In this paper, we use the following notations:
Here, we first give some remarks on the normalized polynomials.
Firstly, we introduce some notations which are related to the earlier works by (among others) Carlitz [1, 2], Bodin , Roman [, pp.1-125]. We recall from the work of Bodin : Let p be a prime number and . For , we denote by the finite field having q elements. denotes the multiplicative group of non-zero elements of .
Let be a polynomial of degree exactly d
f is said to be normalized if the first non-zero term in the sequence is equal to 1. Any polynomial g can be written
where f is a normalized polynomial and (cf. ).
We recall the work of Carlitz [, p.60]: Let k be a fixed integer >1 and let denote (complex) numbers such that
Let or 0 and let be distinct numbers. Then consider the functional equation
where denotes a normalized polynomial of degree m (that is, a polynomial with the highest coefficient 1). Here is completely determined by (1); moreover, form an Appell set of polynomials (cf. ).
Theorem 1.1 Let k be a fixed integer >1 and let be complex numbers such that
Let or 0 and let be distinct numbers. Then equation (1) is satisfied by a unique set of normalized polynomials which form an Appell set (cf. ).
Every Appell set satisfies an equation of the form (1) (cf. ).
If is a normalized polynomial, then it satisfies the following formula:
If y is an even positive integer, some normalized polynomials satisfy the following equation (cf. ):
where and denote the normalized polynomials of degree and n, respectively.
We give some Hermite base polynomials of higher order, which are defined as follows (cf.  and ):
where denotes Hermite base Bernoulli polynomials of higher order,
where denotes Hermite base Euler polynomials of higher order and
where denotes Hermite base Genocchi polynomials of higher order.
The proof of polynomials which satisfied (2) was given in various ways. In this paper, we study normalized polynomials which are defined above by using the umbral algebra and umbral calculus methods. We also recall from the work of Roman  the following.
Let P be the algebra of polynomials in the single variable x over the field complex numbers. Let be the vector space of all linear functionals on P. Let
be the action of a linear functional L on a polynomial . Let denote the algebra of formal power series in the variable t over ℂ. The formal power series
defines a linear functional on P by setting
for all . In a special case,
This kind of algebra is called an umbral algebra (cf. ). Any power series
is a linear operator on P defined by
Here, each plays three roles in the umbral calculus: a formal power series, a linear functional and a linear operator. For example, let and
As a linear functional, satisfies the following property:
As a linear operator, satisfies the following property:
Let , be in , then
for all polynomials . The order of a power series is the smallest integer k for which the coefficient of does not vanish. If , . A series for which
is called a delta series. A series for which
is called an invertible series (for details, see ).
Theorem 1.2 [, p.20, Theorem 2.3.6]
Let be a delta series and let be an invertible series. Then there exists a unique sequence of polynomials satisfying the orthogonality conditions
for all .
The sequence in (6) is the Sheffer polynomials for a pair . The Sheffer polynomials for a pair are the Appell polynomials or Appell sequences for (cf. ).
The Appell polynomials are defined by means of the following generating function (cf. ):
The Appell polynomials satisfy the following relations:
the derivative formula
the multiplication formula
In the next section, we need the following generalized multinomial identity.
Lemma 1.3 (Generalized multinomial identity [, p.41, Equation (12m)])
If are commuting elements of a ring (, ), then we have for all real or complex variables α:
the last summation takes places over all positive or zero integers , where
are called generalized multinomial coefficients defined by [, p.27, Equation (10c″)], where and .