PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 96 (110) (2014), 67–83 DOI: 10.2298/PIM1410067C

AN ALGEBRAIC EXPOSITION OF UMBRAL CALCULUS WITH APPLICATION TO GENERAL LINEAR INTERPOLATION PROBLEM – A SURVEY

Francesco Aldo Costabile and Elisabetta Longo

Abstract. A systematic exposition of Sheffer polynomial sequences via determinantal form is given. A general linear interpolation problem related to Sheffer sequences is considered. It generalizes many known cases of linear interpolation. Numerical examples and conclusions close the paper.

1. The modern umbral calculus

In the 1970s Rota and his collaborators [17,19,20] began to construct a completely rigorous foundation for the classical umbral calculus, consisted primarily of a symbolic technique for the manipulation of numerical and polynomial sequences. The theory of Rota et al. was based on the relatively modern ideas of linear functional, linear operator and adjoint. This theory followed that less efficient of generating function methods; in fact, Appell [1], Sheffer [22] and Steffensen [23] based their theories on formal power series. These theories can be criticized both for their formalism not suitable for nonspecialists and for insufficient computational tools. The umbral calculus, because of its numerous applications in many branches of mathematics, physics, chemistry and engineering [24], has received many attentions from researchers. Recently, Di Bucchianico and Loeb [14] summarized and documented more than five hundred old and new findings related to umbral calculus. In last years attention has centered on finding novel approaches. For example, in [21] the connection between Sheffer polynomials and Riordan array is sketched and in [16] the isomorphism between the Sheffer group and the Riordan Group is proved. In [5,27] two different matrix approaches to Appell polynomials are given, in [9,26], these methods have been extended to Sheffer polynomials, and in [11], to binomial polynomial sequences. Recently, the relation between the umbral calculus and the general linear interpolation problem has been highlighted [6–8,10,11]. In this survey we give a unitary matrix approach to Sheffer polynomials, including Appell and binomial type polynomial sequences. Our theory of Sheffer sequences assumes binomial type polynomial sequences, therefore the paper is organized as follows: in Section 2 we give the preliminaries on binomial type sequences; in Section 3 we provide the new determinantal definition of Sheffer polynomials; in Section 4 we give the relationship with δ-operator and functional, finding the connection with Rota’s theory; in Section 5 we recall the determinantal definition of Yang [26]; in Section 6 we define the umbral interpolation problem and in Section 7 we provide classical and nonclassical examples; in Section 8 we furnish some numerical examples and finally, in Section 9, conclusions close the paper.

2010 Mathematics Subject Classification: 11B83; 65F40.

Key words and phrases: Sheffer, Appell, binomial, umbral, determinant, interpolation.

2. Preliminaries

In order to render the work self-contained we provide the necessary preliminary tools on binomial polynomial sequences. Let P_{n} be the commutative algebra of all polynomials in a simple variable x, with coefficients in a field 𝕂 (typically ℝ or ℂ).

Definition 2.1. [19] A polynomial sequence {p_{n}(x)}_{n∈ℕ} , where n is the degree of p_{n}(x), is said to be of binomial type (b.t.) if and only if it satisfies the binomial identity

(2.1)  p_{n}(x + y) = ∑_{k=0}^{n} ( {n \atop k}) p_{k}(x)p_{n−k} (y),

for all n ∈ ℕ and x, y ∈ ℝ.

Well-known examples of polynomial sequences can been found in [17–19] and references therein. For a polynomial sequence p_{n}(x) we could consider its coefficients a_{n,k} in its decomposition over power monomials x^{k}, that is

(2.2)  p_{n}(x) = ∑_{k=0}^{n} a_{n,k} x^{k}.

Relation (2.1), defining polynomial sequences of b.t., can be expressed in terms of coefficients a_{n,k} as follows [19, p. 111]

(2.3)  ( {i + j \atop i} ) a_{n,i+j} = ∑_{l=0}^{n} ( {n \atop l}) a_{l,i} a_{n−l,j} .

From (2.1) we can observe that a_{n,0} = δ_{n,0} , and from (2.3) that all the coefficients a_{n,k} , k = 1, …, n, n = 0, 1, …, are completely determined by a sequence {b_{n}}_{n∈ℕ} , with b_{0} = 0, b_{1} ≠ 0, as follows

a_{n,1} = b_{n}, a_{n,k} = \frac{1}{k} ∑_{i=1}^{n−k+1} ( {n \atop i}) a_{i,1} a_{n−i,k−1} , k = 2, …, n,

We also have [2]

a_{n,k} = \frac{1}{k!} ∑ ( {n \atop ν_{1}, …, ν_{k}} ) b_{ν_{1}} ⋯ b_{ν_{k}} , k = 2, …, n,

where the summation is over all k-tuples (ν_{1}, …, ν_{k}) with ν_{i} ⩾ 1 and ∑ ν_{i} = n. Finally, for the coefficients a_{n,k} defined in (2.2) we have

a_{n,0} = δ_{n,0} , a_{n,1} = b_{n}, (2.4)

a_{n,k} = \frac{1}{k} ∑_{i=1}^{n−k+1} ( {n \atop i}) a_{i,1} a_{n−i,k−1} = \frac{1}{k!} ∑ ( {n \atop ν_{1}, …, ν_{k}} ) b_{ν_{1}} ⋯ b_{ν_{k}} , k = 2, …, n.

Therefore, a polynomial sequence of b.t. p_{n}(x), n = 0, 1, …, is associated to a real numerical sequence b_{n}, n = 0, 1, …, with b_{0} = 0 and b_{1} ≠ 0. Moreover, let us define the formal power series

(2.5)  B(t) = ∑_{n=1}^{∞} b_{n} \frac{t^{n}}{n!},

then we have

∑_{n=0}^{∞} p_{n}(x)\frac{t^{n}}{n!} = e^{xB(t)} ,

that is, the function e^{xB(t)} is the generating function for polynomials p_{n}(x) [19, p. 106]. Moreover, we also have

(B(x))^{k} = ∑_{n=1}^{∞} a_{n,k} \frac{x^{n}}{n!} ,

with a_{n,k} defined by (2.4).

Now, let us set X_{n}(x) = [1, x, …, x^{n}]^{T}, P_{n}(x) = [p_{0}(x), p_{1}(x), …, p_{n}(x)]^{T} (P_{n}(x) which will be called b.t. vector), and

(2.6)  (A)_{i,j} = \{ \begin{matrix}a_{i,j} & i ⩾ j \\ 0 & otherwise \end{matrix}  i, j = 0, …, n,

with a_{i,j} , i, j = 0, …, n, defined by (2.4). Then, from (2.2), we can write

(2.7)  P_{n}(x) = AX_{n}(x),

and, observing that A is invertible,

(2.8)  X_{n}(x) = A^{−1} P_{n}(x).

Remark 2.1. [9] For the calculation of the inverse matrix

Â ≡ A^{−1} = (A^{−1} )_{i,j} ≡ \{ \begin{matrix}â_{i,j} & i ⩾ j \\ 0 & otherwise \end{matrix} i, j = 0, …, n,

we can use the following algorithm.

• determine the sequence b̂_{n}, n = 0, 1, …, as follows [15]

(2.9)  b̂_{0} = 0, b̂_{1} = \frac{1}{a_{1,1}} , b̂_{n} = −\frac{1}{a_{n,n}} ∑_{k=1}^{n−1} b̂_{k}a_{n,k} , n = 2, …,

with a_{n,k} defined by (2.4);

70 COSTABILE AND LONGO

• determine the sequence â_{n,k} , k = 0, …, n, n = 0, 1, …, as follows [7, 9] â_{n,0} = δ_{n,0} , â_{n,1} = b̂_{n}, (2.10)

â_{n,k} = \frac{1}{k} ∑_{i=1}^{n−k+1} ( {n \atop i}) â_{i,1} â_{n−i,k−1} = \frac{1}{k!} ∑ ( {n \atop ν_{1}, …, ν_{k}} ) b̂_{ν_{1}} ⋯ b̂_{ν_{k}}, k = 2, …, n.

Remark 2.2. We observe explicitly that from (2.7) and (2.8) we have

p_{n}(x) = ∑_{k=0}^{n} a_{n,k} x^{k}, x^{n} = ∑_{k=0}^{n} â_{n,k} p_{k}(x),

that is, these two relations are inverse ones. Therefore, p_{n}(x) in a basis for P_{n}. Now, we define the linear operator on P_{n}

(2.11)  Qy = ∑_{n=0}^{∞} \frac{b̂_{n}}{n!}y^{(n)} ,

with b̂_{n} defined by (2.9). Then we have Theorem 2.1. [11] The relation Qp_{n}(x) = np_{n−1} (x), n = 1, 2, … holds. Remark 2.3. According to the Rota theory, after Theorem 2.1 Q is the δ-

operator associated to the polynomial sequence of b.t. p_{n}(x).

3. Sheffer polynomial sequences

Let p_{n}(x), n = 0, 1, …, be a polynomial sequence of b.t. and β_{n}, n = 0, 1, …, a real numerical sequence with β_{0} ≠ 0.

Definition 3.1. [9] The polynomial sequence s_{n}(x), n = 0, 1, …, defined by

(3.1) s_{0}(x) = \frac{1}{β_{0}} ,
s_{n}(x) = \frac{(−1)^{n}}{(β_{0})^{n+1}} |\begin{matrix}p_{0} & p_{1}(x) & ⋯ & p_{n−1} (x) & p_{n}(x) \\ β_{0} & β_{1} & ⋯ & β_{n−1} & β_{n} \\ 0 & β_{0} & ⋯ & \binom{n−1}{1}β_{n−2} & \binom{n}{1}β_{n−1} \\ ⋮ & & ... & ⋮ & ⋮ \\ 0 & ⋯ & 0 & β_{0} & \binom{n}{n−1}β_{1} \end{matrix} | , n = 1, 2, …

is called the Sheffer polynomial sequence for (β_{n}, p_{n}(x)).

Remark 3.1. For p_{n}(x) = x^{n} the sequence s_{n}(x) is the Appell sequence for β_{n} [5]; for β_{0} = 1, and β_{n} = 0, n = 1, 2, …, we get s_{n}(x) = p_{n}(x). In general, a Sheffer sequence s_{n}(x) is the umbral composition [19] of an Appell sequence and a binomial type polynomial sequence.

Now, we give several characterizations for Sheffer polynomial sequences.

Theorem 3.1. [9] The sequence s_{n}(x) is the Sheffer polynomial sequence for (β_{n}, p_{n}(x)) if and only if

s_{n}(x) = α_{n}p_{0} + ( {n \atop 1}) α_{n−1} p_{1}(x) + ( {n \atop 2}) α_{n−2} p_{2}(x) + ⋯+ α_{0}p_{n}(x), n = 0, 1, …

with

(3.2)  α_{0} = \frac{1}{β_{0}} , α_{i} = −\frac{1}{β_{0}} ∑_{k=0}^{i−1} ( {i \atop k}) β_{i−k} α_{k}, i = 1, 2, …, n.

Remark 3.2. [9] Let

(3.3)  (M)_{i,j} = \{ \begin{matrix}\binom{i}{j}β_{i−j} & i ⩾ j \\ 0 & otherwise \end{matrix},  i, j = 0, …, n.

The matrix M is invertible and

(M^{−1} )_{i,j} = \{ \begin{matrix}\binom{i}{j}α_{i−j} & i ⩾ j \\ 0 & otherwise \end{matrix}, i, j = 0, …, n,

with α_{n} defined by (3.2).

Therefore, setting

(3.4)  β_{0} = \frac{1}{α_{0}} , β_{i} = −\frac{1}{α_{0}} ∑_{k=0}^{i−1} ( {i \atop k}) α_{i−k} β_{k}, i = 1, 2, …, n,

Relations (3.2) and (3.4) are inverse ones.

Theorem 3.2 (Recurrence, [9]). The sequence s_{n}(x), n = 0, 1, …, is the Sheffer sequence for (β_{n}, p_{n}(x)) if and only if

s_{n}(x) = \frac{1}{β_{0}} ( p_{n}(x) − ∑_{k=0}^{n−1} ( {n \atop k}) β_{n−k} s_{k}(x)) , n = 1, 2, …

Now, we have the following

Theorem 3.3 (Sheffer identity, [9]). The sequence s_{n}(x) is the Sheffer polynomial sequence for (β_{n}, p_{n}(x)) if and only if

s_{n}(x + y) = ∑_{i=0}^{n} ( {n \atop i}) s_{n−i} (x)p_{i}(y) = ∑_{i=0}^{n} ( {n \atop i}) s_{i}(x)p_{n−i} (y), n = 1, 2, …

Remark 3.3. After Theorem 3.3 the polynomial sequence s_{n}(x) defined by (3.1) is the Sheffer sequence associated to the polynomial sequence of b.t. p_{n}(x), as defined in [19, p. 139].

Theorem 3.4 (Multiplication theorem, [9]). The sequence s_{n}(x), n = 0, 1, …, is the Sheffer sequence for (β_{n}, p_{n}(x)) if and only if

s_{n}(mx) = ∑_{i=0}^{n} ( {n \atop i}) s_{n−i} (x)p_{i}((m − 1)x), \begin{matrix}n = 0, 1, …, \\ m = 1, 2, … \end{matrix}

Now, we give the relationship with generating function and the equivalence with Sheffer theory [22].

Theorem 3.5 (Generating function, [9]). The sequence s_{n}(x) is the Sheffer polynomial sequence for (β_{n}, p_{n}(x)) if and only if there exists a unique numerical sequence α_{n}, n = 0, 1, …, α_{0} ≠ 0, such that, setting

a(t) = α_{0} + α_{1}t + α_{2} \frac{t^{2}}{2} + ⋯+ α_{n} \frac{t^{n}}{n!} + …, α_{0} ≠ 0,

we have

a(t)e^{xB(t)} = ∑_{n=0}^{∞} s_{n}(x)\frac{t^{n}}{n!},

where B(t) is defined in (2.5).

4. Relationship with δ-operator and functional

Let Q be the δ-operator [19] associated to the polynomial sequence of b.t. p_{n}(x), that is given by (2.11). Then we have the following

Theorem 4.1. [9] If s_{n}(x), n = 0, 1, … is the Sheffer sequence for (β_{n}, p_{n}(x)), then

(4.1)  Qs_{n}(x) = ns_{n−1} (x) n = 1, 2, . . ..

Theorem 4.2. [9] If s_{n}(x), n = 0, 1, … be the Sheffer sequence for (β_{n}, p_{n}(x)) and Q the δ-operator associated to the polynomial sequence of b.t. p_{n}(x), then s_{n}(x) satisfies the functional equation

\frac{β_{n}}{n!} Q^{n}y(x) + \frac{β_{n−1}}{(n − 1)!} Q^{n−1} y(x) + ⋯+ \frac{β_{2}}{2!} Q^{2}y(x) + β_{1}Qy(x) + β_{0}y(x) = p_{n}(x)

Let, now, L be the linear functional on P_{n} defined by β_{n} = L(p_{n}(x)), n = 0, 1, . . .. Then we have the following

Theorem 4.3. The sequence s_{n}(x), n = 0, 1, … is the Sheffer sequence for (β_{n}, p_{n}(x)) if and only if

(4.2)  L(Q^{k}s_{n}(x)) = n!δ_{n,k} , k = 0, …, n.

Proof. It follows from (3.1), (4.1) and from the linearity of L. □

Remark 4.1. After Theorem 4.3 we have the equivalence of our matrix approach with Roman theory [18].

Remark 4.2. Putting L_{i} = LQ^{i}, i = 0, …, n, relations (4.2) can be interpreted as a linear interpolation problem on P_{n}, the solution of which is the polynomial sequence s_{n}(x) given by (3.1).

Let us, now, introduce the Sheffer vector.

Definition 4.1. If s_{n}(x) is the Sheffer polynomial sequence for (β_{n}, p_{n}(x)), then the vector of functions S_{n}(x) = [s_{0}(x), …, s_{n}(x)]^{T} is called the Sheffer vector for (β_{n}, p_{n}(x)).

Then we have

Theorem 4.4 (Matrix form, [9]). Let S_{n}(x) be a vector of polynomial function. It is the Sheffer vector for (β_{n}, p_{n}(x)) if and only if

S_{n}(x) = SX_{n}(x), X_{n}(x) = ŜS_{n}(x),

where S = M^{−1} A, Ŝ ≡ S^{−1} = A^{−1} M, with M and A defined by (3.3) and (2.6) respectively.

Remark 4.3. Putting

S ≡ (S)_{i,j} ≡ \{ \begin{matrix}s_{i,j} & i ⩾ j \\ 0 & otherwise \end{matrix}, i, j = 0, …, n,

(4.3)  Ŝ ≡ (Ŝ)_{i,j} ≡ \{ \begin{matrix}ŝ_{i,j} & i ⩾ j \\ 0 & otherwise \end{matrix},  i, j = 0, …, n,

we get

s_{i,j} = ∑_{k=j}^{i} ( {i \atop k}) α_{i−k} a_{k,j} , i = 0, …, n, j = 0, …, i,

(4.4)  ŝ_{i,j} = ∑_{k=j}^{i} ( {i \atop k}) β_{i−k} â_{k,j} = \frac{L(Q^{j} x^{i})}{j!} , i = 0, …, n, j = 0, …, i.

Theorem 4.5 (Representation Theorem, [9]). If s_{n}(x) is the Sheffer sequence for (β_{n}, p_{n}(x)) and q_{n}(x) is a polynomial of degree ⩽ n such that q_{n}(x) = ∑ _{k=0}^{n} c_{k}x^{k}, n = 0, 1, …, then

q_{n}(x) = ∑_{k=0}^{n} \frac{L(Q^{k}q_{n}(x))}{k!}s_{k}(x) = ∑_{k=0}^{n} ( ∑_{j=k}^{n} c_{k} ŝ_{j,k} ) s_{k}(x), n = 0, 1, . . ..

Theorem 4.6. [9] The sequence s_{n}(x) is the Sheffer sequence for (β_{n}, p_{n}(x)) if and only if

s_{n}(x) = \frac{(−1)^{n}}{∏ _{k=0}^{n} ŝ_{k,k}} |\begin{matrix}1 & x & ⋯ & x^{n−1} & x^{n} \\ ŝ_{0,0} & ŝ_{1,0} & ⋯ & ŝ_{n−1,0} & ŝ_{n,0} \\ 0 & ŝ_{1,1} & ⋯ & ŝ_{n−1,1} & ŝ_{n,1} \\ ⋮ & & ... & & ⋮ \\ 0 & ⋯ & 0 & ŝ_{n−1,n−1} & ŝ_{n,n−1} \end{matrix} | , n = 0, 1, …,

with ŝ_{i,j} , i, j = 0, …, n, defined by (4.4).

Remark 4.4. For ŝ_{i,j} = \binom{i}{j}β_{i−j} , s_{n}(x) is the Appell polynomial sequence as defined in [5].

Theorem 4.7 (Binomial sequences, [9]). If p_{n}(x), n = 0, 1, …, is a binomial type polynomial sequence associated to b_{n} then

p_{0}(x) = 1, p_{n}(x) = \frac{(−1)^{n+1}}{∏ _{k=0}^{n} â_{k,k}} |\begin{matrix}x & x^{2} & ⋯ & x^{n−1} & x^{n} \\ â_{1,1} & â_{2,1} & ⋯ & â_{n−1,1} & â_{n,1} \\ 0 & â_{2,2} & ⋯ & â_{n−1,2} & â_{n,2} \\ ⋮ & & ... & & ⋮ \\ 0 & ⋯ & & â_{n−1,n−1} & â_{n,n−1} \end{matrix} | ,

n = 1, 2, …, with â_{i,j} , i, j = 0, …, n, defined by (2.10).

Remark 4.5. In [11] the inverse of Theorem 4.7 is proved.

Theorem 4.8 (Connection constants, [9]). Let S_{n}(x) and T_{n}(x) be the Sheffer vectors for (β_{n}, p_{n}(x)) and (γ_{n}, q_{n}(x)), respectively, with p_{n}(x) = ∑ _{k=0}^{n} a_{n,k} x^{k} and q_{n}(x) = ∑ _{k=0}^{n} d_{n,k} x^{k}. Then T_{n}(x) = CS_{n}(x), where C = T S^{−1} , with S^{−1} defined by (4.3) and (4.4), and

(T)_{i,j} = \{ \begin{matrix}∑ _{k=j}^{i} \binom{i}{k}δ_{i−k} d_{k,j} & i ⩾ j \\ 0 & otherwise \end{matrix}, i, j = 0, …, n,

where δ_{n} are defined by ∑ _{k=0}^{n} \binom{n}{k}δ_{k}γ_{n−k} = \{ \begin{matrix}1, & n=0, \\ 0, & n=1,... \end{matrix}

5. The algebraic approach of Yang

Yang in [26] gives a new determinantal form and a recurrence relation for Sheffer sequences; his approach is based on the production matrix [13] of an invertible lower triangular matrix. There is the following

Theorem 5.1. [26] Let [g(t), f(t)] be an exponential Riordan array with the csequences (c_{i})_{i⩾0} and r-sequences (r_{i})_{i⩾0}. Let (a_{n}(x))_{x⩾0} be the Sheffer polynomial sequence for (g(t), f(t)). Then (a_{n}(x))_{x⩾0} satisfies the recurrence relation

a_{n+1} (x) = (x − c_{0} − nr_{1})a_{n}(x) − \frac{n!}{(n − 1)!} (c_{1} + (n − 1)r_{2})a_{n−1} (x) − …

− \frac{n!}{2!}(c_{n−2} + 2r_{n−1} )a_{2}(x) − \frac{n!}{1!}(c_{n−1} + r_{n})a_{1}(x) − n!c_{n}a_{0}(x),

with initial conditions a_{0}(x) = 1 and a_{1}(x) = x − c_{0}. For n ⩾ 0, a_{n+1} (x) is given by

a_{n+1} (x) = (−1)^{n+1}

× |\begin{matrix}c_{0} − x & 1 & 0 & ⋯ & 0 \\ 1!c_{1} & \frac{1!}{1!} (c_{0} + r_{1}) − x & 1 & ⋯ & 0 \\ 2!c_{2} & \frac{2!}{1!} (c_{1} + r_{2}) & \frac{2!}{2!} (c_{0} + 2r_{1}) − x & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ n!c_{n} & \frac{n!}{1!} (c_{n−1} + r_{n}) & \frac{n!}{2!} (c_{n−2} + 2r_{n−1} ) & ⋯ & \frac{n!}{n!} (c_{0} + nr_{1}) − x \end{matrix} |

= (−1)^{n+1} |P_{n+1} − xI_{n+1} | = |xI_{n+1} − P_{n+1} |.

Thus, the Sheffer sequences are characteristic polynomials of production matrices.

6. The umbral interpolation

The umbral interpolation [10] is the linear interpolation problem which is expressed through a basis of Sheffer polynomials (Appell or binomial polynomials in particular). Let X be the linear space of real functions defined in the interval [a, b] continuous and with continuous derivatives of all necessary orders. Let Q be a δoperator on P_{n} and p_{n}(x), n = 0, 1, …, the polynomial sequence of binomial type associated to it. Let L be a linear functional on X such that L(1) ≠ 0. Let us we set β_{n} = L(p_{n}(x)), n = 0, 1, …, and let s_{n}(x) be the Sheffer polynomial sequence for (β_{n}, p_{n}(x)).

Definition 6.1. [10] The polynomial sequence s_{n}(x), n = 0, 1, …, is a basis for P_{n} and we call it Umbral basis for (L, Q).

Let be ω_{i} ∈ ℝ, i = 0, 1, …, n, and consider the problem
L(Q^{i}P_{n}) = i!ω_{i}, i = 0, …, n, P_{n}(x) ∈ P_{n}.

We call it Umbral interpolation problem.

Theorem 6.1 (Main theorem, [10]). Let s_{i}(x) be the Umbral basis for (L, Q) and ω_{i} ∈ ℝ, i = 0, 1, …, n; the polynomial P_{n}(x) = ∑ _{i=0}^{n} ω_{i}s_{i}(x) is the unique polynomial of degree less than or equal to n such that L(Q^{i}P_{n}) = i!ω_{i}, i = 0, …, n.

Remark 6.1. [10] For each P_{n}(x) ∈ P_{n} we have P_{n}(x) = ∑ _{i=0}^{n} \frac{L(Q^{i} P_{n} )}{i!} s_{i}(x).

Let us consider a function f ∈ X, such that Q^{i}f is defined in X. Then we have

Theorem 6.2. [10] Let f ∈ X such that Q^{i}f ∈ X. The polynomial

(6.1)  P_{n}[f](x) = ∑_{i=0}^{n} \frac{L(Q^{i}f)}{i!} s_{i}(x)

is the unique polynomial of degree less than or equal to n such that

L(Q^{i}P_{n}[f]) = L(Q^{i}f), i = 0, …, n.

Definition 6.2. The polynomial P_{n}[f](x), if not identically zero, is called Umbral interpolation polynomial of the function f for (L, Q); it satisfies

L(Q^{i}P_{n}) = L(Q^{i}f), i = 0, 1, …, n.

Therefore, it is interesting to consider the estimation of the remainder. Let us set R_{n}[f](x) = f(x) − P_{n}[f](x), for all x ∈ [a, b].

Theorem 6.3 (Exactness, [10]). For any f ∈ P_{n} and each x ∈ [a, b],

R_{n}[f](x) = 0 and R_{n}[p_{n+1} (x)] ≠ 0.

For a fixed x, we may consider R_{n}[f](x) as a linear functional which acts on f and annihilates all elements of P_{n}. By Peano’s theorem, if a linear functional has this property, then it must have a simple representation in terms of f^{(n+1)} .

Theorem 6.4. [10] For f ∈ C^{n+1} [a, b], the following relation holds

∀x ∈ [a, b], R_{n}[f](x) = \frac{1}{n!} ∫_{a}^{b}K_{n}(x, t)f^{(n+1)} (t) dt

where

K_{n}(x, t) = R_{n}[(x − t)_{+}^{n} ] = (x − t)_{+}^{n} − ∑_{i=0}^{n} \frac{L_{x}[Q^{i}(x − t)_{+}^{n} ]}{i!} s_{i}(x).

Now, let us fix z ∈ [a, b] and consider the polynomial

(6.2) \overline{P}_{n}[f, z](x) ≡ f(z)+P_{n}[f](x)−P_{n}[f](z) = f(z)+∑_{i=1}^{n} \frac{L(Q^{i}f)}{i!} (s_{i}(x)−s_{i}(z)).

Definition 6.3. The polynomial \overline{P}_{n}[f, z](x) is called Umbral interpolation polynomial of f centered at z.

Theorem 6.5. [10] The polynomial \overline{P}_{n}[f, z](x) is an approximating polynomial of degree n for f(x), i.e.,

∀ x ∈ [a, b], f(x) = \overline{P}_{n}[f, z](x) + \overline{R}_{n}[f](x),

with \overline{R}_{n}[p_{i}(x)] = 0, i = 0, …, n and \overline{R}_{n}[p_{n+1} (x)] ≠ 0.

Theorem 6.6. [10] The polynomial \overline{P}_{n}[f, z](x) satisfies the interpolation conditions \overline{P}_{n}[f, z](z) = f(z), L(Q^{i} \overline{P}_{n}[f, z]) = L(Q^{i}f), i = 1, …, n.

7. Examples

7.1. Appell interpolation [6]. With the previous notation, let Qf = Df = f′(x). Then the associated sequence is the canonical basis x^{n}, n = 0, 1, . . ..

Now, let L be a linear functional verifying L(1) ≠ 0. Then the umbral interpolation polynomials (6.1) and (6.2) become

P_{n}[f](x) = L(f) + ∑_{i=1}^{n} \frac{L(f^{(i)} (x))}{i!} A_{i}(x), (7.1)

\overline{P}_{n}[f, 0](x) = f(0) + ∑_{i=1}^{n} \frac{L(f^{(i)} (x))}{i!} (A_{i}(x) − A_{i}(0)),

where A_{i}(x) is the umbral basis for (L, D), and in particular, it is an Appell sequence.

7.1.1. Taylor interpolation [6, 12]. Let L(f) = f(x_{0}), x_{0} ∈ [a, b]. Then the umbral basis for (L, D) is the sequence A_{0}(x) = 1, A_{n}(x, a) = (x − x_{0})^{n}, n = 1, 2, . . .. The interpolation polynomial (6.1) becomes

T_{n}[f](x) = ∑_{i=0}^{n} \frac{f^{(i)} (x_{0})}{i!} (x − x_{0})^{i}

i.e., the Umbral interpolation is the well known Taylor interpolation [12]. Therefore, (7.1) can be seen as a generalization of Taylor interpolation.

7.1.2. Bernoulli interpolation [3, 6]. Let L(f) = ∫_{0}^{1} f(x) dx. Then the umbral basis for (L, D) is the Bernoulli sequence B_{n}(x) [4, 5]. Interpolation polynomials (6.1) and (6.2) become

B_{n}[f](x) = ∫_{0}^{1} f(x)dx + ∑_{i=1}^{n} \frac{f^{(i−1)} (1) − f^{(i−1)} (0)}{i!} B_{i}(x),

(7.2)  \overline{B}_{n}[f, 0](x) = f(0) + ∑_{i=1}^{n} \frac{f^{(i−1)} (1) − f^{(i−1)} (0)}{i!} (B_{i}(x) − B_{i}(0)).

7.1.3. Euler interpolation [6]. We consider L(f) = \frac{1}{2}(f(0) + f(1)). Then the umbral basis for (L, D) is the Euler sequence E_{n}(x) [5]. Interpolation polynomials (6.1) and (6.2) become

(7.3)  E_{n}[f](x) = \frac{f(0) + f(1)}{2} + ∑_{i=1}^{n} \frac{f^{(i)} (0) + f^{(i)} (1)}{2i!} E_{i}(x),

\overline{E}_{n}[f, 0](x) = f(0) + ∑_{i=1}^{n} \frac{f^{(i)} (0) + f^{(i)} (1)}{2i!} (E_{i}(x) − E_{i}(0)).

7.2. Abel–Sheffer interpolation [10]. With the previous notation, let Qf = D_{a}f = f′(x + a), a ∈ ℝ, a ≠ 0. Then the associated sequence is the Abel sequence A_{0}(x, a) = 1, A_{n}(x, a) = x(x − an)^{n−1} , n = 1, 2, … [19].

Now, let L be a linear functional verifying L(1) ≠ 0. Then umbral interpolation polynomials (6.1) and (6.2) become

P_{n}[f](x) = L(f) + ∑_{i=1}^{n} \frac{L(f^{(i)} (x + ai))}{i!} s_{i}(x), (7.4)

\overline{P}_{n}[f, 0](x) = f(0) + ∑_{i=1}^{n} \frac{L(f^{(i)} (x + ai))}{i!} (s_{i}(x) − s_{i}(0)),

where s_{i}(x) is the umbral basis for (L, D_{a}).

7.2.1. Abel–Gontscharoff interpolation [10, 12]. Let L(f) = f(x_{0}), x_{0} ∈ [a, b]. Then the umbral basis for (L, D_{a}) is the sequence

G̃_{0}(x) = 1, G̃_{n}(x, a) = (x − x_{0})(x − x_{0} − an)^{n−1} , n = 1, 2, …,

that is the classical Abel–Gontscharoff sequence [12] on the equidistant points x_{i} = x_{0} + ai, i = 0, …, n. Interpolation polynomial (6.1) becomes

G̃_{n}[f](x) = ∑_{i=0}^{n} \frac{f^{(i)} (x_{0} + ai)}{i!} G̃_{i}(x, a)

i.e., the Umbral interpolation is the well known Abel–Gontscharoff interpolation [12, 25] on the equidistant points x_{i} = x_{0} + ai, i = 0, …, n. Therefore (7.4) can be seen as a generalization of Abel–Gontscharoff interpolation on the equidistant points.

7.2.2. Abel–Bernoulli interpolation [10]. Let L(f) = ∫_{0}^{1} f(x) dx. Then the umbral basis for (L, D_{a}) is the Bernoulli–Abel sequence B̃_{n} (x, a) [9]. Interpolation polynomials (6.1) and (6.2) become

B̃_{n} [f](x) = ∫_{0}^{1} f(x) dx + ∑_{i=1}^{n} \frac{f^{(i−1)} (1 + ai) − f^{(i−1)} (ai)}{i!} B̃_{i}(x, a),

(7.5) \overline{B̃}_{n}[f, 0](x) = f(0) + ∑_{i=1}^{n} \frac{f^{(i−1)} (1 + ai) − f^{(i−1)} (ai)}{i!} (B̃_{i}(x, a) − B̃_{i}(0, a)).

7.2.3. Abel–Euler interpolation [10]. We consider L(f) = \frac{1}{2}(f(0)+f(1)). Then the umbral basis for (L, D_{a}) is the Euler–Abel sequence Ẽ_{n} (x, a) [9]. Interpolation polynomials (6.1) and (6.2) become

(7.6)  Ẽ_{n} [f](x) = \frac{f(0) + f(1)}{2} + ∑_{i=1}^{n} \frac{f^{(i)} (ai) + f^{(i)} (1 + ai)}{2i!} Ẽ_{i}(x, a),

\overline{Ẽ}_{n}[f, 0](x) = f(0) + ∑_{i=1}^{n} \frac{f^{(i)} (ai) + f^{(i)} (1 + ai)}{2i!} (Ẽ_{i}(x, a) − Ẽ_{i}(0, a)).

7.3. Δ_{h}-Appell interpolation [8]. Let Qf = Δ_{h}f(x) = f (x + h) − f(x).

Moreover, let Δ_{h}^{−1} be the inverse operator of Δ_{h}, such that

Δ_{h}^{−1} φ(x) = f(x) ⇔ Δ_{h}f(x) = φ(x).

Then the associated sequence to Δ_{h} is the falling factorial sequence [19]

(x)_{0} = 1, (x)_{n} = x(x − h)(x − 2h) ⋯ (x − (n − 1)h), n = 1, 2, . . ..

Now, let L be a linear functional verifying L(1) ≠ 0. Then umbral interpolation polynomials (6.1) and (6.2) become

P_{n}[f](x) = ∑_{i=0}^{n} \frac{L(Δ_{h}^{i}f)}{h^{i}i!} A_{i}(x), (7.7)

\overline{P}_{n}[f, 0](x) = f(0) + ∑_{i=1}^{n} \frac{L(Δ_{h}^{i}f)}{h^{i}i!} (A_{i}(x) − A_{i}(0)),

where A_{i}(x) is the umbral basis for (L, Δ_{h}) and, in particular, it is a Δ_{h}-Appell sequence [8].

As in the previous example we can consider the following cases.

7.3.1. Newton interpolation on equidistant points [8, 12]. Let L(f) = f(x_{0}). The umbral basis for (L, Δ_{h}) is (x − x_{0})_{n}. Interpolation polynomial (6.1) becomes

P_{n}[f](x) = f(x_{0}) + ∑_{i=1}^{n} \frac{Δ_{h}^{i}f(x_{0})}{h^{i}i!} (x − x_{0})_{i}.

It is known as Newton interpolation on equidistant points, therefore (7.7) can be seen as a generalization of Newton interpolation on equidistant points.

7.3.2. Bernoulli interpolation of second kind [8]. Let L(f) = (DΔ_{h}^{−1} f)_{x=0} . We call the umbral basis for (L, Δ_{h}) Bernoulli polynomial sequence of second kind B_{n}^{II} (x) [8]. Interpolation polynomials (6.1) and (6.2) become

B_{n}^{II} [f](x) = (DΔ_{h}^{−1} f)_{x=0} + ∑_{i=1}^{n} \frac{Δ_{h}^{i−1} f′(0)}{h^{i}i!} B_{i}^{II} (x),

(7.8)  \overline{B}_{n}^{II} [f, 0](x) = f(0) + ∑_{i=1}^{n} \frac{Δ_{h}^{i−1} f′(0)}{h^{i}i!} (B_{i}^{II} (x) − B_{i}^{II} (0)).

7.3.3. Boole interpolation [8]. Let L(f) = (M_{h}f)_{x=0} = \frac{1}{2}(f(0)+f(h)). We call the umbral basis for (L, Δ_{h}) Boole polynomial sequence E_{n}^{II} (x) [8]. Interpolation polynomials (6.1) and (6.2) become

(7.9)  E_{n}^{II} [f](x) = \frac{f(0) + f(h)}{2} + ∑_{i=1}^{n} \frac{Δ_{h}^{i}(f(0) + f(h))}{2h^{i}i!} E_{i}^{II} (x),

\overline{E}_{n}^{II} [f, 0](x) = f(0) + ∑_{i=1}^{n} \frac{Δ_{h}^{i}(f(0) + f(h))}{2h^{i}i!} (E_{i}^{II} (x) − E_{i}^{II} (0)).

7.4. δ_{h}-Sheffer interpolation [10]. Let

Qf = δ_{h}f(x) = \frac{1}{h}( f( x + \frac{1}{2}h) − f( x − \frac{1}{2}h) )

and let δ_{h}^{−1} be the inverse operator of δ_{h}, such that

δ_{h}^{−1} φ(x) = f(x) ⇔ δ_{h}f(x) = φ(x).

Then the associated sequence to δ_{h} is the sequence [19]

x^{[0]} = 1, x^{[n]} ≡ x( x + ( \frac{n}{2} − 1) h)_{n−1} = x( x + ( \frac{n}{2} − 1) h) ⋯ ( x + ( − \frac{n}{2} + 1) h) ,

n = 1, 2, . . .. Now, let L be a linear functional verifying L(1) ≠ 0. Then umbral interpolation polynomials (6.1) and (6.2) become

(7.10)  P_{n}[f](x) = ∑_{i=0}^{n} \frac{L(δ_{h}^{i}f)}{i!} s_{i}(x),

\overline{P}_{n}[f, 0](x) = f(0) + ∑_{i=1}^{n} \frac{L(δ_{h}^{i}f)}{i!} (s_{i}(x) − s_{i}(0)),

where s_{i}(x) is the umbral basis for (L, δ_{h}).

As in the previous example we can consider the following cases.

7.4.1. δ_{h}-central interpolation [10,12]. Let L(f) = f(0). The umbral basis for (L, δ_{h}) is s_{n}(x) = x^{[n]}. Interpolation polynomial (6.1) becomes

P_{n}[f](x) = f(0) + ∑_{i=1}^{n} \frac{δ_{h}^{i}f(0)}{i!} x^{[i]}.

It is known as interpolation formula with central differences [23, p. 32], therefore (7.10) can be seen as a generalization of central interpolation.

7.4.2. δ_{h}-Bernoulli interpolation [10]. Let L(f) = (Dδ_{h}^{−1} f) _{x=0} . We call the umbral basis for (L, δ_{h}) δ_{h}-Bernoulli polynomial sequence B̂_{n} (x). Interpolation polynomials (6.1) and (6.2) become

B̂_{n} [f](x) = (Dδ_{h}^{−1} f) _{x=0} + ∑_{i=1}^{n} \frac{δ_{h}^{i−1} f′(0)}{i!} B̂_{i}(x),

(7.11)  \overline{B̂}_{n}[f](x) = f(0) + ∑_{i=1}^{n} \frac{δ_{h}^{i−1} f′(0)}{i!} (B̂_{i}(x) − B̂_{i}(0)).

7.4.3. δ_{h}-Euler interpolation [10]. Let

L(f) = (M_{h}f)_{x=0} = \frac{1}{2}(f(h/2) + f(−h/2)).

We call the umbral basis for (L, δ_{h}) δ_{h}-Euler polynomial sequence Ê_{n} (x). Interpolation polynomials (6.1) and (6.2) become

(7.12)  Ê_{n} [f](x) = \frac{f(h/2) + f(−h/2)}{2} + ∑_{i=1}^{n} \frac{δ_{h}^{i}(f(h/2) + f(−h/2))}{2i!} Ê_{i}(x),

\overline{Ê}_{n}[f, 0](x) = f(0) + ∑_{i=1}^{n} \frac{δ_{h}^{i}((h/2) + f(−h/2))}{2i!} (Ê_{i}(x) − Ê_{i}(0)).

8. Numerical examples

Now we consider some interpolation test problem and report the numerical results obtained by using an ad hoc “Mathematica" code. We compare the error in approximating a given function with Appell, Abel–Sheffer, Δ_{h}-Appell and δ_{h}-Sheffer interpolation polynomials. In particular we compare numerical results obtained by applying:

• Abel-Bernoulli interpolation polynomial \overline{B̃}_{n}[f, 0](x) (7.5)

• Bernoulli interpolation polynomial \overline{B}_{n}[f, 0](x) (7.2)

• δ_{h}-Bernoulli interpolation polynomial \overline{B̂}_{n}[f, 0](x) (7.11)

• Bernoulli interpolation polynomial of second kind \overline{B}_{n}^{II} [f, 0](x) (7.8)

• Abel-Euler interpolation polynomial Ẽ_{n} [f](x) (7.6)

• Euler interpolation polynomial E_{n}[f](x) (7.3)

• δ_{h}-Euler interpolation polynomial Ê_{n} [f](x) (7.12)

• Boole interpolation polynomial E_{n}^{II} [f](x) (7.9)

We emphasize that the compared polynomials of the same degree have the same degree of exactness.

Example 8.1. For the function f(x) = e^{(x+1)/2} , x ∈ [0, 1], the interpolation error is reported in the following tables.

\overline{B̃}_{n}[f, 0](x) \overline{B}_{n}[f, 0](x) \overline{B̂}_{n}[f, 0](x) \overline{B}_{n}^{II} [f, 0](x)
n = 5 2.774 ∗ 10^{−6} 1.102 ∗ 10^{−6} 1.628 ∗ 10^{−5} 6.949 ∗ 10^{−7}
n = 6 1.460 ∗ 10^{−7} 8.619 ∗ 10^{−8} 8.814 ∗ 10^{−7} 1.733 ∗ 10^{−8}
n = 7 1.463 ∗ 10^{−8} 6.885 ∗ 10^{−9} 4.160 ∗ 10^{−8} 4.354 ∗ 10^{−10}
n = 8 7.589 ∗ 10^{−10} 5.458 ∗ 10^{−10} 1.742 ∗ 10^{−9} 8.609 ∗ 10^{−12}
Ẽ_{n} [f](x)  E_{n}[f](x)  Ê_{n} [f](x)  E_{n}^{II} [f](x)
n = 5 5.978 ∗ 10^{−5} 4.460 ∗ 10^{−5} 2.606 ∗ 10^{−6} 1.318 ∗ 10^{−7}
n = 6 9.432 ∗ 10^{−6} 7.103 ∗ 10^{−6} 1.171 ∗ 10^{−7} 2.925 ∗ 10^{−9}
n = 7 1.483 ∗ 10^{−6} 1.130 ∗ 10^{−6} 4.715 ∗ 10^{−9} 5.817 ∗ 10^{−11}
n = 8 2.354 ∗ 10^{−7} 1.799 ∗ 10^{−7} 1.726 ∗ 10^{−10} 1.041 ∗ 10^{−12}

Example 8.2. For the function f(x) = ln(x^{2} +10), x ∈ [0, 1], the interpolation error is reported in the following tables.

\overline{B̃}_{n}[f, 0](x) \overline{B}_{n}[f, 0](x) \overline{B̂}_{n}[f, 0](x) \overline{B}_{n}^{II} [f, 0](x)
n = 5 1.994 ∗ 10^{−6} 4.526 ∗ 10^{−6} 1.311 ∗ 10^{−4} 2.823 ∗ 10^{−6}
n = 6 2.482 ∗ 10^{−6} 1.760 ∗ 10^{−6} 1.737 ∗ 10^{−6} 3.744 ∗ 10^{−7}
n = 7 5.442 ∗ 10^{−7} 3.457 ∗ 10^{−7} 5.477 ∗ 10^{−6} 1.579 ∗ 10^{−8}
n = 8 1.267 ∗ 10^{−7} 2.559 ∗ 10^{−7} 6.832 ∗ 10^{−8} 4.487 ∗ 10^{−9}
Ẽ_{n} [f](x)  E_{n}[f](x)  Ê_{n} [f](x)  E_{n}^{II} [f](x)
n = 5 1.183 ∗ 10^{−4} 2.138 ∗ 10^{−4} 1.974 ∗ 10^{−5} 4.435 ∗ 10^{−7}
n = 6 1.257 ∗ 10^{−4} 1.410 ∗ 10^{−4} 1.559 ∗ 10^{−6} 6.225 ∗ 10^{−8}
n = 7 6.482 ∗ 10^{−5} 8.666 ∗ 10^{−5} 5.843 ∗ 10^{−7} 1.624 ∗ 10^{−9}
n = 8 5.736 ∗ 10^{−5} 7.829 ∗ 10^{−5} 4.823 ∗ 10^{−8} 5.279 ∗ 10^{−10}

Example 8.3. For the function f(x) = 10 cos(x) + \frac{1}{10} sin^{2}(x), x ∈ [0, 1], the interpolation error is reported in the following tables.

\overline{B̃}_{n}[f, 0](x) \overline{B}_{n}[f, 0](x) \overline{B̂}_{n}[f, 0](x) \overline{B}_{n}^{II} [f, 0](x)
n = 5 4.652 ∗ 10^{−4} 2.319 ∗ 10^{−4} 4.227 ∗ 10^{−3} 1.483 ∗ 10^{−4}
n = 6 1.882 ∗ 10^{−5} 2.451 ∗ 10^{−6} 3.613 ∗ 10^{−6} 5.090 ∗ 10^{−7}
n = 7 1.3378 ∗ 10^{−5} 2.695 ∗ 10^{−6} 1.224 ∗ 10^{−5} 9.767 ∗ 10^{−8}
n = 8 1.981 ∗ 10^{−6} 2.058 ∗ 10^{−6} 4.386 ∗ 10^{−7} 3.399 ∗ 10^{−8}
Ẽ_{n} [f](x)  E_{n}[f](x)  Ê_{n} [f](x)  E_{n}^{II} [f](x)
n = 5 9.016 ∗ 10^{−3} 9.265 ∗ 10^{−3} 6.613 ∗ 10^{−4} 2.737 ∗ 10^{−5}
n = 6 1.880 ∗ 10^{−3} 3.195 ∗ 10^{−4} 2.770 ∗ 10^{−6} 6.299 ∗ 10^{−8}
n = 7 1.490 ∗ 10^{−3} 7.061 ∗ 10^{−4} 1.201 ∗ 10^{−6} 1.661 ∗ 10^{−8}
n = 8 6.494 ∗ 10^{−4} 6.746 ∗ 10^{−4} 3.200 ∗ 10^{−7} 4.111 ∗ 10^{−9}

9. Conclusions

We have given a construction of Sheffer sequences via determinantal form, proving the equivalence with previous theories (Rota et al., Sheffer). Moreover, all the main known and some new properties have been shown. Another recent determinantal form has been mentioned. Afterwards, a general linear interpolation problem has been proposed and solved, giving many examples. Further developments both in the multivariate case and in computational applications (stability, conditioning, etc.) are possible.

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Department of Mathematics University of Calabria Rende Italy

francesco.costabile@unical.it
longo@mat.unical.it