Some Identities Involving the Higher-Order Changhee Numbers and Polynomials ()
1. Introduction
Recently, many works have been devoted to the study of Changhee number identities by various method [1] [2] [3] [4]. In [1], D.S.Kim and T.Kim give various identities of the higher-order Changhee numbers and polynomials which are derived from umbral calculas. In [3], J.Kwon consider Witt-type formula for the weighted Changhee numbers and polynomials. In [4], D.S.Kim and T.Kim also introduced the non-linear Changhee differential equations and these differential equations turned out to be very useful for special polynomials and mathematical physics and so on. In the present paper, we make use of the Riordan arrays method in a constructive way to establish some general summation formulas, from which series of Changhee numbers and polynomials identities can be obtained. In particular, besides the Changhee numbers, some identities also involve the Stirling numbers of both kinds, Daehee numbers of both kinds, Lah numbers, Harmonic numbers, Genocchi numbers and polynomials and Euler polynomials. It can be found that no Changhee number identities presented in [1] [2] [3] [4] referred to above have other special combinatorial sequences, and actually, there are not many identities involving both Changhee numbers and other combinatorial numbers in the literature. From this point of view, our results extend the range of Changhee number and polynomials identities.
The study of this paper follows D.S.Kim and T.Kim, s result [1]. Let p be an odd prime number. Throughout this paper,
,
,
will denote the ring of p-adic integers, the field of p-adic numbers and the completion of the algebraic closure of
. The p-adic norm
is normalized as
. Let
be the space of continuous functions on
.
For
, the fermionic p-adic integral on
is defined by Kim to be
. (1)
For
, we have
. (2)
For
with
, the Changhee polynomials of the first kind are
given by the fermionic p-adic integral on
:
, (3)
In special case, when
are called the Changhee numbers of the first kind.
From Equation (1), we note that
, (4)
where
.
For
, Changhee polynomials of the first kind with order r by defined by the generating function as follows:
(5)
where n is a nonnegative integer. In special case, when
are called the Changhee numbers of the first kind with order r.
It is not difficult to show that
. (6)
From Equation (5) and Equation (6), we have
. (7)
For
and
, Changhee polynomials of the second kind with
order
are defined by the generating function to be
(8)
In special case, when
are called the Changhee numbers of the second kind with order r.
Let
be a formal power series in the indeterminate t; then
has the form
. (9)
As usual, the coefficient of
in
may be denoted by
.
A Riordan array is a pair
of formal power series with
. It defines an infinite lower triangular array
according to the rule:
. [5] [6] [7] [8] [9] (10)
Hence we write
.
Lemma 1 If
is an Riordan array and
is the generating function of the sequence
, i.e.,.
or
. Then we have
. [7] (11)
For convenience, we recall some definitions in the paper. The generalized Stirling numbers of the second kind
have the following exponential generating function [6]:
. (12)
The higher-order Changhee polynomials
may be related to the generalized Genocchi polynomials
and the generalized Genocchi numbers
, which are defined by the generating function [7] to be:
, (13)
. (14)
Another two interesting numbers, associated with the higher-order Daehee numbers of both kinds are defined by the generating function [8] to be:
, (15)
. (16)
where
are the higher-order Daehee numbers of the first kind and
are the higher-order Daehee numbers of the second kind.
The generating functions of generalized harmonic polynomials
are given by [5]:
. (17)
The generating functions of higher-order Euler polynomials
are defined by [3]:
. (18)
2. Identities of Changhee Numbers and Special Combinatorial Sequences
Theorem 2.1 For
, the following relations hold:
, (19)
. (20)
Proof An interesting Riordan arrays, associated with the Lah numbers
are defined by
. (21)
Then applying the summation property (11) to the Riordan array (21) and the generating function (5) yields
from which we can establish Equation (19).
Similarly, from the Riordan array (21) and the generating function (8), we can get the Equation (20). Then the proof is complete.
For
, when
, the combinatorial numbers
defined by the following generating functions:
, (22)
then Equation (22) is equivalent to
, (23)
Based on the generating function (22), we obtain the next Riordan arrays, to which we pay particular attention in the next Theorem:
. (24)
Theorem 2.2 For
, we have
. (25)
Proof Applying the summation property (11) to the Riordan array (24) and the generating function (5) yields
Which gives Equation (25).
Theorem 2.3 For
, we have
.(26)
Proof Based on the generating functions (5), (6) and (7), we obtain the next Riordan arrays:
. (27)
To obtain Equation (26), apply the summation property (11) to the Riordan array (27) and the generating functions (23), we have
which gives Equation (26).
Corollary 2.1 For
, the following relations hold:
(28)
Proof From the generating functions of unsigned Stirling numbers of the first kind
,
we obtain the next Riordan arrays
, (29)
From Equation (29), it can be verified that
.
Thus, comparing with Equation (22), we obtain the following connection between the numbers
and the unsigned Stirling numbers of the first
kind:
, Finally, the substitution
in Equation
(26) gives Equation (28).
Corollary 2.2 For
, we have
, (30)
, (31)
(32)
Proof Setting
in Theorem 2.3 gives Equation (30), (31), (32), respectively.
Let
are the generalized Stirling polynomials of the first kind defined by [2]:
,
with the alternative representation:
.
Now, let us define the infinite lower triangular martrice
by:
,
It is easy to show that
may be expressed by the Riordan array:
. (33)
Theorem 2.4 For
, the following relations hold:
(34)
Proof To obtain Equation (34), from the Riordan array (33) and the generating function (14), (17), we have
Similarly, from the Riordan array (33) and the generating function (14), then we can get the Equation (34),
Which completes the proof.
3. Identities Involving the Changhee Polynomials
Theorem 3.1 For
, we have
. (35)
Proof From generating functions (5),we have
Comparing the coefficients of
on both sides, we obtain Equation (35).
Which completes the proof.
Theorem 3.2 For
, we have
(36)
Proof From the generating functions (5), we have
Comparing the coefficients of
on both sides, we obtain Equation (36).
Which completes the proof.
Corollary 3.1 The following relations hold:
(37)
Proof Setting
in Theorem 3.2, we can get Equation (37).
The generalized harmonic numbers
have the following exponential generating function:
Let us define
to be an infinite lower triangular array, it is easy to show that
does not constitute a Riordan array but
, (38)
is a Riordan array.
Theorem 3.3 Let
, then
(39)
Proof From the Riordan array (38) and the generating function (18), we have
Which completes the proof.
Corollary 3.2 When
, we have
Theorem 3.4 For
, the higher-order Euler polynomials
may be expressed by means of the Changhee polynomials of both kinds,
(40)
(41)
Proof From Equation (12), we note that the generalized Stirling numbers of second kind
may be expressed by the Riordan array:
(42)
To obtain Equation (40), from the Riordan array (42) and the generating function (5), we have
Similarly, from the Riordan array (42) and the generating function (18), we can get the Equation (41).Then the proof is complete.
For a sequence
and
, Using the inverse relation
, we get
,
.
Theorem 2.5 Let
, we have
. (43)
Proof Firstly, from Equation (11), (38) and (13), we have
Secondly, from Equation (15) and (42), we have
Acknowledgements
The research is supported by the Natural Science Foundation of China under Grant 11461050 and Natural Science Foundation of Inner Mongolia under Grant 2016MS0104.