1. Introduction
The harmonic numbers play an important role in combinatorial problem and numbers theory, and they also frequently appear in the analysis of algorithms and probabilistic statistical calculation. The objective of this paper is using Riordan arrays and generating function to discover identities on the generalized harmonic numbers. The harmonic numbers
are defined by
and the generating function of
is
The first few harmonic numbers are
. The harmonic numbers
have been generalized by several authors. For other generalizations of the harmonic numbers, one can consult [1] [2]. One of them is the generalized harmonic numbers
defined by see [3] [4]:
are integers,
are real numbers, and
.
(1)
For convenience, we recall some definitions involved in the paper as following [5] - [17].
High order Changhee polynomial of the first kind
and the second kind
has the following generating function
(2)
(3)
when
,
and
are called the high order Changhee numbers of the first kind and the second kind.
High order Daehee polynomial of the first kind
and the second kind
has the following generating function
(4)
(5)
when
,
and
are called the high order Daehee numbers of the first kind and the second kind.
High order Apostol Changhee polynomial
has the following generating function
(6)
when
,
are called the high order Apostol Changhee numbers.
High order Apostol Daehee polynomial
has the following generating function
(7)
when
,
are called the high order Apostol Daehee numbers.
High order Apostol Bernoulli polynomial
has the following generating function
(8)
when
,
are called the high order Apostol Bernoulli numbers.
High order Apostol Euler polynomial
has the following generating function
(9)
when
,
are called the high order Apostol Euler numbers.
High order Apostol Genocchi polynomial
has the following generating function
(10)
when
,
are called the high order Apostol Genocchi numbers.
When
in (10), we get the generating function of high order Genocchi polynomials
(11)
The degenerate Changhee polynomials of the second kind
have the following generating function
(12)
The degenerate Euler polynomial
has the following generating function
(13)
The degenerate Genocchi polynomial
has the following generating function
(14)
The degenerate Changhee-Genocchi polynomial of the second kind
has the following generating function
(15)
The degenerate stirling numbers of first kind and second kind have following generating function
(16)
(17)
Generalized Harmonic polynomial
has the following generating function
(18)
with
, and we also obtain when
,
.
Let
, the combinatorial numbers
has the following generating function
(19)
Lemma 1. If
is a Riordan array and
is the generating function of the sequence
, then we have ( [18])
(20)
2. Some Identities Involving Generalized Harmonic Numbers
In this part, using generating functions and coefficient method we discuss some interesting relationships of generalized harmonic numbers
.
Theorem 2.1. Let n is a nonnegative integer, we have
(21)
(22)
Proof. By (1) and (2), we get
Comparing the coefficients of
in both sides of the last equation, we get the identity. (22) can be obtained in the same way.
Corollary 2.1. For
in Theorem 2.1, we obtain the following identities
(23)
(24)
Corollary 2.2. For
in Corollary 2.1, we obtain the following identities
(25)
(26)
Theorem 2.2. Let n is a nonnegative integer, we have
(27)
(28)
Proof. By (1) and (2), we have
Comparing the coefficients of
in both sides of the last equation, we get the identity. (28) can be obtained in the same way.
Corollary 2.3 For
in Theorem 2.2, we obtain the following identities
(29)
(30)
For
in (27), the Theorem 2.3 in [5] is as follows
(31)
Theorem 2.3. Let n be a nonnegative integer, we have
(32)
(33)
Proof. By (1) and (4), we get
Comparing the coefficients of
in both sides of the last equation, we get the identity. (33) can be obtained in the same way.
Corollary 2.4. For
in Theorem 2.3, we obtain the following identities
(34)
(35)
Corollary 2.5. For
in Corollary 2.4, we obtain the following identities
(36)
(37)
Theorem 2.4. Let n be a nonnegative integer, we have
(38)
(39)
Proof. By (1) and (4), we get
Comparing the coefficients of
in both sides of the last equation, we get the identity. (39) can be obtained in the same way.
Corollary 2.7. For
in Theorem 2.4, we obtain the following identities
(40)
(41)
Corollary 2.8. For
in Corollary 2.7, we obtain the following identities
(42)
(43)
Theorem 2.5. Let
be a nonnegative integer, we have
(44)
Proof. By (1), (2) and (4), we get
Comparing the coefficients of
in both sides of the last equation, we get the identity.
Corollary 2.9. For
in Theorem 2.5, we obtain the following identities
(45)
3. Identities about Generalized Harmonic Number
In this part, using Riordan arrays, we derive some new equalities between Generalized Harmonic number
and Apostol Bernoulli polynomials, Apostol Euler polynomials, Apostol Genocchi polynomials.
Theorem 3.1. Let n be a nonnegative integer, we have
(46)
Proof. An interesting Riordan arrays, associated with the
are defined by
(47)
On the one hand, by (8), (47) and Lemma 1, we get
On the other hand, we get
which completes the proof.
Corollary 3.1. For
in Theorem 3.1, we obtain the following identities
(48)
Corollary 3.2. For
in Corollary 3.1, we obtain the following identities
(49)
Corollary 3.3. For
in Corollary 3.2, we obtain the following identities
(50)
Theorem 3.2. Let n be a nonnegative integer, we have
(51)
Proof. By (8), (47) and Lemma 1, we get
which completes the proof.
Corollary 3.4. For
in Theorem 3.2, we obtain the following identities
(52)
Corollary 3.5. For
in Corollary 3.4, we obtain the following identities
(53)
Corollary 3.6. For
in Corollary 3.5, we obtain the following identities
(54)
Theorem 3.3. Let
be a nonnegative integer, we have
(55)
Proof. On the one hand, by (9), (47) and Lemma 1, we get
On the other hand, we get
which completes the proof.
Corollary 3.7. For
in Theorem 3.3, we obtain the following identities
(56)
Corollary 3.8. For
in Corollary 3.7, we obtain the following identities
(57)
Corollary 3.9. For
in Corollary 3.8, we obtain the following identities
(58)
Theorem 3.4. Let n be a nonnegative integer, we have
(59)
Proof. By (10), (47) and Lemma 1, we get
which completes the proof.
Corollary 3.10. For
in Theorem 3.4, we obtain the following identities
(60)
Corollary 3.11. For
in Corollary 3.10, we obtain the following identities
(61)
Corollary 3.12. For
in Corollary 3.11, we obtain the following identities
(62)
Theorem 3.5. Let n be a nonnegative integer, we have
(63)
Proof. By (11), (47) and Lemma 1, we get
which completes the proof.
Corollary 3.13. For
in Theorem 3.5, we obtain the following identities
(64)
Corollary 3.14. For
in Corollary 3.13, we obtain the following identities
(65)
4. Identities about Generalized Harmonic Number
In this part, using generating functions and coefficient method, we derive the new identities involving Generalized Harmonic number
, with Degenerate Changhee polynomials, Degenerate Genocchi polynomials, Degenerate Changhee-Genocchi polynomials, Two kinds of degenerate Stirling numbers and so on.
Theorem 4.1. Let n be a nonnegative integer, we have
(66)
Proof. By (1) and (12), we get
Comparing the coefficients of
in both sides of the last equation, we get the identity.
Theorem 4.2. Let n be a nonnegative integer, we have
(67)
The
-analogue of falling factorial sequence is given by
,
.
Proof. By (14), we get
Comparing the coefficients of
in both sides of the last equation, we get the identity.
Theorem 4.3. Let n be a nonnegative integer, we have
(68)
Proof. By (14), we get
Comparing the coefficients of
in both sides of the last equation, we get the identity.
Theorem 4.4. Let n be a nonnegative integer, we have
(69)
Proof. By (1) and (13), we get
Comparing the coefficients of
in both sides of the last equation, we get the identity.
Theorem 4.5. Let n be a nonnegative integer, we have
(70)
Proof. By (1) and (14), we get
Comparing the coefficients of
in both sides of the last equation, we get the identity.
The combined inversion relations are introduced below (see [19]):
(*)
where
and
are two sequences, expressed as following
Theorem 4.6. Let
be a nonnegative integer, we have
(71)
(72)
Proof. Let
, by (1) and (16), we get
Comparing the coefficients of
in both sides of the last equation, we get the identity.
In addition, from the inversion formula (*), we can get (72).
Theorem 4.7. Let
be a nonnegative integer, we have
(73)
(74)
Proof. By (16), we get
Comparing the coefficients of
in both sides of the last equation, we get the identity.
In addition, from the inversion formula (*), we can get (74).
Supported
Supported by the National Natural Science Foundation of China under Grant 11461050 and Natural Science Foundation of Inner Mongolia 2020MS01020.