In 1880 P. E. Appell () introduced and widely studied sequences of -degree polynomials
satisfying the differential relation
Sequences of polynomials, verifying the (▭), nowadays called Appell polynomials, have been well studied because of their remarkable applications not only in different branches of mathematics (, ) but also in theoretical physics and chemistry (, ). In 1936 an initial bibliography was provided by Davis (p. 25). In 1939 Sheffer () introduced a new class of polynomials which extends the class of Appell polynomials; he called these polynomials of type zero, but nowadays they are called Sheffer polynomials. Sheffer also noticed the similarities between Appell polynomials and the umbral calculus, introduced in the second half of the 19th century with the work of such mathematicians as Sylvester, Cayley and Blissard (for examples, see ). The Sheffer theory is mainly based on formal power series. In 1941 Steffensen () published a theory on Sheffer polynomials based on formal power series too. However, these theories were not suitable as they did not provide sufficient computational tools. Afterwards Mullin, Roman and Rota (, , ), using operators method, gave a beautiful theory of umbral calculus, including Sheffer polynomials. Recently, Di Bucchianico and Loeb () summarized and documented more than five hundred old and new findings related to Appell polynomial sequences. In last years attention has centered on finding a novel representation of Appell polynomials. For instance, Lehemer () illustrated six different approaches to representing the sequence of Bernoulli polynomials, which is a special case of Appell polynomial sequences. Costabile (, ) also gave a new form of Bernoulli polynomials, called determinantal form, and later these ideas have been extended to Appell polynomial sequences. In fact, in 2010, Costabile and Longo () proposed an algebraic and elementary approach to Appell polynomial sequences. At the same time, Yang and Youn () also gave an algebraic approach, but with different methods. The approach to Appell polynomial sequences via linear algebra is an easily comprehensible mathematical tool, specially for non-specialists; that is very good because many polynomials arise in physics, chemistry and engineering. The present work concerns with these topics and it is organized as follows: in Section ▭ we mention the Appell method (); in Section ▭ we provide the determinantal approach () and prove the equivalence with other definitions; in Section ▭ classical and non-classical examples are given; in Section ▭, by using elementary tools of linear algebra, general properties of Appell polynomials are provided; in Section ▭ we mention Appell polynomials of second kind (, ) and, in Section ▭ two classical examples are given; in Section ▭ we provide an application to general linear interpolation problem(), giving, in Section ▭, some examples; in Section ▭ the Yang and Youn approach () is sketched; finally, in Section ▭ conclusions close the work.
2. The Appell approach
Let be a sequence of -degree polynomials satisfying the differential relation (▭). Then we have
Remark 1 There is a one-to-one correspondence of the set of such sequences and the set of numerical sequences given by the explicit representation
Equation (▭), in particular, shows explicitly that for each the polynomial is completely determined by and by the choice of the constant of integration .
Remark 2 Given the formal power series
with real coefficients, the sequence of polynomials, , determined by the power series expansion of the product , i.e.
The function is said, by Appell, 'generating function' of the sequence .
Appell also noticed various examples of sequences of polynomials verifying (▭).
He also considered () an application of these polynomial sequences to linear differential equations, which is out of this context.
3. The determinantal approach
Let be with
We give the following
Definition 1 The polynomial sequence defined by
is called Appell polynomial sequence for
Then we have
Theorem 1 If is the Appell polynomial sequence for the differential relation (▭) holds.
Using the properties of linearity we can differentiate the determinant (▭), expand the resulting determinant with respect to the first column and recognize the factor after multiplication of the -th row by and -th column by
Theorem 2 If is the Appell polynomial sequence for we have the equality (▭) with
From (▭), by expanding the determinant with respect to the first row, we obtain the (▭) with given by (▭) and the determinantal form in (▭); this is a determinant of an upper Hessenberg matrix of order (), then setting for we have
By virtue of the previous setting, (▭) implies
and the proof is concluded.
and that for each sequence of Appell polynomials there exist two sequences of numbers and related by (▭).
Corollary 1 If is the Appell polynomial sequence for we have
Follows from Theorem ▭ being
Remark 4 For computation we can observe that is a -order determinant of a particular upper Hessenberg form and it's known that the algorithm of Gaussian elimination without pivoting for computing the determinant of an upper Hessenberg matrix is stable (p. 27).
we have that satisfies the (▭), i.e. is the Appell polynomial sequence for .
with as in (▭). Then we have , where the product is intended in the Cauchy sense, i.e.:
Let us multiply both hand sides of equation
for and, in the same equation, replace functions and by their Taylor series expansion at the origin; then (▭) becomes
By multiplying the series on the left hand side of (▭) according to the Cauchy-product rules, previous equality leads to the following system of infinite equations in the unknown
From the first one of (▭) we obtain the first one of (▭). Moreover, the special form of the previous system (lower triangular) allows us to work out the unknown operating with the first equations, only by applying the Cramer rule:
By transposition of the previous, we have
that is exactly the second one of (▭) after circular row exchanges: more precisely, the -th row moves to the -th position for , the -th row goes to the first position.
Theorem 4 (Circular) If is the Appell polynomial sequence for we have
Follows from Theorem ▭.
Follows ordering the Cauchy product of the developments and with respect to the powers of and recognizing polynomials , expressed in form (▭), as coefficients of .
Follows from Theorem ▭.
4. Examples of Appell polynomial sequences
The following are classical examples of Appell polynomial sequences.
The following are non-classical examples of Appell polynomial sequences.
5. General properties of Appell polynomials
By elementary tools of linear algebra we can prove the general properties of Appell polynomials.
Let , be a polynomial sequence and with
Theorem 5 (Recurrence) is the Appell polynomial sequence for if and only if
Follows observing that the following holds:
In fact, if is the Appell polynomial sequence for , from (▭), we can observe that is a determinant of an upper Hessenberg matrix of order () and, proceeding as in Theorem ▭, we can obtain the (▭).
Corollary 2 If is the Appell polynomial sequence for then
Follows from (▭).
Corollary 3 Let be the space of polynomials of degree and be an Appell polynomial sequence, then is a basis for .
If we have
then, by Corollary ▭, we get
Let be with
Let us consider the Appell polynomial sequences and for and , respectively, and indicate with the polynomial that is obtained replacing in the powers , respectively, with the polynomials Then we have
Theorem 6 The sequences
are sequences of Appell polynomials again.
Follows from the property of linearity of determinant.
Expanding the determinant with respect to the first row we obtainuid79
We observe that
and hence (▭) becomesuid80
Differentiating both hand sides of (▭) and since is a sequence of Appell polynomials, we deduceuid81
Let us, now, introduce the Appell vector.
Definition 3 If is the Appell polynomial sequence for the vector of functions is called Appell vector for .
Then we have
Theorem 7 (Matrix form) Let be a vector of polynomial functions. Then is the Appell vector for if and only if, putting
and the following relation holds
being the inverse matrix of .
If is the Appell vector for the result easily follows from Corollary ▭.
Vice versa, observing that the matrix defined by (▭) is invertible, setting
Theorem 8 (Connection constants) Let and be the Appell vectors for and , respectively. Then
From Theorem ▭ we have
with as in (▭) or, equivalently,
with as in (▭).
Always from Theorem ▭ we get
from which, setting , we have the thesis.
Theorem 9 (Inverse relations) Let be the Appell polynomial sequence for then the following are inverse relations:
Let us remember that
where the coefficients and are related by (▭).
Moreover, setting and , from (▭) we have
and, from (▭) we get
i.e. (▭) are inverse relations.
Theorem 10 (Inverse relation between two Appell polynomial sequences) Let and be the Appell vectors for and , respectively. Then the following are inverse relations:
Follows from Theorem ▭, after observing that
Theorem 11 (Binomial identity) If is the Appell polynomial sequence for we have
Starting by the Definition ▭ and using the identity
We divide, now, each th column, , for and multiply each th row, , for . Thus we finally obtain
Theorem 12 (Generalized Appell identity) Let and be the Appell polynomial sequences for and respectively. Then, if is the Appell polynomial sequence for with
where and are related to and , respectively, by relations similar to (▭), we have
Starting from (▭) we have
Then, applying (▭) and the well-known classical binomial identity, after some calculation, we obtain the thesis.
Theorem 13 (Combinatorial identities) Let and be the Appell polynomial sequences for and respectively. Then the following relations holds:
If is the Appell polynomial sequence for defined as in (▭), from the generalized Appell identity, we have
Theorem 14 (Forward difference) If is the Appell polynomial sequence for we have
The desired result follows from (▭) with .
Theorem 15 (Multiplication Theorem) Let be the Appell vector for .
The following identities hold:
and defined as in (▭).
Theorem 16 (Differential equation) If is the Appell polynomial sequence for then satisfies the linear differential equation:
From Theorem ▭ we have
From Theorem ▭ we find that
and replacing in the (▭) we obtain
Differentiating both hand sides of the last one and replacing with , after some calculation we obtain the thesis.
6. Appell polynomial sequences of second kind
Let and be the finite difference operator (), i.e.:
we define the finite difference operator of order , with , as
meaning and , where is the identity operator.
Let the sequence of falling factorial defined by
we give the following
Definition 4 Let , with . The polynomial sequence
is called Appell polynomial sequence of second kind.
Then, we have
Theorem 17 For Appell polynomial sequences of second kind we get
By the well-known relation ()
applying the operator to the definition (▭) and using the properties of linearity of we have
We can expand the determinant in (▭) with respect to the first column and, after multiplying the -th row by and the -th column by we can recognize the factor .
We can observe that the structure of the determinant in (▭) is similar to that one of the determinant in (▭). In virtue of this it is possible to obtain a dual theory of Appell polynomials of first kind, in the sense that similar properties can be proven ().
For example, the generating function is
where is an invertible formal series of power.
7. Examples of Appell polynomial sequences of second kind
The following are classical examples of Appell polynomial sequences of second kind.
8. An application to general linear interpolation problem
Let be the linear space of real functions defined in the interval continuous and with continuous derivatives of all necessary orders. Let be a linear functional on such that . If in (▭) and respectively in (▭) we set
and will be said Appell polynomial sequences of first or of second kind related to the functional and denoted by and , respectively.
Remark 8 The generating function of the sequence is
and for is
where means that the functional is applied to the argument as a function of .
For if with we have
For , the proof similarly follows. Then, we have
Theorem 18 Let the polynomials
are the unique polynomials of degree less than or equal to such that
where is the Kronecker symbol.
Remark 9 For every linear functional on , are basis for and, , we have
Let us consider a function Then we have the following
Theorem 19 The polynomials
are the unique polynomial of degree such that
Setting the result follows from Theorem ▭.
Now it is interesting to consider the estimation of the remainders
Remark 10 For any
For a fixed we may consider the remainder and as linear functionals which act on and annihilate all elements of . From Peano's Theorem (p. 69) if a linear functional has this property, then it must also have a simple representation in terms of . Therefore we have
Theorem 20 Let the following relations hold
After some calculation, the results follow by Remark ▭ and Peano's Theorem.
Remark 11 (Bounds) If and with then we apply the Hölder's inequality so that
The two most important cases are and
for we have the estimatesuid174
for we have thatuid178
A further polynomial operator can be determined as follows:
for any fixed we consider the polynomial
Then we have the following
Theorem 21 The polynomials , are approximating polynomials of degree for , i.e.:
and for any fixed , from (▭), we have
Proceeding in the same manner we can prove the result for the polynomial .
Remark 12 The polynomials , satisfy the interpolation conditions
9. Examples of Appell interpolation polynomials
Taylor interpolation and classical interpolation on equidistant points:
the polynomials and are, respectively, the Taylor interpolation polynomial and the classical interpolation polynomial on equidistant points;
Bernoulli interpolation of first and of second kind:
the interpolation polynomials and becomeuid197uid198
Bernoulli interpolation of second kind ():
where denote the indefinite summation operator and is defined as the linear operator inverse of the finite difference operator , the interpolation polynomials and becomeuid201uid202
and are the Bernoulli polynomials of second kind ();
Euler and Boole interpolation:
10. The algebraic approach of Yang and Youn
Yang and Youn () also proposed an algebraic approach to Appell polynomial sequences but with different methods. In fact, they referred the Appell sequence, , to an invertible analytic function g(t):
and called Appell vector for the vector
Then, they proved that
being and the generalized Pascal functional matrix of () defined by
Expressing the (▭) in matrix form we have
We have presented an elementary algebraic approach to the theory of Appell polynomials. Given a sequence of real numbers , a polynomial sequence on determinantal form, called of Appell, has been built. The equivalence of this approach with others existing was proven and, almost always using elementary tools of linear algebra, most important properties od Appell polynomials were proven too. A dual theory referred to the finite difference operator has been proposed. This theory has provided a class of polynomials called Appell polynomials of second kind. Finally, given a linear functional , with , and defined
the linear interpolation problem
has been considered and its solution has been expressed by the basis of Appell polynomials related to the functional by (▭). This problem can be extended to appropriate real functions, providing a new approximating polynomial, the remainder of which can be estimated too. This theory is susceptible of extension to the more general class of Sheffer polynomials and to the bi-dimensional case.