Abstract
In this paper, we consider the poly-Cauchy polynomials and numbers of the second kind which were studied by Komatsu. We note that the poly-Cauchy polynomials of the second kind are the special generalized Bernoulli polynomials of the second kind. The purpose of this paper is to give various identities of the poly-Cauchy polynomials of the second kind which are derived from umbral calculus.
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1 Introduction
As is well known, the Bernoulli polynomials of the second kind are defined by the generating function to be
When , are called the Bernoulli numbers of the second kind (see [[1], p.131]).
Let be the polylogarithm factorial function, which is defined by
The poly-Cauchy polynomials of the second kind (, ) are defined by the generating function to be
When , are called the poly-Cauchy numbers of the second kind, defined by
In particular, if we take , then we have
Thus, we note that
where are the Bernoulli polynomials of order α (see [8]) as their numbers [[9], p.257 and p.259].
When , , where are the Bernoulli numbers of order α.
The falling factorial is defined by
where is the signed Stirling number of the first kind.
For , it is well known that
For with , the Frobenius-Euler polynomials of order r are defined by the generating function to be
In this paper, we investigate the properties of the poly-Cauchy numbers and polynomials of the second kind with umbral calculus viewpoint. The purpose of this paper is to give various identities of the poly-Cauchy polynomials of the second kind which are derived from umbral calculus.
2 Umbral calculus
Let C be the complex number field and let ℱ be the set of all formal power series in the variable t:
Let and let be the vector space of all linear functionals on ℙ. is the action of the linear functional L on the polynomial , and we recall that the vector space operations on are defined by , , where c is a complex constant in C. For , let us define the linear functional on ℙ by setting
Then, by (9) and (10), we get
where is Kronecker’s symbol.
For , we have . That is, . The map is a vector space isomorphism from onto ℱ. Henceforth, ℱ denotes 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 call ℱ the umbral algebra and the umbral calculus is the study of umbral algebra. The order of a power series (≠0) is the smallest integer k for which the coefficient of does not vanish. If , then is called a delta series; if , then is called an invertible series (see [10, 14, 15]). For with and , there exists a unique sequence () such that for . The sequence is called the Sheffer sequence for which is denoted by (see [10, 15]).
For and , we have
and
Thus, by (13), we get
Let us assume that . Then the generating function of is given by
where is the compositional inverse of with (see [10, 15]).
For , we have the following equation:
and
where (see [[10], p.21]).
Let us assume that , . Then the transfer formula is given by
For , , let us assume that
Then we have
3 Poly-Cauchy numbers and polynomials of the second kind
From (3), we note that is the Sheffer sequence for the pair
that is,
Because for , using the formula (15), we get
which is the generating function of in (3).
From (21), we have
and
By (22) and (23), we get
By (17) and (21), we get
Now, we observe that
From (25) and (26), we have
which is the same as the expression in (24). From (1), we note that
For , by (19) and (28), we get
Thus, by (29), we see that
Therefore, by (27) and (30), we obtain the following theorem.
Theorem 1 For , , we have
In addition, for , we have
From (18), we note that
where .
By (22) and (23), we get
Thus, from (31) and (32), we have
By (14), (16), and (21), we get
For , the recurrence formula for is given by
By (21) and (34), we get
We observe that
Therefore, by (35) and (36), we obtain the following theorem.
Theorem 2 For , we have
From (11), we note that
where .
Since , we get
By (37) and (38), we see that
From (1), (6), and (38), we note that
It is not difficult to show that . Since , by (40), we obtain the following theorem.
Theorem 3 For , we have
For , we compute
in two different ways.
On the one hand,
On the other hand, we get
Now, we observe that
By (42) and (43), we get
Therefore, by (41) and (44), we obtain the following theorem.
Theorem 4 For , we have
In particular, if we take , then we get
Remark For , it is known that
By (21) and (45), we easily show that
which is a special case of Proposition 2 in [4].
Let us consider the following two Sheffer sequences:
and
Suppose that
By (20), we see that
Therefore, by (47) and (48), we obtain the following theorem.
Theorem 5 For , we have
Remark The Narumi polynomials of order a are defined by the generating function to be
Indeed, , .
By (48) and (49), we get
From (47) and (50), we have
By (1), we easily show that
From (47) and (52), we can derive the following equation:
For (20) and (24), let
where, by (20), we get
We observe that
Thus, by (55) and (56), we get
Therefore, by (54) and (57), we obtain the following theorem.
Theorem 6 For , we have
For , and , let us assume that
From (20), we note that
Therefore, by (58) and (59), we obtain the following theorem.
Theorem 7 For , we have
References
Jordan C: Sur des polynomes analogues aux polynomes de Bernoulli et sur des formules de sommation analogues à celle de MacLaurin-Euler. Acta Sci. Math. 1928/1929, 4: 130-150.
Komatsu T: Poly-Cauchy numbers. RIMS Kokyuroku 2012, 1806: 42-53. Available at http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1806-06.pdf
Komatsu T: Poly-Cauchy numbers. Kyushu J. Math. 2013, 67: 143-153. 10.2206/kyushujm.67.143
Komatsu T: Poly-Cauchy numbers with a q parameter. Ramanujan J. 2013, 31: 353-371. 10.1007/s11139-012-9452-0
Komatsu T: Sums of products of Cauchy numbers including poly-Cauchy numbers. J. Discrete Math. 2013., 2013: Article ID 373927
Komatsu T, Liptai K, Szalay L: Some relationships between poly-Cauchy type numbers and poly-Bernoulli type numbers. East-West J. Math. 2012, 14(2):114-120.
Komatsu T, Luca F: Some relationships between poly-Cauchy numbers and poly-Bernoulli numbers. Ann. Math. Inform. 2013, 41: 99-105.
Nörlund NE: Vorlesungen über Differenzenrechnung. Springer, Berlin; 1924.
Erdélyi A, Magnus W, Overhettinger F, Tricomi FG 3. In Higher Transcendental Functions. McGraw-Hill, New York; 1955.
Roman S: The Umbral Calculus. Dover, New York; 2005.
Araci S, Acikgoz M: A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials. Adv. Stud. Contemp. Math. 2012, 22(3):399-406.
Kim DS, Kim T, Lee SH: Poly-Cauchy numbers and polynomials with umbral calculus wiewpoint. Int. J. Math. Anal. 2013, 7: 2235-2253.
Kim T: Some identities on the q -Euler polynomials of higher order and q -Stirling numbers by the fermionic p -adic integral on . Russ. J. Math. Phys. 2009, 16(4):484-491. 10.1134/S1061920809040037
Carliz L: A note on Bernoulli and Euler polynomials of the second kind. Scr. Math. 1961, 23: 323-330.
Roman SM, Rota G-C: The umbral calculus. Adv. Math. 1978, 27(2):95-188. 10.1016/0001-8708(78)90087-7
Acknowledgements
The authors express their sincere gratitude to the referees for their valuable suggestions and comments. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MOE) (No. 2012R1A1A2003786).
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Kim, D.S., Kim, T. Poly-Cauchy numbers and polynomials of the second kind with umbral calculus viewpoint. Adv Differ Equ 2014, 36 (2014). https://doi.org/10.1186/1687-1847-2014-36
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DOI: https://doi.org/10.1186/1687-1847-2014-36