Extended finite operator calculus -an example of algebraization of analysis

A.K. Kwaśniewski

Institute of Computer Science, Białystok University PL-15-887 Biał ystok, ul. Sosnowa 64, POLAND e-mail: kwandr@noc.uwb.edu.pl

April 25, 2001

Abstract

“A Calculus of Sequences” started in 1936 by Ward constitutes the general scheme for extensions of classical operator calculus of Rota - Mullin considered by many afterwards and after Ward. Because of the notation we shall call the Wards calculus of sequences in its afterwards elaborated form - a ψ-calculus. The ψ-calculus in parts appears to be almost automatic, natural extension of classical operator calculus of Rota - Mullin or equivalently - of umbral calculus of Roman and Rota. At the same time this calculus is an example of the algebraization of the analysis

- here restricted to the algebra of polynomials. Many of the results of ψ-calculus may be extended to Markowsky Q-umbral calculus where Q stands for a generalized difference operator i.e. the one lowering the degree of any polynomial by one. The article is supplemented by the short indicatory glossaries of terms and notation used by Ward, Viskov, Markowsky, Roman on one side and the Rota-oriented notation on the other side.

KEY WORDS: extended umbral calculus , Graves-Heisenberg-Weyl algebra MCS (2000) : 05A40 , 81S99

1 Introduction

We shall call the Wards calculus of sequences [1] in its afterwards last century elaborated form - a ψ-calculus because of the notation [2]-[7]. The efficiency of the Rota oriented language and our notation used has been already exemplified by easy proving of ψ-extended counterparts of all representation independent statements of ψ-calculus [2]. Here these are ψ-labelled representations of Graves-Heisenberg-Weyl (GHW) algebra of linear operators acting on the algebra P of polynomials.

As a matter of fact ψ-calculus becomes in parts almost automatic extension of Rota - Mullin calculus or equivalently - of umbral calculus of Roman and Rota [8, 9, 10]. The ψextension relies on the notion of ∂_{ψ}-shift invariance of operators with ψ-derivatives ∂_{ψ} staying for equivalence classes representatives of special differential operators lowering degree of polynomials by one [6, 7, 11]. Many of the results of ψ-calculus may be extended to Markowsky Q-umbral calculus [11] where Q stands for arbitrary generalized difference operator i.e. the one lowering the degree of any polynomial by one. Q-umbral calculus [11] - as we call it - includes also those generalized difference operators, which are not series in ψ-derivative ∂_{ψ} whatever an admissible ψ sequence would be.

The note is at the same time the operator formulation of “A Calculus of Sequences” started in 1936 by Ward [1] with the indication of the role the ψ-representations of Graves-Heisenberg-Weyl (GHW) algebra in formulation and derivation of principal statements of the ψ-extension of finite operator calculus of Rota.

Restating what was said above we observe that all statements of standard finite operator calculus of Rota are valid also in the case of ψ-extension under the almost automatic replacement of {D, x̂, id} generators of GHW by their ψ-representation correspondents {∂_{ψ}, x̂_{ψ}, id} - see definitions 2.1 and 2.5. Naturally any specification of admissible ψ - for example the famous one defining q-calculus - has its own characteristic properties not pertaining to the standard case of Rota calculus realisation. Nevertheless the overall picture and system of statements depending only on GHW algebra is the same modulo some automatic replacements in formulas demonstrated in the sequel. The large part of that kind of job was already done in [2, 3].

The aim of this presentation is to give a general picture of the algebra of linear operators on polynomial algebra. The picture that emerges discloses the fact that any ψ-representation of finite operator calculus or equivalently - any ψ-representation of GHW algebra makes up an example of the algebraization of the analysis - naturally when constrained to the algebra of polynomials.

We shall delimit all our considerations to the algebra P of polynomials or sometimes to the algebra of formal series. Therefore the distinction in-between difference and differentiation operators disappears. All linear operators on P are both difference and differentiation operators if the degree of differentiation or difference operator is unlimited.

If all this is extended to Markowsky Q-umbral calculus then many of the results of ψcalculus may be extended to Q-umbral calculus [11]. This is achieved under the almost automatic replacement of {D, x̂, id} generators of GHW or their ψ-representation {∂_{ψ}, x̂_{ψ}, id} by their Q-representation correspondents {Q, x̂_{Q}, id} - see definition 2.5.

The article is supplemented by the short indicatory glossaries of terms and notation used by Ward, Viskov, Markowsky, Roman on one side and the Rota-oriented notation on the other side.

2 Primary definitions, notation and general observations

In the following we shall consider the algebra P of polynomials P =F[x] over the field F of characteristic zero. All operators or functionals studied here are to be understood as linear operators on P. It shall be easy to see that they are always well defined.

Throughout the note while saying “polynomial sequence {p_{n}}_{o}^{∞}” we mean deg p_{n} = n; n ≥ 0 and we adopt also the convention that deg p_{n} < 0 iff p ≡ 0.

Consider ℑ - the family of functions‘ sequences (in conformity with Viskov notation ) such that: ℑ = {ψ; R ⊃ [a, b] ; q ∈ [a, b] ; ψ (q) : Z → F ; ψ_{0} (q) = 1 ; ψ_{n} (q) ≠ 0; ψ_{−n} (q) = 0; n ∈ N}. We shall call ψ = {ψ_{n} (q)}_{n≥0} ; ψ_{n} (q) ≠ 0; n ≥ 0 and ψ_{0} (q) = 1 an admissible sequence. Let now n_{ψ} denotes [2, 3]

n_{ψ} ≡ ψ_{n−1} (q) ψ_{n}^{−1} (q) .

Then

n_{ψ}! ≡ ψ_{n}^{−1} (q) ≡ n_{ψ} (n − 1)_{ψ} (n − 2)_{ψ} (n − 3)_{ψ} ....2_{ψ}1_{ψ}; 0_{ψ}! = 1

n\underline{k} ; ψ = n_{ψ} (n − 1)_{ψ} ... (n − k + 1)_{ψ} and ( \begin{matrix}n \\ k \end{matrix} )_{ψ} ≡ \frac{n\underline{k} ; ψ}{k_{ψ}!} and exp_{ψ}\{y\} = ∑_{k=0}^{∞} \frac{y^{k}}{k_{ψ}!}.

Definition 2.1. Let ψ be admissible. Let ∂_{ψ} be the linear operator lowering degree of polynomials by one defined according to ∂_{ψ}x^{n} = n_{ψ}x^{n−1} ; n ≥ 0. Then ∂_{ψ} is called the ψderivative.

Remark 2.1. The choice ψ_{n} (q)=[R (q^{n})!]^{−1} and R (x) = \frac{1−x}{1−q} results in the well known qfactorial n_{q}! = n_{q} (n − 1)_{q}!; 1_{q}! = 0_{q}! = 1 while the ψ-derivative ∂_{ψ} becomes now (n_{ψ} = n_{q}) the Jackson’s derivative [2, 3] ∂_{q}:

(∂_{q}φ) (x) = \frac{φ(x)−φ(qx)}{(1−q)x}.

Note also that that if ψ = {ψ_{n} (q)}_{n≥0} and φ = {φ_{n} (q)}_{n≥0} are two admissible sequences then [∂_{ψ} , ∂_{φ}]= 0 iff ψ = φ.

Definition 2.2. Let E^{y} (∂_{ψ}) ≡ exp_{ψ}\{y∂_{ψ}\} = ∑_{k=0}^{∞} \frac{y^{k}∂_{ψ}^{k}}{k_{ψ}!} . E^{y} (∂_{ψ}) is called the generalized

translation operator.

Note 2.1. [2, 3]

E^{a} (∂_{ψ}) f(x) ≡ f(x +_{ψ} a) ; (x +_{ψ} a)^{n} ≡ E^{a} (∂_{ψ}) x^{n} ; E^{a} (∂_{ψ}) f = ∑_{n≥0} \frac{a^{n}}{n_{ψ}!}∂_{ψ}^{n}f;

and in general (x +_{ψ} a)^{n} ≠ (x +_{ψ} a)^{n−1}(x +_{ψ} a). Note also that in general (1 +_{ψ} (−1))^{2n+1} ≠ 0 ; n ≥ 0 though (1 +_{ψ} (−1))^{2n} = 0 ; n ≥ 1.

Note 2.2. [1] exp_{ψ} (x +_{ψ} y) ≡ exp_{ψ}{x} exp_{ψ}{y} - while in general exp_{ψ}{x + y} ≠ exp_{ψ}{x} exp_{ψ}{y}.

Possible consequent utilisation of the identity exp_{ψ} (x +_{ψ} y) ≡ exp_{ψ}{x} exp_{ψ}{y} is quite encouraging. It leads among others to “ψ-trigonometry” either ψ-elliptic or ψ-hyperbolic via introducing cos_{ψ}, sin_{ψ} [1], cosh_{ψ} , sinh_{ψ} or in general ψ-hyperbolic functions of m-th order {h_{j}^{(ψ)} (α)}_{j∈Z_{m}}defined according to [12]

R ∋ α → h_{j} (α) = \frac{1}{m} ∑_{k∈Z_{m}} ω^{−kj} exp_{ψ} (ω^{k}α) ; j ∈ Z_{m}, ω = exp (i\frac{2π}{m} ) .

where 1 < m ∈ N and Z_{m} = {0, 1, ..., m − 1}.

Definition 2.3. A polynomial sequence {p_{n}}_{o}^{∞} is of ψ-binomial type if it satisfies the recurrence

E^{y} (∂_{ψ}) p_{n} (x) ≡ p_{n} (x +_{ψ} y) ≡ ∑_{k≥0} ( \begin{matrix}n \\ k \end{matrix} )_{ψ}p_{k} (x) p_{n−k} (y) .

Polynomial sequences of ψ-binomial type [2, 3] are known to correspond in one-to-one manner to special generalized differential operators Q, namely to those Q = Q (∂_{ψ}) which are ∂_{ψ}-shift invariant operators [2, 3]. We shall deal in this note mostly with this special case i.e. with ψ-umbral calculus. However before to proceed let us supply a basic information referring to this general case of Q-umbral calculus.

Definition 2.4. Let P = F[x]. Let Q be a linear map Q : P → P such that: ∀_{p∈P} deg (Qp) = (deg p) − 1 (with the convention deg p = −1 means p = const = 0). Q is then called a generalized difference-tial operator [11] or Gel‘fond-Leontiev [7] operator.

Right from the above definitions we infer that the following holds.

Observation 2.1. Let Q be as in Definition 2.4. Let Qx^{n} = ∑_{k=1}^{n} b_{n,k}x^{n−k} where b_{n,1} ≠ 0

of course. Without loose of generality take b_{1,1} = 1. Then ∃ {q_{k}}_{q≥2} ⊂ F and ∃ admissible ψ such that

Q = ∂_{ψ} + ∑_{k≥2} q_{k}∂_{ψ}^{k}  (2.1)

if and only if

b_{n,k} = ( \begin{matrix}n \\ k \end{matrix} )_{ψ} b_{k,k}; n ≥ k ≥ 1; b_{n,1} ≠ 0; b_{1,1} = 1.  (2.2)

If {q_{k}}_{q≥2} and an admissible ψ exist then these are unique.

Notation 2.1. In the case (2.2) is true we shall write : Q = Q (∂_{ψ}).

Remark 2.2. Note that operators of the (2.1) form constitute a group under superposition of formal power series (compare with the formula (S) in [13]). Of course not all generalized difference-tial operators satisfy (2.1) i.e. are series just only in corresponding ψ-derivative ∂_{ψ} (see Proposition 3.1 ). For example [14] let Q = \frac{1}{2}Dx̂D − \frac{1}{3}D^{3}. Then Qx^{n} = \frac{1}{2}n^{2}x^{n−1} − \frac{1}{3}n\underline{3}x^{n−3} so according to Observation 2.1 n_{ψ} = \frac{1}{2}n^{2} and there exists no admissible ψ such that Q = Q (∂_{ψ}).

Observation 2.2. From theorem 3.1 in [11] we infer that generalized differential operators give rise to subalgebras ∑_{Q} of linear maps (plus zero map of course) commuting with a given generalized difference-tial operator Q. The intersection of two different algebras ∑_{Q_{1}} and ∑_{Q_{2}} is just zero map added.

The importance of the above Observation 2.2 as well as the definition below may be further fully appreciated in the context of the Theorem 2.1 and the Proposition 3.1 to come.

Definition 2.5. Let {p_{n}}_{n≥0} be the normal polynomial sequence [11] i.e. p_{0} (x) = 1 and p_{n} (0) = 0 ; n ≥ 1. Then we call it the ψ-basic sequence of the generalized difference-tial operator Q if in addition Q p_{n} = n_{ψ}p_{n−1}. Parallely we define a linear map x̂_{Q}: P → P such that x̂_{Q}p_{n} = \begin{matrix}( & n+1 & ) \\ (n+1) & _{ψ} \end{matrix} p_{n+1}; n ≥ 0. We call the operator x̂_{Q} the dual to Q operator.

When Q = Q (∂_{ψ}) = ∂_{ψ} we write for short: x̂_{Q(∂_{ψ})} ≡ x̂_{∂_{ψ}} ≡ x̂_{ψ} (see: Definition 2.9).

Of course [Q, x̂_{Q}]= id therefore {Q, x̂_{Q}, id} provide us with a continuous family of generators of GHW in - as we call it - Q-representation of Graves-Heisenberg-Weyl algebra.

In the following we shall restrict to special case of generalized differential operators Q, namely to those Q = Q (∂_{ψ}) which are ∂_{ψ}-shift invariant operators [2, 3] (see: Definition 2.6).

At first let us start with appropriate ψ-Leibnitz rules for corresponding ψ-derivatives. ψ-Leibnitz rules:

It is easy to see that the following hold for any formal series f and g: for ∂_{q}: ∂_{q} (f · g) = (∂_{q}f) · g + (Q̂f) · (∂_{q}g), where (Q̂f) (x) = f (qx);

for ∂_{R} = R (qQ̂) ∂_{o}: ∂_{R}(f • g)(z) = R (qQ̂) {(∂_{o}f)(z) • g(z) + f(0)(∂_{o}g)(z)}

where - note - R (qQ̂) x^{n−1} = n_{R}x^{n−1} ; (n_{ψ} = n_{R} = n_{R(q)} = R (q^{n})) and finally for ∂_{ψ} = n̂_{ψ}∂_{o}:

∂_{ψ}(f • g)(z) = n̂_{ψ}{(∂_{o}f)(z) • g(z) + f(0)(∂_{o}g)(z)}

where n̂_{ψ}x^{n−1} = n_{ψ}x^{n−1} ; n ≥ 1.

Example 2.1. Let Q (∂_{ψ}) = Dx̂D, where x̂f(x) = xf(x) and D = \frac{d}{dx}. Then ψ = {[(n^{2})!]^{−1}}_{n≥0} and Q = ∂_{ψ}. Let Q (∂_{ψ}) R(qQ̂) ∂_{0} ≡ ∂_{R}. Then ψ = {[R (q^{n})!]^{−1}}_{n≥0}

and Q = ∂_{ψ} ≡ ∂_{R}. Here R(z) is any formal Laurent series; Q̂f(x) = f(qx) and n_{ψ} = R(q^{n}). ∂_{0} is q = 0 Jackson derivative which as a matter of fact - being a difference operator is the differential operator of infinite order at the same time:

∂_{o} = ∑_{n=1}^{∞} ( −1)^{n+1} \frac{x^{n−1}}{n!} \frac{d^{n}}{dx^{n}} .  (2.3)

Naturally with the choice ψ_{n} (q) = [R (q^{n})!]^{−1} and R (x) = \frac{1−x}{1−q} the ψ-derivative ∂_{ψ} becomes the Jackson’s derivative [2, 3] ∂_{q}:

(∂_{q}φ) (x) = \frac{1 − qQ̂}{(1 − q)}∂_{0}φ (x) .

The equivalent to (2.3) form of Bernoulli-Taylor expansion one may find [15] in Acta Eruditorum from November 1694 under the name “series univeralissima”.

(Taylor‘s expansion was presented in his “Methodus incrementorum directa et inversa” in 1715 - edited in London).

Definition 2.6. Let us denote by End(P) the algebra of all linear operators acting on the algebra P of polynomials. Let

∑_{ψ} = {T ∈ End(P); ∀ α ∈ F; [T; E^{α} (∂_{ψ})] = 0}.

Then ∑_{ψ} is a commutative subalgebra of End(P) of F -linear operators. We shall call these operators T : ∂_{ψ}-shift invariant operators.

We are now in a position to define further basic objects of “ψ-umbral calculus” [2, 3].

Definition 2.7. Let Q (∂_{ψ}) : P → P; the linear operator Q (∂_{ψ}) is a ∂_{ψ}-delta operator iff

1. Q (∂_{ψ}) is ∂_{ψ} - shift invariant;

2. Q (∂_{ψ}) (id) = const ≠ 0

The strictly related notion is that of the ∂_{ψ}-basic polynomial sequence:

Definition 2.8. Let Q (∂_{ψ}) : P → P; be the ∂_{ψ}-delta operator. A polynomial sequence {p_{n}}_{n≥0}; deg p_{n}= n such that:

1. p_{o} (x) = 1;

2. p_{n} (0) = 0; n > 0;

3. Q (∂_{ψ}) p_{n} = n_{ψ}p_{n−1} is called the ∂_{ψ}-basic polynomial sequence of the ∂_{ψ}-delta operatorQ (∂ψ).

Identification 2.1. It is easy to see that the following identification takes place: ∂_{ψ}-delta operator Q (∂_{ψ}) = ∂_{ψ}-shift invariant generalized differential operator Q. Of course not every generalized differential operator might be considered to be such.

Note: Let Φ (x; λ) = ∑_{n≥0} \frac{λ^{n}}{n_{ψ}!}p_{n} (x) denotes the ψ-exponential generating function of

the ∂_{ψ}-basic polynomial sequence {p_{n}}_{n≥0} of the ∂_{ψ}-delta operator Q ≡ Q (∂_{ψ}) and let Φ (0; λ) = 1. Then QΦ (x; λ) = λΦ (x; λ) and Φ is the unique solution of this eigenvalue problem. In view of Observation 2.2 we affirm that then exists such an admissible sequence φ that Φ (x; λ) = exp_{φ} [λx].

The notation and naming established by Definitions 2.7 and 2.8 serve the target to preserve and to broaden simplicity of Rota‘s finite operator calculus also in its extended “ψumbral calculus” case [2, 3]. As a matter of illustration of such notation efficiency let us quote after [2] the important Theorem 2.1 which might be proved using the fact that ∀ Q (∂_{ψ}) ∃! invertible S ∈ Σ_{ψ} such that Q (∂_{ψ}) = ∂_{ψ}S. ( For Theorem 2.1 see also Theorem 4.3. in [11], which holds for operators, introduced by the Definition 2.5).

Theorem 2.1. (ψ-Lagrange and ψ-Rodrigues formulas) Let {p_{n} (x)}_{n=0}^{∞} be ∂_{ψ}-basic polynomial sequence of the ∂_{ψ}-delta operator Q (∂_{ψ}). Let Q (∂_{ψ}) = ∂_{ψ}S_{∂_{ψ}}. Then for n > 0:

1. p_{n}(x) = Q (∂_{ψ})’ S_{∂_{ψ}}^{−n−1} x^{n} ;

2. p_{n}(x) = S_{∂_{ψ}}^{−n}x^{n} − \frac{n_{ψ}}{n} (S_{∂_{ψ}}^{−n} )’x^{n−1};

3. p_{n}(x) = \frac{n_{ψ}}{n} x̂_{ψ}S_{∂_{ψ}}^{−n}x^{n−1};

4. p_{n}(x) = \frac{n_{ψ}}{n} x̂_{ψ}(Q (∂_{ψ})’ )^{−1}p_{n−1}(x) (← Rodrigues ψ-formula ).

For the proof one uses typical properties of the Pincherle ψ-derivative defined bellow as well as x̂_{ψ} operator.

Definition 2.9. (compare with (17) in [7] ) The Pincherle ψ-derivative i.e. the linear map ’ : Σ_{ψ} → Σ_{ψ};

T ’ = T x̂_{ψ} - x̂_{ψ}T ≡[T_{∂_{ψ}} , x̂_{ψ}]

where the linear map x̂_{ψ} : P → P; is defined in the basis {x^{n}}_{n≥0} as follows

x̂_{ψ}x^{n} = \frac{ψ_{n+1} (q) (n + 1)}{ψ_{n} (q)}x^{n+1} = \begin{matrix}( & n + 1 & ) \\ (n + 1) & ψ \end{matrix} x^{n+1}; n ≥ 0

Observation 2.3. [2,3] The triples {∂_{ψ}, x̂_{ψ}, id} for any admissible ψ-constitute the set of generators of the ψ-labelled representations of Graves-Heisenberg-Weyl (GHW) algebra [16, 17, 18]. Namely, as easily seen [∂_{ψ}, x̂_{ψ}] = id. (compare with Definition 2.5)

Observation 2.4. In view of the Observation 2.3 the general Leibnitz rule in ψ-representation of Graves-Heisenberg-Weyl algebra may be written (compare with 2.2.2 Proposition in [17]) as follows

∂_{ψ}^{n} x̂_{ψ}^{m} = ∑_{k≥0} ( \begin{matrix}n \\ k \end{matrix} ) ( \begin{matrix}m \\ k \end{matrix} ) k! x̂_{ψ}^{m−k} ∂_{ψ}^{n−k}.  (2.4)

One derives the above ψ-Leibnitz rule from ψ-Heisenberg-Weyl exponential commutation rules exactly the same way as in {D, x̂, id} GHW representation - (compare with 2.2.1 Proposition in [17] ). ψ-Heisenberg-Weyl exponential commutation relations read:

exp{t∂_{ψ}}exp{ax̂_{ψ}} = exp{at}exp{ax̂_{ψ}}exp{t∂_{ψ}}.  (2.5)

To this end let us introduce a pertinent ψ-multiplication ∗_{ψ} of functions as specified below.

Notation 2.2.

x ∗_{ψ} x^{n} = x̂_{ψ}(x^{n}) = \begin{matrix}( & n+1 & ) \\ (n+1) & _{ψ} \end{matrix} x^{n+1}; n ≥ 0 hence x ∗_{ψ}1 = 1_{ψ} x ≢ x

x^{n}∗_{ψ} x = x̂_{ψ}^{n}( x ) = \frac{(n+1)!}{(n+1)_{ψ}!}x^{n+1}; n ≥ 0 hence 1∗_{ψ} x = 1_{ψ} x ≢ x therefore x ∗_{ψ}α1 = α1∗_{ψ} x = x ∗_{ψ}α = α∗_{ψ} x = α1_{ψ} x and ∀ x,α ∈ ; f(x)∗_{ψ} x^{n} = f(x̂_{ψ})x^{n}.

For k ≠ n x^{n}∗_{ψ} x^{k} ≠ x^{k}∗_{ψ} x^{n} as well as x^{n}∗_{ψ} x^{k} ≠ x^{n+k} - in general i.e. for arbitrary admissible ψ; compare this with (x +_{ψ}a)^{n} ≠ ( x +_{ψ}a)^{n−1}(x +_{ψ}a). In order to facilitate in the future formulation of observations accounted for on the basis of ψ-calculus representation of GHW algebra we shall use what follows.

Definition 2.10. With Notation 2.2 adopted let us define the ∗_{ψ} powers of x according to x^{n∗_{ψ}} ≡ x ∗_{ψ}x^{(n−1)∗_{ψ}} = x̂_{ψ}(x^{(n−1)∗_{ψ}}) = x ∗_{ψ} x ∗_{ψ} ... ∗_{ψ} x = \frac{n!}{n_{ψ}!}x^{n}; n ≥ 0.

Note that x^{n∗_{ψ}} ∗_{ψ} x^{k∗_{ψ}} = \frac{n!}{n_{ψ}!}x^{(n+k)∗_{ψ}} ≠ x^{k∗_{ψ}} ∗_{ψ} x^{n∗_{ψ}} = \frac{k!}{k_{ψ}!}x^{(n+k)∗_{ψ}} for k ≠ n and x^{0∗_{ψ}} = 1. This noncommutative ψ-product ∗_{ψ} is deviced so as to ensure the following observations:

Observation 2.5.

1. ∂_{ψ}x^{n∗_{ψ}} = nx^{(n−1)∗_{ψ}}; n ≥ 0

2. exp_{ψ}[αx] ≡exp {αx̂_{ψ}}1

3. exp [αx] ∗_{ψ} exp_{ψ}{βx̂_{ψ}}=exp_{ψ}{[α + β]x̂_{ψ}}

4. ∂_{ψ}(x^{k} ∗_{ψ} x^{n∗_{ψ}}) = (Dx^{k}) ∗_{ψ} x^{n∗_{ψ}} + x^{k} ∗_{ψ} (∂_{ψ}x^{n∗_{ψ}}) hence

5. ∂_{ψ}(f ∗_{ψ} g) = (Df) ∗_{ψ} g + f ∗_{ψ} (∂_{ψ}g) ; f, g - formal series 6. f(x̂_{ψ})g(x̂_{ψ}) 1 = f(x) ∗_{ψ} g̃(x) ; g̃(x) = g(x̂_{ψ})1.

Now the consequences of Leibnitz rule (e) for difference-ization of the product are easily feasible. For example the ψ-Poisson process distribution p_{m}(x);

∑_{m≥0} p_{m}(x) = 1;

p_{m}(x) = \frac{(λx)^{m}}{m!} ∗_{ψ} exp_{ψ}[−λx]  (2.6)

is the unique solution of its corresponding ∂_{ψ}-difference equation

∂_{ψ}p_{m}(x) + λp_{m}(x) = λp_{m−1}(x)m > 0 ; ∂_{ψ}p_{0}(x) = −λp_{0}(x)  (2.7)

As announced - the rules of ψ -product ∗_{ψ} are accounted for - as a matter of fact - on the basis of ψ-calculus representation of GHW algebra. Indeed; it is enough to consult Observation 2.5 and to introduce ψ-Pincherle derivation ˆ∂_{ψ} of series in powers of the symbol x̂_{ψ} as below. Then the correspondence between generic relative formulas turns out evident.

Observation 2.6. Let ˆ∂_{ψ} ≡ \frac{∂}{∂x̂_{ψ}} be defined according to ˆ∂_{ψ}f(x̂_{ψ}) = [∂_{ψ}, f(x̂_{ψ})]. Then ˆ∂_{ψ}x̂_{ψ}^{n} = nx̂_{ψ}^{n−1} ; n ≥ 0 and ˆ∂_{ψ}x̂_{ψ}^{n} 1 = ∂_{ψ}x^{n∗_{ψ}} hence [ˆ∂_{ψ}f(x̂_{ψ})]1 = ∂_{ψ}f(x) where f is a formal series in powers of x̂_{ψ} or equivalently in ∗_{ψ} powers of x.

As an example of application note how the solution of ?? is obtained from the obvious solution _{m}(x̂_{ψ}) of the ˆ∂_{ψ}-Pincherle differential equation ?? formulated within G-H-W algebra generated by {∂_{ψ}, x̂_{ψ}, id}

ˆ∂_{ψ}_{m}(x̂_{ψ}) + λ_{m}(x̂_{ψ}) = λ_{m−1}(x̂_{ψ}) m > 0 ; ∂_{ψ}_{0}(x̂_{ψ}) = −λ_{0}(x̂_{ψ})  (2.8)

Namely : due to Observation 2.5 (f) p_{m}(x) = _{m}(x̂_{ψ})1, where

m(x̂_{ψ}) = \frac{(λx̂_{ψ})^{m}}{m!} exp_{ψ}[−λx̂_{ψ}].  (2.9)

3 The general picture

The general picture from the title above relates to the general picture of the algebra End(P) of operators on P as in the following we shall consider the algebra P of polynomials P = F[x] over the field F of characteristic zero.

We shall draw an over view picture of the situation distinguished by possibility to develop umbral calculus for any polynomial sequences {p_{n}}_{o}^{∞} instead of those of traditional binomial type only.

In 1901 it was proved [19] that every linear operator mapping P into P may be represented as infinite series in operators x̂ and D. In 1986 the authors of [20] supplied the explicit expression for such series in most general case of polynomials in one variable ( for many variables see: [21] ). Thus according to Proposition 1 from [20] one has:

Proposition 3.1. Let Q be a linear operator that reduces by one each polynomial. Let \{q_{n} \binom{}{x̂}\}_{n≥0} be an arbitrary sequence of polynomials in the operator x̂. Then T̂ = ∑_{n≥0} q_{n} \binom{}{x̂}Q^{n}

defines a linear operator that maps polynomials into polynomials. Conversely, if T̂ is linear operator that maps polynomials into polynomials then there exists a unique expansion of the form

T̂ = ∑_{n≥0} q_{n} \binom{}{x̂}Q^{n}.

It is also a rather matter of an easy exercise to prove the Proposition 2 from [20]:

Proposition 3.2. “ Let Q be a linear operator that reduces by one each polynomial. Let \{q_{n} \binom{}{x̂}\}_{n≥0}be an arbitrary sequence of polynomials in the operator x̂. Let a linear operator that maps polynomials into polynomials be given by

T̂ = ∑_{n≥0} q_{n} \binom{}{x̂}Q^{n}.

Let P (x; λ) = ∑_{n≥0} q_{n} (x)λ^{n} denotes indicator of T̂. Then there exists a unique formal

series Φ (x; λ); Φ (0; λ) = 1such that:

QΦ (x; λ) = λΦ (x; λ) .

Then also P (x; λ) = Φ (x; λ)^{−1} T̂Φ (x; λ).

Example 3.1. Note that ∂_{ψ}exp_{ψ}{λx} = λexp_{ψ}{λx}; exp_{ψ} [x] |_{x=0} = 1. (*) Hence for indicator of T̂; T̂ = ∑_{n≥0} q_{n} \binom{}{x̂}∂_{ψ}^{n} we have:

P (x; λ) = [exp_{ψ}{λx}]^{−1}T̂exp_{ψ}{λx}. (**)

After choosing ψ_{n} (q) = [n_{q}!]^{−1} we get exp_{ψ}{x} =exp_{q}{x}. In this connection note that exp_{o} (x) = \frac{1}{1−x} and exp(x) are mutual limit deformations for |x| < 1 due to:

\frac{exp_{o}(z)−1}{z} = exp_{o} (z) ⇒ exp_{o} (z) = \frac{1}{1−z} = ∑_{k=0}^{∞} z^{k}; |z| < 1 i.e.

exp (x) ⟵_{1←q} exp_{q} (x) = ∑_{n=0}^{∞} \frac{x^{n}}{n_{q}!} ⟶_{q→0} \frac{1}{1 − x}.

Therefore corresponding specifications of (*) such as exp_{o} (λx) = \frac{1}{1−λx} or exp(λx) lead to corresponding specifications of (**) for divided difference operator ∂_{0} and D operator including special cases from [20].

To be complete let us still introduce [2, 3] an important operator x̂_{Q(∂_{ψ})} dual to Q (∂_{ψ}).

Definition 3.1. (see Definition 2.5) Let {p_{n}}_{n≥0} be the ∂_{ψ}-basic polynomial sequence of the ∂_{ψ}-delta operator Q (∂_{ψ}). A linear map x̂_{Q(∂_{ψ})}: P → P ; x̂_{Q(∂_{ψ})} = \begin{matrix}( & n+1 & ) \\ (n+1) & _{ψ} \end{matrix} p_{n+1}; n ≥ 0 is called the operator dual to Q (∂_{ψ}).

Comment 3.1. Dual in the above sense corresponds to adjoint in ψ-umbral calculus language of linear functionals’ umbral algebra (compare with Proposition 1.1.21 in [22] ).

It is now obvious that the following holds.

Proposition 3.3. Let \{q_{n} \binom{}{x̂_{Q(∂_{ψ})}}\}_{n≥0} be an arbitrary sequence of polynomials in the

operator x̂_{Q(∂_{ψ})}. Then T = ∑_{n≥0} q_{n} \binom{}{x̂_{Q(∂_{ψ})}}Q (∂_{ψ})^{n} defines a linear operator that maps

polynomials into polynomials. Conversely, if T is linear operator that maps polynomials into polynomials then there exists a unique expansion of the form

T = ∑_{n≥0} q_{n} \binom{}{x̂_{Q(∂_{ψ})}}Q (∂_{ψ})^{n} .  (3.1)

Comment 3.2. The pair Q (∂_{ψ}) , x̂_{Q(∂_{ψ})} of dual operators is expected to play a role in the description of quantum-like processes apart from the q-case now vastly exploited [2, 3].

Naturally the Proposition 3.2 for Q (∂_{ψ}) and x̂_{Q(∂_{ψ})} dual operators is also valid. Summing up: we have the following picture for End(P) - the algebra of all linear operators acting on the algebra P of polynomials.

Q(P) ≡ ⋃_{Q} ∑_{Q} ⊂ End(P)

and of course Q(P) ≠ End(P) where the subfamily Q(P) (with zero map) breaks up into sum of subalgebras ∑_{Q} according to commutativity of these generalized difference-tial operators Q (see Definition 2.4 and Observation 2.2). Also to each subalgebra ∑_{ψ} i.e. to each Q (∂_{ψ}) operator there corresponds its dual operator x̂_{Q(∂_{ψ})}

x̂_{Q(∂_{ψ})} ∉ ∑_{ψ}

and both Q (∂_{ψ}) & x̂_{Q(∂_{ψ})} operators are sufficient to build up the whole algebra End(P) according to unique representation given by (3.1) including the ∂_{ψ} and x̂_{ψ} case. Summarising: for any admissible ψ we have the following general statement. General statement:

End(P) =[{∂_{ψ},x̂_{ψ}}] = [{Q (∂_{ψ}) , x̂_{Q(∂_{ψ})}}] = [{Q , x̂_{Q}}]

i.e. the algebra End(P) is generated by any dual pair {Q , x̂_{Q}} including any dual pair {Q (∂_{ψ}) , x̂_{Q(∂_{ψ})}} or specifically by {∂_{ψ},x̂_{ψ}} which in turn is determined by a choice of any admissible sequence ψ.

As a matter of fact and in another words: we have bijective correspondences between different commutation classes of ∂_{ψ}-shift invariant operators from End(P), different abelian subalgebras ∑_{ψ}, distinct ψ-representations of GHW algebra, different ψ-representations of the reduced incidence algebra R(L(S)) - isomorphic to the algebra Φ_{ψ} of ψ-exponential formal power series [2] and finally - distinct ψ-umbral calculi [7, 11, 14, 23, 2]. These bijective correspondences may be naturally extended to encompass also Q-umbral calculi, Q-representations of GHW algebra and abelian subalgebras ∑_{Q}. (Recall: R(L(S)) is the reduced incidence algebra of L(S) where L(S)={A; A⊂S; |A| < ∞}; S is countable and (L(S); ⊆) is partially ordered set ordered by inclusion [10, 2] ).

This is the way the Rota‘s devise has been carried into effect. The devise “much is the iteration of the few” [10] - much of the properties of literally all polynomial sequences - as well as GHW algebra representations - is the application of few basic principles of the ψ-umbral difference operator calculus [2]. ψ− Integration Remark : Recall : ∂_{o}x^{n} = x^{n−1}. ∂_{o} is identical with divided difference operator. ∂_{o} is identical with ∂_{ψ} for ψ = {ψ (q)_{n}}_{n≥0} ; ψ (q)_{n} = 1 ; n ≥ 0 . Let Q̂f(x)f(qx). Recall also that there corresponds to the “∂_{q} difference-ization” the q-integration [24, 25, 26] which is a right inverse operation to “q-difference-ization”. Namely

F (z) :≡ (∫_{q} φ) (z) := (1 − q) z ∑_{k=0}^{∞} φ (q^{k}z) q^{k}  (3.2)

i.e.

F (z) ≡ (∫_{q} φ) (z) = (1 − q) z (∑_{k=0}^{∞} q^{k}Q̂^{k}φ) (z) = ((1 − q) z \frac{1}{1 − qQ̂}φ) (z) .  (3.3)

Of course

∂_{q} ∘ ∫_{q} = id  (3.4)

as

\frac{1 − qQ̂}{(1 − q)}∂_{0} ((1 − q) ẑ\frac{1}{1 − qQ̂}) = id.  (3.5)

Naturally (3.5) might serve to define a right inverse operation to “q-difference-ization”

(∂_{q}φ) (x) = \frac{1−qQ̂}{(1−q)}∂_{0}φ (x)

and consequently the “q-integration “ as represented by (3.2) and (3.3). As it is well known the definite q-integral is an numerical approximation of the definite integral obtained in the q → 1 limit. Following the q-case example we introduce now an R-integration (consult Remark 2.1).

∫_{R} x^{n} = (x̂\frac{1}{R (qQ̂)}) x^{n} = \frac{1}{R (q^{n+1})}x^{n+1}; n ≥ 0  (3.6)

Of course ∂_{R} ∘ ∫_{R} = id as

R (qQ̂) ∂_{o} (x̂\frac{1}{R (qQ̂)}) = id.  (3.7)

Let us then finally introduce the analogous representation for ∂_{ψ} difference-ization

∂_{ψ} = n̂_{ψ}∂_{o}; n̂_{ψ}x^{n−1} = n_{ψ}x^{n−1}; n ≥ 1.  (3.8)

Then

∫_{ψ} x^{n} = (x̂\frac{1}{n̂_{ψ}} ) x^{n} = \frac{1}{(n + 1)_{ψ}} x^{n+1}; n ≥ 0  (3.9)

and of course

∂_{ψ} ∘ ∫_{ψ} = id  (3.10)

Closing Remark:

The picture that emerges discloses the fact that any ψ-representation of finite operator calculus or equivalently - any ψ-representation of GHW algebra makes up an example of the algebraization of the analysis - naturally when constrained to the algebra of polynomials. We did restricted all our considerations to the algebra P of polynomials. Therefore the distinction in-between difference and differentiation operators disappears. All linear operators on P are both difference and differentiation operators if the degree of differentiation or difference operator is unlimited. For example \frac{d}{dx} = ∑_{k≥1} \frac{d_{k}}{k!} Δ^{k} where d_{k} = [\frac{d}{dx}x\underline{k}]_{x=0} = ( −1)^{k−1} (k − 1)!

or Δ = ∑_{n≥1} \frac{δ_{n}}{n!} \frac{d^{n}}{dx^{n}} where δ_{n} = [Δx^{n}]_{x=0} = 1. Thus the difference and differential operators

and equations are treated on the same footing.

An interesting task (which seems to be still ahead) is to investigate the Q representation of finite operator calculus as an example of the algebraization of the analysis - naturally when constrained to the algebra of polynomials.

4 Glossary

Here now come short indicatory glossaries of terms and notation used by Ward [1], Viskov [6, 7], Markowsky [11], Roman [27]- [31] on one side and the Rota-oriented notation on the other side.

Ward  Rota - oriented (this note)

[n] ; [n]!  n_{ψ}; n_{ψ}!
basic binomial coefficient [n, r] = \frac{[n]!}{[r]![n−r]!} ψ-binomial coefficient \binom{n}{k}_{ψ} ≡ \frac{n\underline{k} ; ψ}{k_{ψ}!}

D = D_{x} - the operator D  ∂_{ψ} - the ψ-derivative
D x^{n} = [n] x^{n−1}  ∂_{ψ} x^{n} = n_{ψ} x^{n−1}

(x + y)^{n}  (x +_{ψ} y)^{n}
(x + y)^{n} ≡ ∑_{r=0}^{n} [n, r] x^{n−r}y^{r}  (x +_{ψ} y)^{n} = ∑_{k=0}^{n} \binom{n}{k}_{ψ}x^{k}y^{n−k}

Ward  Rota - oriented (this note)

basic displacement symbol  generalized shift operator
E^{t}; t ∈Z  E^{y} (∂_{ψ}) ≡ exp_{ψ}{y∂_{ψ}}; y ∈F
Eφ(x) = φ(x + 1)  E(∂_{ψ})φ(x) = φ(x +_{ψ} 1)
E^{t}φ(x) = φ (x + \overline{t})  E^{y}(∂_{ψ})x^{n} ≡ (x +_{ψ} y)^{n}

basic difference operator  ψ-difference delta operator
Δ = E − id  Δ_{ψ} = E^{y}(∂_{ψ}) − id

Δ = ε(D) − id = ∑_{n=0}^{∞} \frac{D^{n}}{[n]!} − id

Roman  Rota - oriented (this note)

t; tx^{n} = nx^{n−1}  ∂_{ψ} - the ψ-derivative
∂_{ψ}x^{n} = n_{ψ}x^{n−1}
⟨t^{k}|p(x)⟩ = p^{(k)}(0)  [∂_{ψ}^{k}p(x)]|_{x=0}

evaluation functional  generalized shift operator
ϵ_{y}(t) = exp {yt}  E^{y}(∂_{ψ}) = exp_{ψ} {y∂_{ψ}}

⟨t^{k}|x^{n}⟩ = n!δ_{n,k}

⟨ϵ_{y}(t)|p(x)⟩ = p(y)  [E^{y}(∂_{ψ})p_{n}(x)]|_{x=0} = p_{n}(y)
ϵ_{y}(t)x^{n} = ∑_{k≥0} \binom{n}{k}x^{k}y^{n−k}  E^{y}(∂_{ψ})p_{n}(x) = ∑_{k≥0} \binom{n}{k}_{ψ}p_{k}(x)p_{n−k}(y)

Roman  Rota - oriented (this note)

formal derivative  Pincherle derivative
f′(t) ≡ \frac{d}{dt}f(t)  [Q(∂_{ψ})]‘≡ \frac{d}{d∂_{ψ}} Q(∂_{ψ})
\overline{f}(t) compositional inverse of  Q^{−1}(∂_{ψ}) compositional inverse of
formal power series f(t)  formal power series Q(∂_{ψ})

θ_{t}; θ_{t}x^{n} = x^{n+1}; n ≥ 0  x̂_{ψ}; x̂_{ψ}x^{n} = \frac{n+1}{(n+1)_{ψ}} x^{n+1}; n ≥ 0
θ_{t}t = x̂D  x̂_{ψ}∂_{ψ} = x̂D = N̂

∑_{k≥0} \frac{s_{k}(x)}{k_{ψ}!}t^{k} =  ∑_{k≥0} \frac{s_{k}(x)}{k_{ψ}!}z^{k} =

[g(\overline{f}(z))]^{−1} exp \{x\overline{f}(t)\}  s(q^{−1}(z)) exp_{ψ} \{xq^{−1}(z)\}
{s_{n}(x)}_{n≥0} - Sheffer sequence  q(t), s(t) indicators
for (g(t), f(t))  of Q(∂_{ψ}) and S_{∂_{ψ}}

g(t) s_{n}(x) = q_{n}(x) - sequence  s_{n}(x) = S_{∂_{ψ}}^{−1} q_{n}(x) - ∂_{ψ} - basic
associated for f(t)  sequence of Q(∂_{ψ})

The expansion theorem:  The First Expansion Theorem
h(t) = ∑_{k=0}^{∞} \frac{⟨h(t)|p_{k}(x)⟩}{k!}f(t)^{k}  T_{∂_{ψ}} = ∑_{n≥0} \frac{[T_{∂_{ψ}} p_{n}(z)]|_{z=0}}{n_{ψ}} Q(∂_{ψ})^{n}

p_{n}(x) - sequence associated for f(t) ∂_{ψ} - basic polynomial sequence {p_{n}}_{0}^{∞}

exp\{y\overline{f}(t)\} = ∑_{k=0}^{∞} \frac{p_{k}(y)}{k!}t^{k}  exp_{ψ}\{xQ^{−1}(x)\} = ∑_{k≥0} \frac{p_{k}(y)}{k!}z^{k}

Roman  Rota - oriented (this note)

The Sheffer Identity:  The Sheffer ψ-Binomial Theorem:
s_{n}(x + y) = ∑_{k=0}^{n} \binom{n}{k}p_{n}(y)s_{n−k}(x)  s_{n}(x +_{ψ} y) = ∑_{k≥0} \binom{n}{k}_{ψ}s_{k}(x)q_{n−k}(y)

Viskov  Rota - oriented (this note)

θ_{ψ} - the ψ-derivative  ∂_{ψ} - the ψ-derivative
θ_{ψ} x^{n} = \frac{ψ_{n−1}}{ψ_{n}} x^{n−1}  ∂_{ψ} x^{n} = n_{ψ} x^{n−1}

A_{p} (p = {p_{n}}_{0}^{∞})  Q
A_{p} p_{n} = p_{n−1}  Q p_{n} = n_{ψ}p_{n−1}

B_{p} (p = {p_{n}}_{0}^{∞})  x̂_{Q}
B_{p} p_{n} = (n + 1) p_{n+1}  x̂_{Q} p_{n} = \frac{n+1}{(n+1)_{ψ}} p_{n+1}

E_{p}^{y} (p = {p_{n}}_{0}^{∞})  E^{y} (∂_{ψ}) = exp_{ψ}{y∂_{ψ}}
E_{p}^{y} p_{n}(x) = ∑_{k=0}^{n} p_{n−k}(x)p_{k}(y)  E^{y} (∂_{ψ}) p_{n}(x) =

= ∑_{k≥0} \binom{n}{k}_{ψ} p_{k}(x)p_{n−k}(y)

T − ε_{p}-operator:  E^{y} - shift operator:
T A_{p} = A_{p} T  E^{y}φ(x) = φ(x +_{ψ} y)

T - ∂_{ψ}-shift invariant operator:

∀_{y∈F} T E_{p}^{y} = E_{p}^{y}T  ∀_{α∈F} [T, E^{α}(∂_{ψ})] = 0

Viskov  Rota - oriented (this note)

Q - δ_{ψ}-operator:  Q(∂_{ψ}) - ∂_{ψ}-delta-operator:
Q - ϵ_{p}-operator and  Q(∂_{ψ}) - ∂_{ψ}-shift-invariant and
Qx = const ≠ 0  Q(∂_{ψ})(id) = const ≠ 0

{p_{n}(x), n ≥ 0} - (Q, ψ)-basic  {p_{n}}_{n≥0} -∂_{ψ}-basic
polynomial sequence of the  polynomial sequence of the
δ_{ψ}-operator Q  ∂_{ψ}-delta-operator Q(∂_{ψ})

ψ-binomiality property  ψ-binomiality property
Ψ_{y}s_{n}(x) =  E^{y}(∂_{ψ})p_{n}(x) =
= ∑_{m=0}^{n} \frac{ψ_{n}ψ_{n−m}}{ψ_{n}} s_{m}(x)p_{n−m}(y)  = ∑_{k≥0} \binom{n}{k}_{ψ}p_{k}(x)p_{n−k}(y)

T = ∑_{n≥0} ψ_{n}[V T p_{n}(x)]Q^{n}  T = ∑_{n≥0} \frac{[T p_{n}(z)]|_{z=0}}{n_{ψ}!} Q(∂_{ψ})^{n}

T Ψ_{y}p(x) =  T p(x +_{ψ} y) =
∑_{n≥0} ψ_{n}s_{n}(y)Q^{n}ST p(x)  ∑_{k≥0} \frac{s_{k}(y)}{k_{ψ}!}Q(∂_{ψ})^{k}S_{∂_{ψ}}T p(x)

Markowsky  Rota - oriented (this note)

L - the differential operator  Q
L p_{n} = p_{n−1}  Q p_{n} = n_{ψ}p_{n−1}

M  x̂_{Q}
M p_{n} = p_{n+1}  x̂_{Q} p_{n} = \frac{n+1}{(n+1)_{ψ}} p_{n+1}

Markowsky  Rota - oriented (this note)
L_{y}  F^{y} (Q) = ∑_{k≥0} \frac{p_{k}(y)}{k_{ψ}!}Q^{k}
L_{y} p_{n}(x) =  F^{y} (Q) p_{n}(x) =
= ∑_{k=0}^{n} \binom{n}{k}p_{k}(x)p_{n−k}(y)  = ∑_{k≥0} \binom{n}{k}_{ψ} p_{k}(x)p_{n−k}(y)
E^{a} - shift-operator:  E^{y} - ∂_{ψ}-shift operator:
E^{a} f(x) = f(x + a)  E^{y}φ(x) = φ(x +_{ψ} y)
G - shift-invariant operator: T - ∂_{ψ}-shift invariant operator:
EG = GE  ∀_{α∈F} [T, E^{α}(∂_{ψ})] = 0
G - delta-operator:  Q(∂_{ψ}) - ∂_{ψ}-delta-operator:
G - shift-invariant and  Q(∂_{ψ}) - ∂_{ψ}-shift-invariant and
Gx = const ≠ 0  Q(∂_{ψ})(id) = const ≠ 0
D_{L}(G)  G′ = [G(Q), x̂_{Q}]
L - Pincherle derivative of G  Q - Pincherle derivative

D_{L}(G) = [G, M]

{Q_{0}, Q_{1}, ...} - basic family  {p_{n}}_{n≥0} -ψ-basic
for differential operator L  polynomial sequence of the

generalized difference operator Q

Markowsky  Rota - oriented (this note)
binomiality property  Q - ψ-binomiali property
P_{n}(x + y) =  F^{y}(Q)p_{n}(x) =
= ∑_{i=0}^{n} \binom{n}{i}P_{i}(x)P_{n−1}(y)  = ∑_{k≥0} \binom{n}{k}_{ψ}p_{k}(x)p_{n−k}(y)

Acknowledgements

Discussions with Ewa Gra̧dzka are highly acknowledged. The author expresses his gratitude to her also for preparation of the LaTeX version of this contribution.

References

[1] M. Ward: Amer. J. Math. 58, 255 (1936).

[2] A. K. Kwaśniewski “Towards ψ-extension of Finite Operator Calculus of Rota” Rep. Biał ystok Univ. Inst. Comp. Sci. UwB/Preprint#05/July/2000; Rep. Math. Phys. in press

[3] A. K. Kwaśniewski “On extended finite operator calculus of Rota and quantum groups” Bia^{3}ystok Univ. Inst. Comp. Sci. UwB/Preprint#04/July/2000 Integral Transforms and Special Functions - in press

[4] R. P. Boas and Jr. R. C. Buck: Am. Math. Monthly 63, 626 (1959).

[5] R. P. Boas and Jr. R. C. Buck: Polynomial Expansions of Analytic Functions, Springer, Berlin 1964.

[6] O.V. Viskov: Soviet Math. Dokl. 16, 1521 (1975).

[7] O.V. Viskov: Soviet Math. Dokl. 19, 250 (1978).

[8] G.-C. Rota and R. Mullin: On the foundations of combinatorial theory, III. Theory of Binomial Enumeration in ”Graph Theory and Its Applications”, Academic Press, New York 1970.

[9] G. C. Rota, D.Kahaner and A. Odlyzko: J. Math. Anal. Appl. 42, 684 (1973).

[10] G. C. Rota: Finite Operator Calculus, Academic Press, New York 1975.

[11] G. Markowsky: J. Math. Anal. Appl. 63, 145 (1978).

[12] A. K. Kwaśniewski: Advances in Applied Clifford Algebras 9, 41 (1999).

[13] O.V. Viskov: Trudy Matiematicz‘eskovo Instituta AN SSSR 177, 21 (1986).

[14] A. Di Bucchianico and D.Loeb: J. Math. Anal. Appl. 92, 1 (1994).

[15] N. Ya. Sonin: Izw. Akad. Nauk 7, 337 (1997).

[16] C. Graves: Proc. Royal Irish Academy 6, 144 (1853-1857).

[17] P. Feinsilver and R. Schott: Algebraic Structures and Operator Calculus, Kluwer Academic Publishers, New York 1993.

[18] O.V. Viskov: Integral Transforms and Special Functions 1, 2 (1997).

[19] S. Pincherle and U. Amaldi: Le operazioni distributive e le loro applicazioni all‘analisi, N. Zanichelli, Bologna 1901.

[20] S. G. Kurbanov and V. M. Maximov: Dokl. Akad. Nauk Uz. SSSR 4, 8 (1986).

[21] A. Di Bucchianico and D.Loeb: Integral Transforms and Special Functions 4, 49 (1996).

[22] P. Kirschenhofer: Sitzunber. Abt. II Oster. Ackad. Wiss. Math. Naturw. Kl. 188, 263 (1979).

[23] A. Di Bucchianico and D.Loeb: J. Math. Anal. Appl. 199, 39 (1996).

[24] F. H. Jackson: Quart. J. Pure and Appl. Math. 41, 193 (1910).

[25] F. H. Jackson: Messenger of Math. 47, 57 (1917).

[26] F. H. Jackson: Quart. J. Math. 2, 1 (1951).

[27] S. M. Roman: J. Math. Anal. Appl. 87, 58 (1982).

[28] S. M. Roman: J. Math. Anal. Appl. 89, 290 (1982).

[29] S.M. Roman: J. Math. Anal. Appl. 95, 528 (1983).

[30] S. M. Roman: The umbral calculus, Academic Press, New York 1984.

[31] S. R. Roman: J. Math. Anal. Appl. 107, 222 (1985).