Abstract
In this paper, we investigate some properties of polynomials related to Sheffer sequences. Finally, we derive some identities of higher-order Frobenius-Euler polynomials.
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1 Introduction
Let . The higher-order Frobenius-Euler polynomials are defined by the generating function to be
with the usual convention about replacing by . In the special case, , are called the n th Frobenius-Euler numbers of order α ().
From (1) we have
By (2) we get
It is not difficult to show that
Let us define the λ-difference operator as follows:
From (5) we can derive the following equation:
As is well known, the Stirling numbers of the second kind are defined by the generating function to be
and
By (7) and (8), we get
where .
Now, we consider the λ-analogue of the Stirling numbers of the second kind as follows:
and
From (10) and (11), we have
and
From (4) and (5), we have
Let ℱ be the set of all formal power series in the variable t over ℂ with
ℙ indicates the algebra of polynomials in the variable x over ℂ, and is the vector space of all linear functionals on ℙ (see [5, 11]). In [11], denotes the action of the linear functional L on a polynomial , and we remind that the vector space structure on is defined by
where c is a complex constant.
The formal power series
defines a linear functional on ℙ by setting
From (15) and (16), we have
Let . From (17) we have
By (18) we get . It is known in [11] that the map is a vector space isomorphism from onto ℱ. Henceforth, ℱ will denote both the algebra of formal power series in t and the vector space of all linear functionals on ℙ, and so an element of ℱ will be thought of as both a formal power series and a linear functional. We will call ℱ the umbral algebra. The umbral calculus is the study of umbral algebra (see [5, 11]).
The order of the nonzero power series is the smallest integer k for which the coefficient of does not vanish. A series has is called a delta series and a series has is called an invertible series (see [5, 11]). By (16) and (17), we get , and so (see [5, 11]). For and , we have
Let . Then we see that
For and , it is easy to show that
From (19), we can derive the following equation:
By (22) we get
Thus, from (23) we have
Let be polynomials in the variable x with degree n, and let be a delta series and be an invertible series. Then there exists a unique sequence of polynomials with (), where is the Kronecker symbol. The sequence is called the Sheffer sequence for , which is denoted by . If , then is called the associated sequence for . If , then is called the Appell sequence for (see [5, 11]). For , the following equations (25)-(27) are known in [5, 11]:
and
For , we have
where is the compositional inverse of .
In contrast to the higher-order Euler polynomials, the more general higher-order Frobenius-Euler polynomials have never been studied in the context of umbral algebra and umbral calculus.
In this paper, we investigate some properties of polynomials related to Sheffer sequences. Finally, we derive some identities of higher-order Frobenius-Euler polynomials.
2 Associated sequences
Let and . Then, for , we note that
Let us take (). Then we see that , .
From (27), we can derive the associated sequence for as follows:
where .
Therefore, by (34) we obtain, for ,
We get the following:
From (35), we can derive the equation
By (19) we get
and
where is the Stirling numbers of the first kind.
Therefore, by (37) and (38), we obtain the following theorem.
Lemma 1 For , we have
From (31) we note that
And by (32) we get
As is well known, the n th Frobenius-Euler polynomials are defined by the generating function to be
Thus, by (42) we see that . So, we note that
It is easy to show that (see Eq. (17)). Thus, from (42) we have
3 Frobenius-Euler polynomials of order α
From (1) and (31), we note that
and
From (32), we have
Let us take the operator on both sides of (46).
Then we have
and by (46) we get
Therefore, by (48) we obtain the following proposition.
Proposition 2 For and , we have
Thus, we have
Thus, by (49) we get
Let us take . Then, from Proposition 2, we have
Therefore, by (51) we obtain the following corollary.
Corollary 3 For , we have
In the special case, , we have
Let . We get
Thus, from (52) we have
Therefore, by (51), (52) and (53), we obtain the following theorem.
Theorem 4 For and , we have
From (19), we have
and
By (54) and (55), we also get
4 Further remark
Let us take in (34). Then we have , .
For , by (33) we get
where is the Stirling numbers of the second kind.
From (56) we have
where
Thus, by (58) we get
From (57) and (59), we can derive the following equation:
and
Therefore, by (60) and (61), we obtain the following lemma.
Lemma 5 For , we have
Let . Then we write . From (34), we note that
Thus, by (19) and (62), we get
For , we have
From (9), we have
Therefore, by (63), (64) and (65), we obtain the following theorem.
Theorem 6 For , we have
From the recurrence formula of the Appell sequence, we note that
Therefore, by (66) we obtain the following theorem.
Theorem 7 For , we have
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Acknowledgements
Dedicated to Professor Hari M Srivastava.
The authors express their sincere gratitude to the referees for their valuable suggestions and comments. This paper is supported in part by the Research Grant of Kwangwoon University in 2013.
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Kim, D.S., Kim, T., Lee, SH. et al. A note on the higher-order Frobenius-Euler polynomials and Sheffer sequences. Adv Differ Equ 2013, 41 (2013). https://doi.org/10.1186/1687-1847-2013-41
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DOI: https://doi.org/10.1186/1687-1847-2013-41