see for instance [23, ch. 3] where they are studied in the case qn (x) = xn! . The extension to other basic sets is straightforward. Keeping in mind the definition of the conjugate, we can restate Theorem 7 as follows: Theorem 9. Let A ∈ A. Then − A= qn (x)Pn

(5.26)

n≥0

where Pn ∈ K[[Q]] has Q-indicator Pn (t) = A∗ t n . ∑ Using Theorem 3, Pn = k≥0 bnk Q k and we can also expand A as Corollary 7. − A= bn (x)Q n n≥0

(5.27)

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where the polynomials − bn (x) = bnk qk (x)

(5.28)

k≥0

are defined by bnk = Sc0 Pn ϕk (x).

(5.29)

6. The Pincherle derivatives Let A be an arbitrary linear operator. Recall that the (first) Pincherle derivative of A is defined by Rota et al. [26]: A′ = Ax − x A.

(6.1)

We write A′ = ∂1 A = A′1 . It can be used to expand A in a “Taylor’s” series as follows (see Corollary 6): − A= an (x)D n , (6.2) n≥0 (n)

1 1 where an (x) = n! (∂1n A)1 = n! A1 1. Since this result can alternatively be written as − A= x n Pn ,

(6.3)

n≥0

it is natural to define a “second” Pincherle derivative of A which directly gives the “coefficients” Pn . This is done as follows: ∂2 A = A′2 = D A − AD.

(6.4)

The Pincherle derivatives have the following straightforward properties. Proposition 8. Let A be an arbitrary operator with indicator a(x, t), P a shift invariant operator and p(x) a polynomial. Then ∂1 A = 0 if and only if A is a polynomial. ∂2 A = 0 if and only if A is shift invariant. ∂2 p(x) = p ′ (x), where p ′ (x) is the ordinary derivative of p(x). ∂1 P = P ′ for every shift invariant operator P. Here P ′ = P ′ (D) is the ordinary derivative of the formal series P(D). (5) ∂2 (∂1 A) = ∂1 (∂2 A). (6) ∂t∂ a(x, t) is the indicator of ∂1 A. (7) ∂∂x a(x, t) is the indicator of ∂2 A. (1) (2) (3) (4)

(8)

∂ n+m a t) ∂xn ∂tm (x,

is the indicator of ∂2n ∂1m A.

Now, the following result is obvious.

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Theorem 10. Every linear operator A can be expanded as: − ank x k D n , A=

2127

(6.5)

n,k≥0

where ank =

  1 Sc0 ∂2n ∂1k A 1. n!k!

(6.6)

Acknowledgments The author thanks the referees for their valuable comments and suggestions. References [1] W. Al-Salam, Orthogonal polynomials. Theory and practice, in: P. Nevai (Ed.), Characterization Theorems for Orthogonal Polynomials, Kluwer Academic Publishers, Dordrecht, Netherlands, 1990, pp. 1–24. [2] K. Belbahri, Scale invariant operators and combinatorial expansions, Adv. Appl. Math. (2010). doi:10.1016/j.aam.2010.01.010. [3] C. Berge, Principles of Combinatorics, Academic Press, New York, 1971. [4] C. Brezinski, P. Maroni, The algebra of linear functionals on polynomials, with applications to Pad´e approximation, in: Orthogonal Polynomials and Numerical Analysis, Numer. Algorithms 11 (1996) 25–33 (special issue). [5] H. Cohen, F.R. Villegas, D. Zagier, Convergence acceleration of alternating series, Experiment. Math. 9 (1) (2000) 3–12. [6] L. Comtet, Analyse Combinatoire, Presses Universitaires de France, Paris, 1970. [7] P.J. Davis, Interpolation and Approximation, Dover, New York, 1975. [8] A. Di Bucchianico, D.E. Loeb, A Simpler characterization of Sheffer polynomials, Stud. Appl. Math. 92 (1994) 1–15. [9] A. Di Bucchianico, D.E. Loeb, Sequences of binomial type with persistent roots, J. Math. Anal. Appl. 199 (1996) 39–58. [10] G.M. Fichtenholz, Infinite Series: Ramifications, Gordon and Breach, 1970, Revised English Edition Translated and Adapted by R.A. Silverman. [11] J.M. Freeman, Transforms of operators, Congr. Numer. 48 (1985) 115–132. [12] B. Gabuttil, J.N. Lyness, Some generalizations of the Euler–Knopp transformation, Numer. Math. 48 (1986) 199–220. [13] A.O. Gelfond, Calculus of Finite Differences, Hindustan Publishing Corporation, Delhi, 1971, Translated from the Russian. [14] C.W. Groetsch, Lanczos’ generalized derivative, AMM 105 (4) (1998) 320–326. [15] R.W. Hamming, Numerical Methods for Scientists and Engineers, McGraw-Hill, Inc., 1973. [16] G.W. Hardy, Divergent Series, Clarendon Press, Oxford, 1949. [17] S.G. Kurbanov, V.M. Maksimov, Mutual expansions of differential operators and divided difference operators, Dokl. Akad. Nauk USSR 4 (8–9) (1986). [18] G. Markowsky, Differential operators and the theory of binomial enumeration, J. Math. Anal. Appl. 63 (1978) 145–155. [19] R. Mullin, G.-C. Rota, Theory of binomial enumeration, in: B. Harris (Ed.), Graph Theory and its Applications, Academic Press, New York, 1970, pp. 167–213. [20] R.E. Powell, S.M. Shah, Summability Theory and Applications, Van Nostrand Reinhold, London, 1972. [21] H. Prodinger, Some information about the binomial transform, Fibonacci Quart. 32 (1994) 412–415. [22] J. Riordan, Combinatorial Identities, John Wiley, New-York, 1968. [23] S.M. Roman, The Umbral Calculus, Academic Press, London, 1984. [24] S.M. Roman, The theory of the umbral calculus I, J. Math. Anal. Appl. 87 (1982) 58–115. [25] S.M. Roman, G.C. Rota, The umbral calculus, Adv. Math. 27 (1978) 95–188. [26] G.C. Rota, D. Kahaner, A. Odlyzko, Finite operator calculus, J. Math. Anal. Appl. 42 (1973) 684–760.

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Further reading [1] M. Craciun, A. Di Bucchianico, Sheffer sequences, probability distributions and approximation operators, SPOR Report, 2005–04, Eindhoven University of Technology, The Netherlands. [2] E.C. Popa, On an expansion theorem in the finite operator calculus of G–C Rota, Gen. Math. 16 (2008) 149–154.