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Some identities on generalized harmonic numbers and generalized harmonic functions

  • Dae San Kim , Hyekyung Kim EMAIL logo and Taekyun Kim EMAIL logo
Published/Copyright: May 25, 2023
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 56 Issue 1

Abstract

The harmonic numbers and generalized harmonic numbers appear frequently in many diverse areas such as combinatorial problems, many expressions involving special functions in analytic number theory, and analysis of algorithms. The aim of this article is to derive some identities involving generalized harmonic numbers and generalized harmonic functions from the beta functions F n ( x ) = B ( x + 1 , n + 1 ) , ( n = 0 , 1 , 2 , … ) using elementary methods. For instance, we show that the Hurwitz zeta function ζ ( x + 1 , r ) and r ! are expressed in terms of those numbers and functions, for every r = 2 , 3 , 4 , 5 .

Keywords: generalized harmonic numbers of order α; generalized harmonic function
MSC 2010: 11B68; 11B83; 26A42

1 Introduction

The harmonic numbers and generalized harmonic numbers have been studied in connection with combinatorial problems, special functions in analytic number theory, and analysis of algorithms [1–14]. As a generalization of harmonic numbers in another direction, the hyperharmonic numbers were introduced by Conway and Guy in 1996 [4,5]. Recently, the study of degenerate versions of many special numbers and special polynomials regained interests of some mathematicians, which began with the pioneering work of Carlitz on the degenerate Bernoulli and degenerate Euler numbers (see [6,7] and the references therein). In this line of study, the degenerate harmonic numbers and the degenerate hyperharmonic numbers were investigated in [8,9].

The aim of this article is to derive some identities relating to generalized harmonic numbers H n ( α ) (see (7)) and generalized harmonic functions H n ( x , α ) (see (11)) by using elementary means like integration, differentiation, and binomial theorem. In more detail, by making use of the beta functions F n ( x ) = B ( x + 1 , n + 1 ) , ( n = 0 , 1 , 2 , … ) , we show that, for every r = 2 , 3 , 4 , 5 ,

∑ k = 0 n n k ( − 1 ) k 1 ( k + x + 1 ) r , ζ ( x + 1 , r ) , r !

are all expressed in terms of the generalized harmonic functions and the generalized harmonic numbers. For example, we show that

2 H n ( x , 3 ) + 3 H n ( x , 1 ) H n ( x , 2 ) + ( H n ( x , 1 ) ) 3 3 ! x + n + 1 n ( x + 1 ) = ∑ k = 0 n n k ( − 1 ) k 1 ( x + k + 1 ) 4 , 1 3 ! ∑ n = 0 ∞ ∑ k = 0 n n k ( − 1 ) k x + k + 1 k ( x + 1 ) ( 2 H k ( x , 3 ) + 3 H k ( x , 1 ) H k ( x , 2 ) + ( H k ( x , 1 ) ) 3 ) = ζ ( x + 1 , 4 ) , ∑ n = 1 ∞ 2 H n + 1 ( 3 ) + 3 H n + 1 H n + 1 ( 2 ) + ( H n + 1 ) 3 n ( n + 1 ) = 4 ! ,

where ζ ( x , s ) is the Hurwitz zeta function (see (4)).

In principle, the methods employed in this article could be continued further for r = 6 , 7 , 8 , … . However, this requires determination of an explicit expression for F n ( r − 1 ) ( x ) in terms of the generalized harmonic functions.

Similar method to the present article is exploited in [10] to find summation formulas involving harmonic numbers and the generalized harmonic numbers (called harmonic numbers of order r there). However, the beta functions G n ( x ) = 1 2 B ( n + 1 , x + 1 2 ) , ( n = 0 , 1 , 2 , … ) are used in [10] instead of F n ( x ) = B ( x + 1 , n + 1 ) , ( n = 0 , 1 , 2 , … ) in this article. In addition, the generalized harmonic functions are not introduced in [10], whereas they play an important role in this article.

The motivation of this article is very simple. We note that G n ( x ) = ∫ 0 1 ( 1 − t 2 ) n t x d t and F n ( x ) = ∫ 0 1 ( 1 − t ) n t x d t . So it is natural to ask what if we replace ( 1 − t 2 ) n by ( 1 − t ) n in the integrand of G n ( x ) . This question led us to carry out the present research and the introduction of the generalized harmonic functions.

Interested readers can refer to [11,12] for the existing literature on the same topic. Indeed, some identities are obtained about certain finite or infinite series involving harmonic numbers and the generalized harmonic numbers by applying an algorithmic method to a known summation formula for the hypergeometric function 5 F 4 ( 1 ) in [11] and by applying a differential operator to a known identity in [12]. For the rest of this section, we recall the facts that are needed throughout this article.

For s ∈ C , the Riemann zeta function is defined by

ζ ( s ) = ∑ n = 1 ∞ 1 n s , ( Re ( s ) > 1 ) , ( see [ 13 , 14 , 15 ] ) .

As is well known, we have

(1) ζ ( 2 n ) = ( − 1 ) n + 1 B 2 n 2 ( 2 n ) ! ( 2 π ) 2 n , ( n ∈ N ) , ( see [ 15 ] ) ,

where B n are the Bernoulli numbers defined by

(2) t e t − 1 = ∑ n = 0 ∞ B n t n n ! = 1 − 1 2 x + 1 6 x 2 2 ! − 1 30 x 4 4 ! + 1 42 x 6 6 ! − 1 30 x 8 8 ! + ⋯ , ( see [ 1–3,5–15 ] ) .

Thus, we see from (1) and (2) that

(3) ζ ( 2 ) = π 2 6 , ζ ( 4 ) = π 4 90 , ζ ( 6 ) = π 6 945 , ζ ( 8 ) = π 8 9,450 , … .

More generally, the Hurwitz zeta function is defined by

(4) ζ ( x , s ) = ∑ n = 0 ∞ 1 ( n + x ) s , ( Re ( s ) > 1 , x > 0 ) , ( see [ 15 ] ) .

The harmonic numbers are defined by

(5) H 0 = 0 , H n = 1 + 1 2 + ⋯ + 1 n , ( n ∈ N ) , ( see [ 3,5,7–9 ] ) .

The generating function of harmonic numbers is given by

(6) − 1 1 − x log ( 1 − x ) = ∑ n = 1 ∞ H n x n , ( see [ 3,5,8,9 ] ) .

For α ∈ N , the generalized harmonic numbers of order α are defined by

(7) H 0 ( α ) = 0 , H n ( α ) = 1 + 1 2 α + 1 3 α + ⋯ + 1 n α , ( see [ 2,9,11,12 ] ) .

For s ∈ C with Re ( s ) > 0 , the gamma function is given by

(8) Γ ( s ) = ∫ 0 ∞ e − t t s − 1 d t , ( see [ 6,14,15 ] ) ,

with Γ ( s + 1 ) = s Γ ( s ) and Γ ( 1 ) = 1 . For Re ( α ) > 0 and Re ( β ) > 0 , the beta function is given by

(9) B ( α , β ) = ∫ 0 1 t α − 1 ( 1 − t ) β − 1 d t , ( see [ 6,14,15 ] ) .

From (8) and (9), we have

(10) B ( α , β ) = Γ ( α ) Γ ( β ) Γ ( α + β ) , ( see [ 15 ] ) .

Finally, the following binomial inversion theorem is well known.

Theorem 1.1

For any integer n ≥ 0 , we have

b n = ∑ k = 0 n n k ( − 1 ) k a k ⇔ a n = ∑ k = 0 n n k ( − 1 ) k b k .

In this article, we derive some expressions for ∑ k = 0 n n k ( − 1 ) k 1 ( k + x + 1 ) r , ζ ( x + 1 , r ) , and r ! in terms of the generalized harmonic functions and the generalized harmonic numbers using elementary methods like the binomial inversion, differentiation, and integration.

2 Some identities on generalized harmonic numbers and generalized harmonic functions

For every α ∈ N , we consider the generalized harmonic function H n ( x , α ) defined by

(11) H n ( x , α ) = ∑ k = 0 n 1 ( k + x + 1 ) α , ( x ≥ 0 ) , ( see [ 2,11,12,16,17 ] ) .

Note that

(12) d d x H n ( x , α ) = − α H n ( x , α + 1 ) .

In particular, for x = 0 , we have H n ( 0 , α ) = H n + 1 ( α ) .

For x > − 1 and n = 0 , 1 , 2 , … , we let

(13) F n ( x ) = ∫ 0 1 ( 1 − t ) n t x d t = B ( x + 1 , n + 1 ) = Γ ( x + 1 ) Γ ( n + 1 ) Γ ( x + n + 2 ) = n ! ( x + n + 1 ) ( x + n ) ⋯ ( x + 1 ) = 1 x + n + 1 n ( x + 1 ) .

By the binomial theorem, (13) is also equal to

(14) F n ( x ) = ∫ 0 1 ( 1 − t ) n t x d t = ∑ k = 0 n n k ( − 1 ) k ∫ 0 1 t k + x d t = ∑ k = 0 n n k ( − 1 ) k 1 x + k + 1 .

We will use the following lemma repeatedly throughout this article.

Lemma 2.1

Let F n ( x ) = B ( x + 1 , n + 1 ) . Then, for any positive integer r, we have

( a ) F n ( r ) ( x ) = ∫ 0 1 ( 1 − t ) n ( log t ) r t x d t = r ! ∑ k = 0 n n k ( − 1 ) k − r 1 ( x + k + 1 ) r + 1 , ( b ) F n ′ ( x ) = − H n ( x , 1 ) F n ( x ) , ( c ) ∑ n = 1 ∞ 1 n ∑ k = 0 n n k ( − 1 ) k 1 ( k + 1 ) r = r .

Proof

  1. Follows from (14).

  2. From (13), we obtain

    F n ′ ( x ) = d d x F n ( x ) = d d x n ! ( x + n + 1 ) ( x + n ) ⋯ ( x + 1 ) = − ∑ k = 0 n 1 k + x + 1 ⋅ n ! ( x + n + 1 ) ( x + n ) ⋯ ( x + 1 ) = − H n ( x , 1 ) F n ( x ) .

  3. The left-hand side of (c) is equal to

    ∑ n = 1 ∞ 1 n ∫ 0 1 ⋯ ∫ 0 1 ( 1 − x 1 x 2 ⋯ x r ) n d x 1 d x 2 ⋯ d x r = − ∫ 0 1 ⋯ ∫ 0 1 ( log x 1 + ⋯ + log x r ) d x 1 d x 2 ⋯ d x r = r ,

    where the integrals are r -multiple ones.□

From Lemma 2.1, (a), (b), and (13), we have

(15) ∑ k = 0 n n k ( − 1 ) k 1 ( k + x + 1 ) 2 = H n ( x , 1 ) F n ( x ) = H n ( x , 1 ) x + n + 1 n ( x + 1 ) .

In particular, for x = 0 , we obtain

(16) ∑ k = 0 n n k ( − 1 ) k 1 ( k + 1 ) 2 = 1 n + 1 H n + 1 , ( n ≥ 0 ) ,

which, by Lemma 2.1 (c), yields

∑ n = 1 ∞ 1 n ( n + 1 ) H n + 1 = 2 .

Therefore, from (15) and (16), by binomial inversion, we obtain the following theorem.

Theorem 2.2

For n ≥ 0 , we have

∑ k = 0 n n k ( − 1 ) k H k ( x , 1 ) x + k + 1 k ( x + 1 ) = 1 ( n + x + 1 ) 2 , a n d ∑ k = 0 n n k ( − 1 ) k H k + 1 k + 1 = 1 ( n + 1 ) 2 .

We note from Theorem 2.2, (3) and (4) that

∑ n = 0 ∞ ∑ k = 0 n n k ( − 1 ) k H k ( x , 1 ) x + k + 1 k ( x + 1 ) = ∑ n = 0 ∞ 1 ( n + x + 1 ) 2 = ζ ( x + 1 , 2 ) , ∑ n = 0 ∞ ∑ k = 0 n n k ( − 1 ) k H k + 1 k + 1 = ∑ n = 0 ∞ 1 ( n + 1 ) 2 = π 2 6 .

From Lemma 2.1 (b), (12) and (13), we note that

(17) F n ( 2 ) ( x ) = d 2 d x 2 F n ( x ) = ( − 1 ) d d x ( H n ( x , 1 ) F n ( x ) ) = ( − 1 ) 2 H n ( x , 2 ) F n ( x ) + ( − 1 ) 2 ( H n ( x , 1 ) ) 2 F n ( x ) = ( − 1 ) 2 ( H n ( x , 2 ) + ( H n ( x , 1 ) ) 2 ) 1 x + n + 1 n ( x + 1 )

and that

(18) F n ( 3 ) ( x ) = d 3 d x 3 F n ( x ) = d d x ( ( − 1 ) 2 H n ( x , 2 ) F n ( x ) + ( − 1 ) 2 ( H n ( x , 1 ) ) 2 F n ( x ) ) = 2 ( − 1 ) 3 H n ( x , 3 ) F n ( x ) + ( − 1 ) 3 H n ( x , 2 ) H n ( x , 1 ) F n ( x ) + 2 ( − 1 ) 3 H n ( x , 1 ) H n ( x , 2 ) F n ( x ) + ( − 1 ) 3 ( H n ( x , 1 ) ) F n ( x ) = 2 ( − 1 ) 3 H n ( x , 3 ) F n ( x ) + 3 ( − 1 ) 3 H n ( x , 1 ) H n ( x , 2 ) F n ( x ) + ( − 1 ) 3 ( H n ( x , 1 ) ) 3 F n ( x ) = ( − 1 ) 3 ( 2 H n ( x , 3 ) + 3 H n ( x , 1 ) H n ( x , 2 ) + ( H n ( x , 1 ) ) 3 ) 1 x + n + 1 n ( x + 1 ) .

By Lemma 2.1 (a), we obtain

(19) F n ( 2 ) ( x ) = 2 ! ∑ k = 0 n n k ( − 1 ) k − 2 1 ( x + k + 1 ) 3 , F n ( 3 ) ( x ) = 3 ! ∑ k = 0 n n k ( − 1 ) k − 3 1 ( x + k + 1 ) 4 .

Therefore, by (17)–(19), we obtain the following theorem.

Theorem 2.3

For n ≥ 0 , we have

H n ( x , 2 ) + ( H n ( x , 1 ) ) 2 x + n + 1 n ( x + 1 ) = 2 ! ∑ k = 0 n n k ( − 1 ) k 1 ( x + k + 1 ) 3

and

2 H n ( x , 3 ) + 3 H n ( x , 1 ) H n ( x , 2 ) + ( H n ( x , 1 ) ) 3 x + n + 1 n ( x + 1 ) = 3 ! ∑ k = 0 n n k ( − 1 ) k 1 ( x + k + 1 ) 4 .

For x = 0 , we also have

2 ! ∑ k = 0 n n k ( − 1 ) k ( k + 1 ) 3 = H n + 1 ( 2 ) + ( H n + 1 ) 2 n + 1

and

3 ! ∑ k = 0 n n k ( − 1 ) k ( k + 1 ) 4 = 2 H n + 1 ( 3 ) + 3 H n + 1 H n + 1 ( 2 ) + ( H n + 1 ) 3 n + 1 .

From Theorem 2.3, by binomial inversion, we obtain

(20) 1 2 ! ∑ k = 0 n n k ( − 1 ) k H k ( x , 2 ) + ( H k ( x , 1 ) ) 2 x + k + 1 k ( x + 1 ) = 1 ( x + n + 1 ) 3 ,

(21) 1 3 ! ∑ k = 0 n n k ( − 1 ) k 2 H k ( x , 3 ) + 3 H k ( x , 1 ) H k ( x , 2 ) + ( H k ( x , 1 ) ) 3 x + k + 1 k ( x + 1 ) = 1 ( x + n + 1 ) 4 .

By (20) and (4), we obtain

1 2 ! ∑ n = 0 ∞ ∑ k = 0 n n k ( − 1 ) k x + k + 1 k ( x + 1 ) ( H k ( x , 2 ) + ( H k ( x , 1 ) ) 2 ) = ζ ( x + 1 , 3 ) .

For x = 0 , we have

1 2 ! ∑ n = 0 ∞ ∑ k = 0 n n k ( − 1 ) k k + 1 ( H k + 1 ( 2 ) + ( H k + 1 ) 2 ) = ζ ( 3 ) .

In addition, from (21) and (4), we also have

1 3 ! ∑ n = 0 ∞ ∑ k = 0 n n k ( − 1 ) k x + k + 1 k ( x + 1 ) ( 2 H k ( x , 3 ) + 3 H k ( x , 1 ) H k ( x , 2 ) + ( H k ( x , 1 ) ) 3 ) = ζ ( x + 1 , 4 ) .

For x = 0 , from (3), we obtain

1 3 ! ∑ n = 0 ∞ ∑ k = 0 n n k ( − 1 ) k k + 1 ( 2 H k + 1 ( 3 ) + 3 H k + 1 H k + 1 ( 2 ) + ( H k + 1 ) 3 ) = ∑ n = 0 ∞ 1 ( n + 1 ) 4 = π 4 90 .

From Theorem 2.3 and Lemma 2.1 (c), we note that

(22) 1 2 ! ∑ n = 1 ∞ H n + 1 ( 2 ) + ( H n + 1 ) 2 n ( n + 1 ) = ∑ n = 1 ∞ 1 n ∑ k = 0 n n k ( − 1 ) k ( k + 1 ) 3 = 3

and that

(23) 1 3 ! ∑ n = 1 ∞ 2 H n + 1 ( 3 ) + 3 H n + 1 H n + 1 ( 2 ) + ( H n + 1 ) 3 n ( n + 1 ) = ∑ n = 1 ∞ 1 n ∑ k = 0 n n k ( − 1 ) k 1 ( k + 1 ) 4 = 4 .

Therefore, by (22) and (23), we obtain the following corollary.

Corollary 2.4

For n ≥ 1 , we have

∑ n = 1 ∞ H n + 1 ( 2 ) + ( H n + 1 ) 2 n ( n + 1 ) = 3 ! , ∑ n = 1 ∞ 2 H n + 1 ( 3 ) + 3 H n + 1 H n + 1 ( 2 ) + ( H n + 1 ) 3 n ( n + 1 ) = 4 ! .

From (21), Lemma 2.1 (b), (12), and (13), we have

(24) F n ( 4 ) ( x ) = d d x ( 2 ( − 1 ) 3 H n ( x , 3 ) F n ( x ) + 3 ( − 1 ) 3 H n ( x , 1 ) H n ( x , 2 ) F n ( x ) + ( − 1 ) 3 ( H n ( x , 1 ) ) 3 F n ( x ) )

= 6 ( − 1 ) 4 H n ( x , 4 ) F n ( x ) + 2 ( − 1 ) 4 H n ( x , 3 ) H n ( x , 1 ) F n ( x ) + 3 ( − 1 ) 4 H n ( x , 2 ) H n ( x , 2 ) F n ( x ) + 6 ( − 1 ) 4 H n ( x , 1 ) H n ( x , 3 ) F n ( x ) + 3 ( − 1 ) 4 ( H n ( x , 1 ) ) 2 H n ( x , 2 ) F n ( x ) + 3 ( − 1 ) 4 ( H n ( x , 1 ) ) 2 H n ( x , 2 ) F n ( x ) + ( − 1 ) 4 ( H n ( x , 1 ) ) 4 F n ( x ) = ( − 1 ) 4 ( 6 H n ( x , 4 ) + 8 H n ( x , 3 ) H n ( x , 1 ) + 3 ( H n ( x , 2 ) ) 2 + 6 ( H n ( x , 1 ) ) 2 H n ( x , 2 ) + ( H n ( x , 1 ) ) 4 ) F n ( x ) = ( − 1 ) 4 6 H n ( x , 4 ) + 8 H n ( x , 3 ) H n ( x , 1 ) + 3 ( H n ( x , 2 ) ) 2 + 6 ( H n ( x , 1 ) ) 2 H n ( x , 2 ) + ( H n ( x , 1 ) ) 4 x + n + 1 n ( x + 1 )

and

(25) F n ( 4 ) ( 0 ) = ( − 1 ) 4 6 H n + 1 ( 4 ) + 8 H n + 1 ( 3 ) H n + 1 + 3 ( H n + 1 ( 2 ) ) 2 + 6 ( H n + 1 ) 2 H n + 1 ( 2 ) + ( H n + 1 ) 4 n + 1 .

On the other hand, by Lemma 2.1 (a), we obtain

(26) F n ( 4 ) ( x ) = 4 ! ∑ k = 0 n n k ( − 1 ) k − 4 1 ( k + x + 1 ) 5 = ∫ 0 1 ( 1 − t ) n ( log t ) 4 t x d t

and

(27) F n ( 4 ) ( 0 ) = 4 ! ∑ k = 0 n n k ( − 1 ) k − 4 1 ( k + 1 ) 5 = ∫ 0 1 ( 1 − t ) n ( log t ) 4 d t .

By (24)–(27), we obtain

(28) ∑ k = 0 n n k ( − 1 ) k 1 ( k + x + 1 ) 5 = 1 4 ! ∫ 0 1 ( 1 − t ) n ( log t ) 4 t x d t = 6 H n ( x , 4 ) + 8 H n ( x , 3 ) H n ( x , 1 ) + 3 ( H n ( x , 2 ) ) 2 + 6 ( H n ( x , 1 ) ) 2 H n ( x , 2 ) + ( H n ( x , 1 ) ) 4 4 ! x + n + 1 n ( x + 1 )

and

(29) ∑ k = 0 n n k ( − 1 ) k 1 ( k + 1 ) 5 = 1 4 ! ∫ 0 1 ( 1 − t ) n ( log t ) 4 d t = 6 H n + 1 ( 4 ) + 8 H n + 1 ( 3 ) H n + 1 + 3 ( H n + 1 ( 2 ) ) 2 + 6 ( H n + 1 ) 2 H n + 1 ( 2 ) + ( H n + 1 ) 4 4 ! ( n + 1 ) .

Therefore, from (28) and (29), by binomial inversion, we obtain the following theorem.

Theorem 2.5

For n ≥ 0 , we have

∑ k = 0 n n k ( − 1 ) k 6 H k ( x , 4 ) + 8 H k ( x , 3 ) H k ( x , 1 ) + 3 ( H k ( x , 2 ) ) 2 + 6 ( H k ( x , 1 ) ) 2 H k ( x , 2 ) + ( H k ( x , 1 ) ) 4 4 ! x + k + 1 k ( x + 1 ) = 1 ( n + x + 1 ) 5

and

1 4 ! ∑ k = 0 n n k ( − 1 ) k 6 H k + 1 ( 4 ) + 8 H k + 1 ( 3 ) H k + 1 + 3 ( H k + 1 ( 2 ) ) 2 + 6 ( H k + 1 ) 2 H k + 1 ( 2 ) + ( H k + 1 ) 4 k + 1 = 1 ( n + 1 ) 5 .

By Theorem 2.5 and (4), we have

∑ n = 0 ∞ ∑ k = 0 n n k ( − 1 ) k 6 H k ( x , 4 ) + 8 H k ( x , 3 ) H k ( x , 1 ) + 3 ( H k ( x , 2 ) ) 2 + 6 ( H k ( x , 1 ) ) 2 H k ( x , 2 ) + ( H k ( x , 1 ) ) 4 4 ! x + k + 1 k ( x + 1 ) = ∑ n = 0 ∞ 1 ( n + x + 1 ) 5 = ζ ( x + 1 , 5 ) .

For x = 0 , we also have

1 4 ! ∑ n = 0 ∞ ∑ k = 0 n n k ( − 1 ) k 6 H k + 1 ( 4 ) + 8 H k + 1 ( 3 ) H k + 1 + 3 ( H k + 1 ( 2 ) ) 2 + 6 ( H k + 1 ) 2 H k + 1 ( 2 ) + ( H k + 1 ) 4 k + 1 = ∑ n = 0 ∞ 1 ( n + 1 ) 5 = ζ ( 5 ) .

From (29) and Lemma 2.1 (c), we note that

1 4 ! ∑ n = 1 ∞ 6 H n + 1 ( 4 ) + 8 H n + 1 ( 3 ) H n + 1 + 3 ( H n + 1 ( 2 ) ) 2 + 6 ( H n + 1 ) 2 H n + 1 ( 2 ) + ( H n + 1 ) 4 n ( n + 1 ) = ∑ n = 1 ∞ 1 n ∑ k = 0 n n k ( − 1 ) k 1 ( k + 1 ) 5 = 5 .

Hence, we have

∑ n = 1 ∞ 6 H n + 1 ( 4 ) + 8 H n + 1 ( 3 ) H n + 1 + 3 ( H n + 1 ( 2 ) ) 2 + 6 ( H n + 1 ) 2 H n + 1 ( 2 ) + ( H n + 1 ) 4 n ( n + 1 ) = 5 ! .

From Lemma 2.1 (a), (b), and (12), we note that

( r + 1 ) ! ∑ k = 0 n n k ( − 1 ) k − r 1 ( k + x + 1 ) r + 2 = d r d x r ( H n ( x , 1 ) F n ( x ) ) = ∑ l = 0 r r l d l d x l H n ( x , 1 ) d r − l d x r − l F n ( x ) = ∑ l = 0 r r l l ! ( − 1 ) l H n ( x , l + 1 ) ∫ 0 1 ( 1 − t ) n ( log t ) r − l t x d t ,

which gives us the following:

(30) ∑ k = 0 n n k ( − 1 ) k 1 ( k + x + 1 ) r + 2 = ( − 1 ) r ( r + 1 ) ! ∑ l = 0 r r l l ! ( − 1 ) l H n ( x , l + 1 ) ∫ 0 1 ( 1 − t ) n ( log t ) r − l t x d t .

From (30), by binomial inversion and summing over n , we obtain

(31) ( − 1 ) r ( r + 1 ) ! ∑ n = 0 ∞ ∑ k = 0 n n k ( − 1 ) k ∑ l = 0 r r l l ! ( − 1 ) l H k ( x , l + 1 ) ∫ 0 1 ( 1 − t ) k ( log t ) r − l t x d t = ∑ n = 0 ∞ 1 ( n + x + 1 ) r + 2 = ζ ( x + 1 , r + 2 ) ,

which, for x = 0 , yields

( − 1 ) r ( r + 1 ) ! ∑ n = 0 ∞ ∑ k = 0 n n k ( − 1 ) k ∑ l = 0 r r l l ! ( − 1 ) l H k + 1 ( l + 1 ) ∫ 0 1 ( 1 − t ) k ( log t ) r − l d t = ∑ n = 0 ∞ 1 ( n + 1 ) r + 2 = ζ ( r + 2 ) .

By (30) with x = 0 and Lemma 2.1 (c), we have

(32) ( − 1 ) r ( r + 1 ) ! ∑ l = 0 r r l l ! ( − 1 ) l ∑ n = 1 ∞ 1 n H n + 1 ( l + 1 ) ∫ 0 1 ( 1 − t ) n ( log t ) r − l d t = ∑ n = 1 ∞ 1 n ∑ k = 0 n n k ( − 1 ) k 1 ( k + 1 ) r + 2 = r + 2 .

Thus, from (30)–(32), we obtain the following theorem.

Theorem 2.6

For n , r ≥ 0 , we have the following identities:

∑ k = 0 n n k ( − 1 ) k 1 ( k + x + 1 ) r + 2 = ( − 1 ) r ( r + 1 ) ! ∑ l = 0 r r l l ! ( − 1 ) l H n ( x , l + 1 ) ∫ 0 1 ( 1 − t ) n ( log t ) r − l t x d t , ( − 1 ) r ( r + 1 ) ! ∑ n = 0 ∞ ∑ k = 0 n n k ( − 1 ) k ∑ l = 0 r r l l ! ( − 1 ) l H k ( x , l + 1 ) ∫ 0 1 ( 1 − t ) k ( log t ) r − l t x d t = ζ ( x + 1 , r + 2 ) ,

and

∑ l = 0 r r l l ! ( − 1 ) l ∑ n = 1 ∞ 1 n H n + 1 ( l + 1 ) ∫ 0 1 ( 1 − t ) n ( log t ) r − l d t = ( − 1 ) r ( r + 2 ) ! .

3 Conclusion

In this article, we used elementary methods to derive some identities involving the generalized harmonic numbers and the generalized harmonic functions from the beta functions F n ( x ) = B ( x + 1 , n + 1 ) , ( n = 0 , 1 , 2 , … ) . In more detail, for every r = 2 , 3 , 4 , 5 , we showed that

(33) ∑ k = 0 n n k ( − 1 ) k 1 ( k + x + 1 ) r , ζ ( x + 1 , r ) , r !

are all expressed in terms of the generalized harmonic functions and the generalized harmonic numbers. The methods employed in this article could be continued further for r = 6 , 7 , 8 , … , which requires to find an explicit expression for F n ( r − 1 ) ( x ) in terms of the generalized harmonic functions. In Theorem 2.6, for any r = 2 , 3 , 4 , … , we found expressions for (33) without deriving the explicit expressions for F n ( r − l ) ( x ) = ∫ 0 1 ( 1 − t ) n ( log t ) r − l d t in terms of the generalized harmonic functions.

Acknowledgments

The author would like to thank the referees for the detailed and valuable comments that helped improve the original manuscript in its present form. The authors also thank the Jangjeon Institute for Mathematical Science for the support of this research.

  1. Funding information: This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF-2021R1F1A1050151).

  2. Conflict of interest: The authors declare no conflict of interest.

  3. Ethics approval and consent to participate: The authors declare that there is no ethical problem in the production of this article.

  4. Consent for publication: The authors want to publish this article in this journal.

  5. Data availability statement: Data sharing is not applicable to the article as no datasets were generated or analyzed during the current study.

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Received: 2022-12-30
Revised: 2023-03-22
Accepted: 2023-04-08
Published Online: 2023-05-25

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. A novel class of bipolar soft separation axioms concerning crisp points
  3. Duality for convolution on subclasses of analytic functions and weighted integral operators
  4. Existence of a solution to an infinite system of weighted fractional integral equations of a function with respect to another function via a measure of noncompactness
  5. On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group
  6. Hyers-Ulam stability of isometries on bounded domains-II
  7. Asymptotic study of Leray solution of 3D-Navier-Stokes equations with exponential damping
  8. Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃
  9. Jordan triple (α,β)-higher ∗-derivations on semiprime rings
  10. The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping
  11. Approximation of the image of the Lp ball under Hilbert-Schmidt integral operator
  12. Best proximity points in ℱ-metric spaces with applications
  13. Approximation spaces inspired by subset rough neighborhoods with applications
  14. A numerical Haar wavelet-finite difference hybrid method and its convergence for nonlinear hyperbolic partial differential equation
  15. A novel conservative numerical approximation scheme for the Rosenau-Kawahara equation
  16. Fekete-Szegö functional for a class of non-Bazilevic functions related to quasi-subordination
  17. On local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) -preinvexity involving local fractional integral operators with Mittag-Leffler kernel
  18. On some geometric results for generalized k-Bessel functions
  19. Convergence analysis of M-iteration for 𝒢-nonexpansive mappings with directed graphs applicable in image deblurring and signal recovering problems
  20. Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂn
  21. Graded weakly 1-absorbing primary ideals
  22. The existence and uniqueness of solutions to a functional equation arising in psychological learning theory
  23. Some aspects of the n-ary orthogonal and b(αn,βn)-best approximations of b(αn,βn)-hypermetric spaces over Banach algebras
  24. Numerical solution of a malignant invasion model using some finite difference methods
  25. Increasing property and logarithmic convexity of functions involving Dirichlet lambda function
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  27. Global optimum solutions for a system of (k, ψ)-Hilfer fractional differential equations: Best proximity point approach
  28. The study of solutions for several systems of PDDEs with two complex variables
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  36. Some notes on graded weakly 1-absorbing primary ideals
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  38. Integral transforms involving a generalized k-Bessel function
  39. Ruled real hypersurfaces in the complex hyperbolic quadric
  40. On the monotonic properties and oscillatory behavior of solutions of neutral differential equations
  41. Approximate multi-variable bi-Jensen-type mappings
  42. Mixed-type SP-iteration for asymptotically nonexpansive mappings in hyperbolic spaces
  43. On the equation fn + (f″)m ≡ 1
  44. Results on the modified degenerate Laplace-type integral associated with applications involving fractional kinetic equations
  45. Characterizations of entire solutions for the system of Fermat-type binomial and trinomial shift equations in ℂn#
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  47. On I. Meghea and C. S. Stamin review article “Remarks on some variants of minimal point theorem and Ekeland variational principle with applications,” Demonstratio Mathematica 2022; 55: 354–379
  48. Special Issue on Fixed Point Theory and Applications to Various Differential/Integral Equations - Part II
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https://doi.org/10.1515/dema-2022-0229
Keywords for this article
generalized harmonic numbers of order α; generalized harmonic function
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Articles in the same Issue

  1. Regular Articles
  2. A novel class of bipolar soft separation axioms concerning crisp points
  3. Duality for convolution on subclasses of analytic functions and weighted integral operators
  4. Existence of a solution to an infinite system of weighted fractional integral equations of a function with respect to another function via a measure of noncompactness
  5. On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group
  6. Hyers-Ulam stability of isometries on bounded domains-II
  7. Asymptotic study of Leray solution of 3D-Navier-Stokes equations with exponential damping
  8. Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃
  9. Jordan triple (α,β)-higher ∗-derivations on semiprime rings
  10. The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping
  11. Approximation of the image of the Lp ball under Hilbert-Schmidt integral operator
  12. Best proximity points in ℱ-metric spaces with applications
  13. Approximation spaces inspired by subset rough neighborhoods with applications
  14. A numerical Haar wavelet-finite difference hybrid method and its convergence for nonlinear hyperbolic partial differential equation
  15. A novel conservative numerical approximation scheme for the Rosenau-Kawahara equation
  16. Fekete-Szegö functional for a class of non-Bazilevic functions related to quasi-subordination
  17. On local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) -preinvexity involving local fractional integral operators with Mittag-Leffler kernel
  18. On some geometric results for generalized k-Bessel functions
  19. Convergence analysis of M-iteration for 𝒢-nonexpansive mappings with directed graphs applicable in image deblurring and signal recovering problems
  20. Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂn
  21. Graded weakly 1-absorbing primary ideals
  22. The existence and uniqueness of solutions to a functional equation arising in psychological learning theory
  23. Some aspects of the n-ary orthogonal and b(αn,βn)-best approximations of b(αn,βn)-hypermetric spaces over Banach algebras
  24. Numerical solution of a malignant invasion model using some finite difference methods
  25. Increasing property and logarithmic convexity of functions involving Dirichlet lambda function
  26. Feature fusion-based text information mining method for natural scenes
  27. Global optimum solutions for a system of (k, ψ)-Hilfer fractional differential equations: Best proximity point approach
  28. The study of solutions for several systems of PDDEs with two complex variables
  29. Regularity criteria via horizontal component of velocity for the Boussinesq equations in anisotropic Lorentz spaces
  30. Generalized Stević-Sharma operators from the minimal Möbius invariant space into Bloch-type spaces
  31. On initial value problem for elliptic equation on the plane under Caputo derivative
  32. A dimension expanded preconditioning technique for block two-by-two linear equations
  33. Asymptotic behavior of Fréchet functional equation and some characterizations of inner product spaces
  34. Small perturbations of critical nonlocal equations with variable exponents
  35. Dynamical property of hyperspace on uniform space
  36. Some notes on graded weakly 1-absorbing primary ideals
  37. On the problem of detecting source points acting on a fluid
  38. Integral transforms involving a generalized k-Bessel function
  39. Ruled real hypersurfaces in the complex hyperbolic quadric
  40. On the monotonic properties and oscillatory behavior of solutions of neutral differential equations
  41. Approximate multi-variable bi-Jensen-type mappings
  42. Mixed-type SP-iteration for asymptotically nonexpansive mappings in hyperbolic spaces
  43. On the equation fn + (f″)m ≡ 1
  44. Results on the modified degenerate Laplace-type integral associated with applications involving fractional kinetic equations
  45. Characterizations of entire solutions for the system of Fermat-type binomial and trinomial shift equations in ℂn#
  46. Commentary
  47. On I. Meghea and C. S. Stamin review article “Remarks on some variants of minimal point theorem and Ekeland variational principle with applications,” Demonstratio Mathematica 2022; 55: 354–379
  48. Special Issue on Fixed Point Theory and Applications to Various Differential/Integral Equations - Part II
  49. On Cauchy problem for pseudo-parabolic equation with Caputo-Fabrizio operator
  50. Fixed-point results for convex orbital operators
  51. Asymptotic stability of equilibria for difference equations via fixed points of enriched Prešić operators
  52. Asymptotic behavior of resolvents of equilibrium problems on complete geodesic spaces
  53. A system of additive functional equations in complex Banach algebras
  54. New inertial forward–backward algorithm for convex minimization with applications
  55. Uniqueness of solutions for a ψ-Hilfer fractional integral boundary value problem with the p-Laplacian operator
  56. Analysis of Cauchy problem with fractal-fractional differential operators
  57. Common best proximity points for a pair of mappings with certain dominating property
  58. Investigation of hybrid fractional q-integro-difference equations supplemented with nonlocal q-integral boundary conditions
  59. The structure of fuzzy fractals generated by an orbital fuzzy iterated function system
  60. On the structure of self-affine Jordan arcs in ℝ2
  61. Solvability for a system of Hadamard-type hybrid fractional differential inclusions
  62. Three solutions for discrete anisotropic Kirchhoff-type problems
  63. On split generalized equilibrium problem with multiple output sets and common fixed points problem
  64. Special Issue on Computational and Numerical Methods for Special Functions - Part II
  65. Sandwich-type results regarding Riemann-Liouville fractional integral of q-hypergeometric function
  66. Certain aspects of Nörlund ℐ-statistical convergence of sequences in neutrosophic normed spaces
  67. On completeness of weak eigenfunctions for multi-interval Sturm-Liouville equations with boundary-interface conditions
  68. Some identities on generalized harmonic numbers and generalized harmonic functions
  69. Study of degenerate derangement polynomials by λ-umbral calculus
  70. Normal ordering associated with λ-Stirling numbers in λ-shift algebra
  71. Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operators
  72. Compositions of positive integers with 2s and 3s
  73. Kinematic-geometry of a line trajectory and the invariants of the axodes
  74. Hahn Laplace transform and its applications
  75. Discrete complementary exponential and sine integral functions
  76. Special Issue on Recent Methods in Approximation Theory - Part II
  77. On the order of approximation by modified summation-integral-type operators based on two parameters
  78. Bernstein-type operators on elliptic domain and their interpolation properties
  79. A class of strongly convergent subgradient extragradient methods for solving quasimonotone variational inequalities
  80. Special Issue on Recent Advances in Fractional Calculus and Nonlinear Fractional Evaluation Equations - Part II
  81. Application of fractional quantum calculus on coupled hybrid differential systems within the sequential Caputo fractional q-derivatives
  82. On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative
  83. A certain class of fractional difference equations with damping: Oscillatory properties
  84. Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals
  85. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems - Part II
  86. The behavior of hidden bifurcation in 2D scroll via saturated function series controlled by a coefficient harmonic linearization method
  87. Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves
  88. Petri net analysis of a queueing inventory system with orbital search by the server
  89. Asymptotic stability of an epidemiological fractional reaction-diffusion model
  90. On the stability of a strongly stabilizing control for degenerate systems in Hilbert spaces
  91. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part I
  92. New conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in terms of generalized conformable fractional operators via majorization
  93. Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equation
  94. Impacts of Brownian motion and fractional derivative on the solutions of the stochastic fractional Davey-Stewartson equations
  95. Some results on fractional Hahn difference boundary value problems
  96. Properties of a subclass of analytic functions defined by Riemann-Liouville fractional integral applied to convolution product of multiplier transformation and Ruscheweyh derivative
  97. Special Issue on Development of Fuzzy Sets and Their Extensions - Part I
  98. The cross-border e-commerce platform selection based on the probabilistic dual hesitant fuzzy generalized dice similarity measures
  99. Comparison of fuzzy and crisp decision matrices: An evaluation on PROBID and sPROBID multi-criteria decision-making methods
  100. Rejection and symmetric difference of bipolar picture fuzzy graph
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