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Euler-type sums involving multiple harmonic sums and binomial coefficients

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Published/Copyright: December 31, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this paper, we mainly show that generalized Euler-type sums of multiple harmonic sums with reciprocal binomial coefficients can be expressed in terms of rational linear combinations of products of classical multiple zeta values (MZVs) and multiple harmonic star sums (MHSSs). Furthermore, applying the stuffle relations, we prove that the Euler-type sums involving products of generalized harmonic numbers and reciprocal binomial coefficients can be evaluated by MZVs and MHSSs.

Keywords: Euler-type sum; multiple zeta value; multiple harmonic (star) sum; binomial coefficient
MSC 2010: 11M32; 11M99

1 Introduction

We begin with some basic notations. Let N (resp. Z ) be the set of positive integers (resp. integers) and N 0 ≔ N ∪ { 0 } . A finite sequence k ≔ ( k 1 , … , k r ) ∈ N r is called a composition. We define its weight and the depth of k , respectively, by

∣ k ∣ ≔ k 1 + ⋯ + k r and dep ( k ) ≔ r .

If k 1 > 1 , k is called admissible.

For a composition k = ( k 1 , … , k r ) and positive integer n , the classical multiple harmonic sums (MHSs) and the classical multiple harmonic star sums (MHSSs) are defined by

(1.1) ζ n ( k ) ≡ ζ n ( k 1 , k 2 , … , k r ) ≔ ∑ n ≥ n 1 > n 2 > ⋯ > n r ≥ 1 1 n 1 k 1 n 2 k 2 ⋯ n r k r ,

(1.2) ζ n ⋆ ( k ) ≡ ζ n ⋆ ( k 1 , k 2 , … , k r ) ≔ ∑ n ≥ n 1 ≥ n 2 ≥ ⋯ ≥ n r ≥ 1 1 n 1 k 1 n 2 k 2 ⋯ n r k r ,

when n < k , then ζ n ( k ) ≔ 0 , and ζ n ( ∅ ) = ζ n ⋆ ( ∅ ) ≔ 1 . For k = ( k ) ∈ N ,

ζ n ( k ) ≡ H n ( k ) = ∑ j = 1 n 1 j k

is the generalized harmonic number of order k , and furthermore, if k = 1 , then H n ≡ H n ( 1 ) is the classical harmonic number. When taking the limit n → ∞ in (1.1) and (1.2), we get the so-called classical multiple zeta values (MZVs) and the classical multiple zeta star values (MZSVs), respectively

(1.3) ζ ( k ) ≔ lim n → ∞ ζ n ( k ) ,

(1.4) ζ ⋆ ( k ) ≔ lim n → ∞ ζ n ⋆ ( k ) ,

defined for an admissible composition k to ensure convergence of the series. The study of multiple zeta (star) values began in the early 1990s with the works of Hoffman [1] and Zagier [2]. For an admissible composition k , Hoffman [1] called (1.3) multiple harmonic series. Zagier [2] called (1.3) MZVs since for r = 1 they generalize the usual Riemann zeta values ζ ( k ) . Identities involving multiple harmonic (star) sums and multiple zeta (star) values have been extensively studied in the literature in the last three decades (see, e.g., [3,4] and references therein), since they play an essential role in number theory, combinatorics, analysis of algorithms and many other areas.

There are also some studies on the sums involving harmonic numbers and binomial coefficients. The readers may consult the works presented in [5,6,7, 8,9,10]. Motivated by Wang-Xu’s paper [8] and Xu et al. paper [10], they discussed the evaluations of some Euler-type sums involving harmonic numbers and binomial coefficients, such as

S π 1 , q ( k ) = ∑ n = 1 ∞ H n ( π 1 ) n q ∏ i = 1 r n + k i k i , S π 1 q ( k ) = ∑ n = 1 ∞ n q H n ( π 1 ) ∏ i = 1 r n + k i k i ,

and some other forms. In particular, they proved that the sums S π 1 , q ( k ) and S π 1 q ( k ) can be expressed as rational linear combinations of products of zeta values, linear Euler sums (or double zeta values), harmonic numbers and double harmonic star sums.

In the present paper, we mainly show that generalized MHSs with reciprocal binomial coefficients of types

(1.5) ∑ m = 1 ∞ ζ m − 1 ( k 2 , … , k r ) m k 1 m + n m

can be expressed in terms of linear combinations of classical MHSs and classical MZVs with depth less than or equal to r . Then applying the stuffle relations, also called quasi-shuffle relations (see [11]), we know that for any composition k = ( k 1 , … , k r ) , the product H n ( k 1 ) ⋯ H n ( k r ) can be expressed in terms of linear combinations of MHSs (for the explicit formula, see [12, equation (2.4)]). Hence, we can prove that the Euler-type sums

(1.6) ∑ m = 1 ∞ H n − 1 ( k 2 ) ⋯ H n − 1 ( k r ) m k 1 m + n m

can be evaluated by a linear combinations of products of the classical MHVs and classical MZVs with depth less than or equal to r . Some illustrative special cases as well as immediate consequences of the main results are also considered. Here ( k 1 , … , k r ) ∈ N r and n ∈ N .

2 Explicit evaluations of Euler-type sums

In this section, we first develop closed form representations for the following integral involving multiple polylogarithm function

∫ 0 1 x n − 1 Li k 1 , k 2 , … , k r ( x ) d x

by using the iterated integral. Then, using the integral, we give some explicit evaluations for the Euler-type sums of MHSs with reciprocal binomial coefficients (1.5). For convenience, for a composition k ≡ k r = ( k 1 , k 2 , … , k r ) , we let k l → ≔ ( k 1 , k 2 , … , k l ) , k 0 → = ∅ , k l ← ≔ ( k l + 1 , k l + 2 , … , k r ) , k l ← ( j ) = ( k l + 1 + 1 − j , k l + 2 , … , k r ) , k ← r = k ← r ( j ) ≔ ∅ and

H n ( k 1 , k 2 , … , k r ) ≔ 1 n ζ n ⋆ ( k 2 , … , k r ) and H n ( k 1 ) ≔ 1 n k 1 .

The theory of iterated integrals was developed first by Chen in the 1960s [13,14]. It has played important roles in the study of algebraic topology and algebraic geometry in the past half century. Its simplest form over R is

∫ a b f p ( t ) d t f p − 1 ( t ) d t ⋯ f 1 ( t ) d t ≔ ∫ a < t p < ⋯ < t 1 < b f p ( t p ) f p − 1 ( t p − 1 ) ⋯ f 1 ( t 1 ) d t 1 d t 2 ⋯ d t p ,

which can be easily extended to iterated path integrals over C .

Theorem 2.1

Let r , n ∈ N and k ≔ ( k 1 , … , k r ) ∈ N r . Then

(2.1) ∫ 0 1 x n − 1 Li k 1 , k 2 , … , k r ( x ) d x = ∑ l = 0 r ( − 1 ) ∣ k l ∣ → − l ∑ j = 1 k l + 1 − 1 ( − 1 ) j − 1 H n ( k l → , j ) ζ ( k l ← ( j ) ) ,

where k r + 1 ≔ 2 and Li k 1 , … , k r ( z ) is the single-variable multiple polylogarithm function defined by

(2.2) Li k 1 , … , k r ( z ) ≔ ∑ n 1 > ⋯ > n r > 0 z n 1 n 1 k 1 ⋯ n r k r , z ∈ [ − 1 , 1 ] and ( k 1 , z ) ≠ ( 1 , 1 ) .

Proof

According to the definition of multiple polylogarithm function, we have

(2.3) d d x Li k 1 , k 2 , … , k r ( x ) = 1 x Li k 1 − 1 , k 2 , … , k r ( x ) , k 1 > 1 , 1 1 − x Li k 2 , … , k r ( x ) , k 1 = 1 .

Hence, applying (2.3) we obtain the following iterated integral expression:

(2.4) Li k ( x ) = ∫ 0 x d t 1 − t d t t k r − 1 ⋯ d t 1 − t d t t k 1 − 1 .

Using integration by parts, by an elementary calculation, we deduce the recurrence relation that

∫ 0 1 x n − 1 Li k r ( x ) d x = ∑ j = 1 k r − 1 ( − 1 ) j − 1 n j ζ ( k r − 1 , k r + 1 − j ) + ( − 1 ) k r − 1 n k r ∑ j = 1 n ∫ 0 1 x j − 1 Li k r − 1 ( x ) d x .

Thus, we arrive at the desired formula by a direct calculation.□

Corollary 2.2

For a composition k = ( k 1 , … , k r ) and n ∈ N , we have

(2.5) ∑ m = 1 ∞ ζ m − 1 ( k 2 , … , k r ) m k 1 ( m + n ) = ∑ l = 0 r ( − 1 ) ∣ k l → ∣ − l ∑ j = 1 k l + 1 − 1 ( − 1 ) j − 1 H n ( k l → , j ) ζ ( k l ← ( j ) ) .

Proof

According to the definition of multiple polylogarithm function, we have

∫ 0 1 x n − 1 Li k 1 , k 2 , … , k r ( x ) d x = ∑ m = 1 ∞ ∫ 0 1 x n − 1 ζ m − 1 ( k 2 , … , k r ) m k 1 x m d x = ∑ m = 1 ∞ ζ m − 1 ( k 2 , … , k r ) m k 1 ( m + n ) .

Then, applying (2.1) yields the desired formula.□

Theorem 2.3

For a composition k = ( k 1 , … , k r ) and n ∈ N with k r + 1 ≔ 2 , we have

(2.6) ∑ m = 1 ∞ ζ m − 1 ( k 2 , … , k r ) m k 1 m + n n = ∑ i = 1 n ( − 1 ) i − 1 i n i ∑ l = 0 r ( − 1 ) ∣ k l → ∣ − l ∑ j = 1 k l + 1 − 1 ( − 1 ) j − 1 H i ( k l → , j ) ζ ( k l ← ( j ) ) .

Proof

For suitably selected sequences { f m } , we remark that [10, p. 951]

∑ m = 1 ∞ f m m p m + n n = ∑ i = 1 n ( − 1 ) i − 1 i n i ∑ m = 1 ∞ f m m p ( m + i ) .

Hence, we obtain

∑ m = 1 ∞ ζ m − 1 ( k 2 , … , k r ) m k 1 m + n n = ∑ i = 1 n ( − 1 ) i − 1 i n i ∑ m = 1 ∞ ζ m − 1 ( k 2 , … , k r ) m k 1 ( m + i ) .

Thus, we finish the proof by using (2.5).□

Let

a 1 + a 2 ⋮ + a n ≔ a 1 + a 2 + ⋯ + a n .

From Theorem 2.3, we can get the following examples.

Example 2.4

Setting r = 1 , 2 and 3 in Theorem 2.3, we have

(2.7) ∑ m = 1 ∞ 1 m k 1 m + n n = ∑ i = 1 n ( − 1 ) i − 1 i n i ∑ j = 1 k 1 − 1 ( − 1 ) j − 1 ⋅ ζ ( k 1 + 1 − j ) i j + ( − 1 ) k 1 − 1 H i i k 1 ,

(2.8) ∑ m = 1 ∞ ζ m − 1 ( k 2 ) m k 1 m + n n = ∑ i = 1 n ( − 1 ) i − 1 i n i ∑ j = 1 k 1 − 1 ( − 1 ) j − 1 ζ ( k 1 + 1 − j , k 2 ) i j + ( − 1 ) k 1 + k 2 ζ i ⋆ ( k 2 , 1 ) i k 1 + ( − 1 ) k 1 − 1 ∑ j = 1 k 2 − 1 ( − 1 ) j − 1 ζ i ( j ) i k 1 ζ ( k 2 + 1 − j ) ,

(2.9) ∑ m = 1 ∞ ζ m − 1 ( k 2 , k 3 ) m k 1 m + n n = ∑ i = 1 n ( − 1 ) i − 1 i n i ∑ j = 1 k 1 − 1 ( − 1 ) j − 1 i j ζ ( k 1 + 1 − j , k 2 , k 3 ) + ( − 1 ) k 1 − 1 ∑ j = 1 k 2 − 1 ( − 1 ) j − 1 ζ i ( j ) i k 1 ζ ( k 2 + 1 − j , k 3 ) + ( − 1 ) k 1 + k 2 ∑ j = 1 k 3 − 1 ( − 1 ) j − 1 ζ i ⋆ ( k 2 , j ) i k 1 ζ ( k 3 + 1 − j ) + ( − 1 ) k 1 + k 2 + k 3 − 1 ζ i ⋆ ( k 2 , k 3 , 1 ) i k 1 .

Now, we give an evaluation for (1.6) with r = 3 . The key idea to study (1.6) is to express H n ( k 1 ) ⋯ H n ( k r ) by using general MHSs of the form (1.1) by applying the stuffle relations, also called quasi-shuffle relations (see [11]). In fact, we know that for any composition k = ( k 1 , … , k r ) , the product H n ( k 1 ) ⋯ H n ( k r ) can be expressed in terms of linear combinations of MHSs (for the explicit formula, see [12, equation (2.4)]), for example

(2.10) H n ( m ) H n ( p ) = ζ n ( m ) ζ n ( p ) = ζ n ( m , p ) + ζ n ( p , m ) + ζ n ( p + m ) .

Hence, the Euler-type sums (1.6) can be expressed in terms of linear combinations of products of MHSSs and MZVs.

Theorem 2.5

For positive integers k 1 , k 2 , k 3 and n , we have

(2.11) ∑ m = 1 ∞ H m − 1 ( k 2 ) H m − 1 ( k 3 ) m k 1 m + n n = ∑ i = 1 n ( − 1 ) i − 1 i n i ∑ j = 1 k 1 − 1 ( − 1 ) j − 1 i j ζ ( k 1 + 1 − j , k 2 , k 3 ) + ζ ( k 1 + 1 − j , k 3 , k 2 ) + ζ ( k 1 + 1 − j , k 2 + k 3 ) + ( − 1 ) k 1 + k 2 + k 3 ζ i ⋆ ( k 2 , k 3 , 1 ) + ζ i ⋆ ( k 3 , k 2 , 1 ) + ζ i ⋆ ( k 2 + k 3 , 1 ) i k 1 + ( − 1 ) k 1 − 1 ∑ j = 1 k 2 − 1 ( − 1 ) j − 1 ζ ( k 2 + 1 − j , k 3 ) ζ i ⋆ ( j ) i k 1 + ( − 1 ) k 1 − 1 ∑ j = 1 k 3 − 1 ( − 1 ) j − 1 ζ ( k 3 + 1 − j , k 2 ) ζ i ⋆ ( j ) i k 1 + ( − 1 ) k 1 − 1 ∑ j = 1 k 2 + k 3 − 1 ( − 1 ) j − 1 ζ ( k 2 + k 3 + 1 − j ) ζ i ⋆ ( j ) i k 1 + ( − 1 ) k 1 + k 2 ∑ j = 1 k 3 − 1 ( − 1 ) j − 1 ζ i ⋆ ( k 2 , j ) i k 1 ζ ( k 3 + 1 − j ) + ( − 1 ) k 1 + k 3 ∑ j = 1 k 2 − 1 ( − 1 ) j − 1 ζ i ⋆ ( k 3 , j ) i k 1 ζ ( k 2 + 1 − j ) .

Proof

The theorem follows immediately from (2.9) and (2.10).□

3 More general sums

From [8, Theorem 3.1], we know that the key idea to study the following more general Euler-type sums

∑ m = 1 ∞ ζ m − 1 ( k 2 , … , k r ) m k 1 ∏ i = 1 p m + n i n i and ∑ m = 1 ∞ H m − 1 ( k 2 ) ⋯ H m − 1 ( k r ) m k 1 ∏ i = 1 p m + n i n i

is to express the sums ∑ m = 1 ∞ ζ m − 1 ( k 2 , … , k r ) m k 1 ( m + n ) q by using the MZVs and multiple harmonic (star) sums, where ( k 1 , … , k r ) ∈ N r , ( n 1 , … , n p ) ∈ N p and n , p , q ∈ N . Moreover, according to the definition of multiple polylogarithm function, we have the identity

(3.1) ∫ 0 1 x n − 1 log p ( x ) Li k 1 , k 2 , … , k r ( x ) d x = p ! ( − 1 ) p ∑ m = 1 ∞ ζ m − 1 ( k 2 , … , k r ) m k 1 ( m + n ) p + 1 .

Hence, we need to evaluate the integral on left-hand side of the aforementioned formula. We can get the following recurrence relation.

Theorem 3.1

For a composition k = ( k 1 , … , k r ) and p , n ∈ N , we have

(3.2) ∫ 0 1 x n − 1 log p ( x ) Li k 1 , k 2 … , k r ( x ) d x = ∑ l = 1 r ( − 1 ) ∣ k l − 1 → ∣ + l − 1 ∑ j = 1 k l ( − 1 ) j ∑ n 0 ≥ n 1 ≥ ⋯ ≥ n l − 1 ≥ 1 p ∫ 0 1 x n l − 1 − 1 log p − 1 ( x ) Li k l − 1 ← ( j ) ( x ) d x n 0 k 1 n 1 k 2 ⋯ n l − 2 k l − 1 n l − 1 j + p ! ∑ l = 1 r ( − 1 ) ∣ k l → ∣ + l + p − 1 ζ ( p + 1 , k l + 1 , … , k r ) n k 1 ζ n ⋆ ( k 2 , … , k r ) + ( − 1 ) ∣ k r → ∣ + r + p p ! ζ n ⋆ ( k 2 , … , k r , p + 1 ) n k 1 ,

where n 0 ≔ n .

Proof

Using integration by parts, we deduce

∫ 0 1 x n − 1 log p ( x ) Li k 1 , k 2 , … , k r ( x ) d x = 1 n ∫ 0 1 log p ( x ) Li k r → ( x ) d x n = − p n ∫ 0 1 x n − 1 log p − 1 ( x ) Li k 1 , … , k r ( x ) d x − 1 n ∫ 0 1 x n − 1 log p ( x ) Li k 1 − 1 , k 2 , … , k r ( x ) d x = ⋯ = ∑ j = 1 k 1 − 1 ( − 1 ) j n j p ∫ 0 1 x n − 1 log p − 1 ( x ) Li k 1 + 1 − j , k 2 , … , k r ( x ) d x + ( − 1 ) k 1 − 1 n k 1 − 1 ∫ 0 1 x n − 1 log p ( x ) Li 1 , k 2 , … , k r ( x ) d x = ∑ j = 1 k 1 ( − 1 ) j n j p ∫ 0 1 x n − 1 log p − 1 ( x ) Li k 1 + 1 − j , k 2 , … , k r ( x ) d x + ( − 1 ) k 1 + p p ! ζ ( p + 1 , k 2 , … , k r ) n k 1 − ( − 1 ) k 1 n k 1 ∑ n 1 = 1 n ∫ 0 1 x n 1 − 1 log p ( x ) Li k 2 , … , k r ( x ) d x = ⋯ = ∑ l = 1 r ( − 1 ) l − 1 + k 1 + ⋯ + k l − 1 ∑ j = 1 k l ( − 1 ) j ∑ n = n 0 ≥ n 1 ≥ ⋯ ≥ n l − 1 ≥ 1 p n 0 k 1 n 1 k 2 ⋯ n l − 2 k l − 1 n l − 1 j ∫ 0 1 x n l − 1 − 1 log p − 1 ( x ) Li k l + 1 − j , k l + 1 , … , k r ( x ) d x + ∑ l = 1 r ( − 1 ) l − 1 ( − 1 ) k 1 + ⋯ + k l + p p ! ζ ( p + 1 , k l + 1 , … , k r ) n k 1 ζ n ( k 2 , … , k r ) + ( − 1 ) k 1 + ⋯ + k r + r n k 1 ∑ n ≥ n 1 ≥ ⋯ ≥ n r ≥ 1 1 n 1 k 1 ⋯ n r − 1 k r ∫ 0 1 x n r − 1 log p ( x ) d x = ∑ l = 1 r ( − 1 ) ∣ k l − 1 → ∣ + l − 1 ∑ j = 1 k l ( − 1 ) j ∑ n 0 ≥ n 1 ≥ ⋯ ≥ n l − 1 ≥ 1 p ∫ 0 1 x n l − 1 − 1 log p − 1 ( x ) Li k l − 1 ← ( j ) ( x ) d x n 0 k 1 n 1 k 2 ⋯ n l − 2 k l − 1 n l − 1 j + p ! ∑ l = 1 r ( − 1 ) ∣ k l → ∣ + l + p − 1 ζ ( p + 1 , k l + 1 , … , k r ) n k 1 ζ n ⋆ ( k 2 , … , k r ) + ( − 1 ) ∣ k r → ∣ + r + p p ! ζ n ⋆ ( k 2 , … , k r , p + 1 ) n k 1 .

Thus, the desired evaluation is obtained.□

Therefore, from Theorems 2.1 and 3.1, we arrive at the conclusion that the integral

∫ 0 1 x n − 1 log p ( x ) Li k 1 , k 2 … , k r ( x ) d x

can be written as linear combinations of products of MZVs and MHSSs. For example, we have

(3.3) ∫ 0 1 x n − 1 log ( x ) Li { 1 } r ( x ) d x = − ζ n ⋆ ( { 1 } r ) n 2 − 1 n ∑ i = 1 r ( ζ n ⋆ ( { 1 } i − 1 , 2 , { 1 } r − i ) − ζ ( i + 1 ) ζ n ⋆ ( { 1 } r − i ) ) = ∑ m = 1 ∞ ζ m − 1 ( { 1 } r − 1 ) m ( n + m ) 2 ,

where { 1 } d is denoted by the sequence of 1’s with d repetitions.

Furthermore, we can get the following corollary.

Corollary 3.2

For positive integers r and n , we have

(3.4) ∑ m = 1 ∞ ζ m − 1 ( { 1 } r − 1 ) m m + n n 2 = ∑ j = 1 n j n j 2 ζ j ⋆ ( { 1 } r ) j 2 − 2 ( H n − j − H j − 1 ) ζ j ⋆ ( { 1 } r ) j + 1 j ∑ i = 1 r ( ζ j ⋆ ( { 1 } i − 1 , 2 , { 1 } r − i ) − ζ ( i + 1 ) ζ j ⋆ ( { 1 } r − i ) ) .

Proof

Applying [8, equation (13)], we can find that

∑ m = 1 ∞ ζ m − 1 ( { 1 } r − 1 ) m m + n n 2 = ∑ j = 1 n j n j 2 ∑ m = 1 ∞ ζ m − 1 ( { 1 } r − 1 ) m ( m + j ) 2 − 2 ( H n − j − H j − 1 ) ∑ m = 1 ∞ ζ m − 1 ( { 1 } r − 1 ) m ( m + j ) .

Using (2.5) and (3.3) yields the desired formula.□

Acknowledgments

The author thanks the anonymous referees for suggestions which led to improvements in the exposition.

  1. Funding information: The National Natural Science Foundation of China (Grant Nos. 12061037, and 41801219) and the High-level Personnel of Special Support Program of Xiamen University of Technology (No. 4010520009).

  2. Author contributions: The author read and approved the final manuscript.

  3. Conflict of interest: The author declares that there are no competing interests.

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Received: 2021-07-25
Revised: 2021-10-30
Accepted: 2021-11-02
Published Online: 2021-12-31

© 2021 Xin Si, published by De Gruyter

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

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  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
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  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0124
Keywords for this article
Euler-type sum; multiple zeta value; multiple harmonic (star) sum; binomial coefficient
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  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
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  89. Global optimization and applications to a variational inequality problem
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  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
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  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
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  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
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  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
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  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
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