In this section, we provide generating functions, related to nonnegative real parameters, for the generalized Eulerian type polynomials and numbers, that is, the so-called generalized Apostol type Frobenius Euler polynomials and numbers. We derive fundamental properties, recurrence relations and many new identities for these polynomials and numbers based on the generating functions, functional equations and differential equations.
These polynomials and numbers have many applications in many branches of mathematics.
The following definition gives us a natural generalization of the Eulerian polynomials.
Definition 4.1 Let (), , and . The generalized Eulerian type polynomials are defined by means of the following generating function:
(13)
( when ; when ).
By substituting into (13), we obtain
where denotes generalized Eulerian type numbers.
By substituting into (13), we have
(14)
From the above equation, we find that
The generalized Eulerian type polynomials of order m, are defined by means of the following generating function:
with, of course
where denotes the generalized Eulerian type numbers of order m.
Remark 4.1 Substituting into (13), we have
a result due to Kurt and Simsek [16]. In the special case when and , the generalized Eulerian type polynomials are reduced to the Eulerian polynomials or Frobenius Euler polynomials which are defined by means of the following generating function:
(15)
with, of course, denotes the so-called Eulerian numbers (cf. [9, 10, 14, 17–23]). Substituting , into (15), we have
where denotes Euler polynomials which are defined by means of the following generating function:
(16)
(cf. [1–56]).
Throughout this paper, we assume that , , and .
The following elementary properties of the generalized Eulerian type polynomials and numbers are derived from their generating functions in (13) and (14).
Theorem 4.2 (Recurrence relation for the generalized Eulerian type numbers)
For , we have . For , following the usual convention of symbolically replacing by , we have
Proof By using (13), we obtain
Comparing the coefficients of on both sides of the above equation, we arrive at the desired result. □
By differentiating both sides of Eq. (13) with respect to the variable x, we obtain the following higher order differential equation:
(17)
From this equation, we arrive at higher order derivative of the generalized Eulerian type polynomials by the following theorem.
Theorem 4.3 Let with . Then we have
Proof Combining (13) and (17), we have
From the above equation, we get
Comparing the coefficients of on both sides of the above equation, we arrive at the desired result. □
Remark 4.2 Setting in Theorem 4.3, we have
In their special case when and , Theorem 4.3 is reduced to the following well-known result:
(cf. [[9], Eq. (3.5)]). Substituting into the above equation, we have
(cf. [[9], Eq. (3.5)], [16]).
Theorem 4.4 The following explicit representation formula holds true:
Proof By using (13) and the umbral calculus convention, we obtain
From the above equation, we get
Comparing the coefficients of on both sides of the above equation, we arrive at the desired result. □
Remark 4.3 By substituting and into Theorem 4.4, we have
(18)
(cf. [[9], Eq. (3.3)]). By setting in the above equation, we have
a result due to Shiratani [51]. By using (18), Carlitz [9] studied on the Mirimanoff polynomial which is defined by
By applying Theorem 4.4, one may generalize the Mirimanoff polynomial.
Theorem 4.5 The following explicit representation formula holds true:
(19)
Proof By using (13), we get
From the above equation, we obtain
Comparing the coefficients of on both sides of the above equation, we arrive at the desired result. □
Remark 4.4 Substituting and into (19), we have
(cf. [9, 10, 14, 16–23]).
Remark 4.5 From (19), we easily get
where after expansion of the right member, is replaced by , we use this convention frequently throughout of this paper.
Theorem 4.6 The following formula holds true:
(20)
Proof By using (13), we have
Therefore,
Comparing the coefficients of on both sides of the above equation, we arrive at the desired result. □
Remark 4.6 In the special case when and , Eq. (20) is reduced to the following result:
(cf. [[9], Eq. (3.6)]). Substituting into the above equation, we get the following well-known result:
(21)
By using (13), we define the following functional equation:
(22)
Theorem 4.7 The following formula holds true:
(23)
Proof Combining (22) and (20), we easily arrive at the desired result. □
Remark 4.7 In the special case when and , Eq. (23) is reduced to the following result:
(cf. [[9], Eq. (3.17)]).
Theorem 4.8 The following formula holds true:
Proof By using (13), we obtain
From the above equation, we get
Therefore
Comparing the coefficients of on both sides of the above equation, we arrive at the desired result. □
Remark 4.8 In their special case when and , Theorem 4.8 is reduced to the following result:
(cf. [[9], Eq. (3.7)]). Substituting into the above equation, we get the following well-known result:
(cf. [[9], Eq. (3.7)], [6, 27, 50, 51]).
Theorem 4.9
Proof By using (13), we get
Comparing the coefficients of on both sides of the above equation, we arrive at the desired result. □
Remark 4.9 When and , Theorem 4.9 is reduced to the following result:
(cf. [[9], Eq. (3.17)]).
4.1 Multiplication formulas for normalized polynomials
In this section, using generating functions, we derive multiplication formulas in terms of the normalized polynomials which are related to the generalized Eulerian type polynomials, the Bernoulli and the Euler polynomials.
Theorem 4.10 (Multiplication formula)
Let . Then we have
(24)
where and denote the Eulerian polynomials and numbers, respectively.
Proof Substituting into (13), we have
(25)
By using the following finite geometric series,
on the right-hand side of (25), we obtain
From this equation, we get
Now by making use of the generating functions (13) and (15) on the right-hand side of the above equation, we obtain
Therefore,
By equating the coefficients of on both sides, we get
Finally, by replacing x by yx on both sides of the above equation, we arrive at the desired result. □
Remark 4.10 By substituting into Theorem 4.10, for , we obtain
(26)
By substituting and into the above equation, we arrive at the multiplication formula for the Eulerian polynomials
(27)
(cf. [10], [[9], Eq. (3.12)]). If , then the above equation reduces to the well known multiplication formula for the Euler polynomials: for y is an odd positive integer, we have
(28)
where denotes the Euler polynomials in the usual notation. If y is an even positive integer, we have
(29)
where and denote the Bernoulli polynomials and Euler polynomials, respectively, (cf. [17, 55]).
To prove the multiplication formula of the generalized Apostol Bernoulli polynomials, we need the following generating function which is defined by Srivastava et al. [[13], pp.254, Eq. (20)]:
Definition 4.11 Let with , and . Then the generalized Bernoulli polynomials of order are defined by means of the following generating functions:
(30)
where
It is easily observe that
(cf. [6, 13, 19, 20, 23, 32, 35, 36, 46, 48, 49, 54]). Moreover, by substituting and into (30), then we arrive at the Apostol-Bernoulli polynomials , which have been introduced and investigated by many mathematicians (cf. [24], [16, 20, 23, 29, 30, 34, 49, 52]). When and into (30), and reduce to the classical Bernoulli numbers and the classical Bernoulli polynomials, respectively, (cf. [1–56]).
Theorem 4.12 Let . Then we have
Proof Substituting and into (30), we get
Therefore,
Comparing the coefficients of on both sides of the above equation, we get
By replacing x by yx on both sides of the above equation, we arrive at the desired result. □
Remark 4.11 Kurt and Simsek [32] proved multiplication formula for the generalized Bernoulli polynomials of order α. When and into Theorem 4.12, we have the multiplication formula for the Bernoulli polynomials given by
(31)
(cf. [2, 9, 13, 17, 20, 23, 24, 33, 34, 36, 46]).
If f is a normalized polynomial which satisfies the formula
(32)
then f is the y th degree Bernoulli polynomial due to (31) (cf. [17, 56]). According to Nielsen [17], if a normalized polynomial satisfies (31) for a single value of , then it is identical with . Consequently, if a normalized polynomial satisfies (26) for a single value of , then it is identical with . The formula (29) is different. Therefore, for y is an even positive integer, Carlitz [[17], Eq. (1.4)] considered the following equation:
where and denote the normalized polynomials of degree and n, respectively. More precisely, as Carlitz has pointed out [[17], p.184], if y is a fixed even integer ≥2 and is an arbitrary normalized polynomial of degree n, then (29) determines as a normalized polynomial of degree . Thus, for a single value y, (29) does not suffice to determine the normalized polynomials and .
Remark 4.12 According to (32), the set of normalized polynomials is an Appell set, (cf. [17]).
We now modify (13) as follows:
(33)
where (, ).
The polynomial is a normalized polynomial of degree m in x. The polynomial may be called Eulerian polynomials with parameter ξ. In particular we note that
since for , , Eq. (33) reduces to the generating function for the Euler polynomials.
By means of Eq. (24), it is easy to verify the following multiplication formulas.
If y is an odd positive integer, then we have
(34)
where
If y is an even positive integer, then we have
(35)
where
where denotes the generalized Euler polynomials, which are defined by means of the following generating function:
(cf. [6, 13, 23, 31, 32, 35, 36, 49, 54]).
Remark 4.13 If we set and , then (34) and (35) reduce to the following multiplication formulas, respectively:
(cf. [[17], Eq. (3.3)]) and
Let and be normalized polynomials in the usual way. Carlitz [[17], Eq. (3.4)] defined the following equation:
where ρ is a fixed primitive r th root of unity, , .
Remark 4.14 If we set , and , then (34) and (35) reduce to (29) and (28).
Remark 4.15 Walum [56] defined multiplication formula for periodic functions as follows:
(36)
where f is periodic with period 1 and under the summation sign indicates that j runs through a complete system of residues mody. Formulas (32), (36) and other multiplication formulas related to periodic functions and normalized polynomials occur in Franel’s formula, in the theory of the Dedekind sums and Hardy-Berndt sums, in the theory of the zeta functions and L-functions and in the theory of periodic bounded variation, (cf. [25, 56]).
4.2 Recurrence relation for the generalized Eulerian type polynomials
In this section, we are going to differentiate (13) with respect to the variable t to derive a recurrence relation for the generalized Eulerian type polynomials. Therefore, we obtain the following partial differential equations:
or
where is defined by
(37)
where . The polynomials are defined by means of the following generating function:
where
with, of course
If we substitute and into (37), then we obtain
By using the above partial differential equations, we obtain recurrence relations for the generalized Eulerian type polynomials by the following theorem:
Theorem 4.13 Let . We have
or
where denotes the generalized Bernoulli numbers.