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The multinomial convolution sum of a generalized divisor function

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Published/Copyright: June 22, 2022
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Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

Abstract

The main theorem of this article is to evaluate and express the multinomial convolution sum of the divisor function σ r ♯ ( n ; N / 4 , N ) in as a simple form as possible, where N / 4 is an arbitrary odd positive integer. This generalizes previous result in combination with Cho and Kim, which is about the case N = 4 . While obtaining our main theorem, we derive some generalizations of other identities to the case that we are dealing with.

Keywords: divisor functions; convolution sums
MSC 2010: 11A25; 11Y70

1 Introduction

For a nonnegative integer r and a positive integer n , let σ r ( n ) be the usual divisor function defined by the equation σ r ( n ) = ∑ d ∣ n d r . The divisor function is an important arithmetic function playing a fundamental role in number theory. This appears naturally as the coefficients of (quasi) modular forms, a number of integer solutions as quadratic forms and in relation to geometry, etc. Let r k ( n ) be the number of representations of a nonnegative number n as the sum of k squares. Jacobi showed in 1834 that r 4 ( n ) = 8 σ 1 ( n ) − 32 σ 1 ( n / 4 ) . Using the result of Jacobi, to compute r 8 ( n ) it is necessary to evaluate the following convolution sum of the divisor function:

∑ m = 1 n − 1 σ 1 ( m ) σ 1 ( n − m ) ,

which was presented by Besge [1]. After that, convolution sums for such divisor functions have become the subject of interest to many mathematicians. Liouville first evaluated the binomial convolution sum of σ r ( n ) as follows.

Theorem 1.1

[2] For every k , n ∈ N we have

∑ s = 0 k − 1 2 k 2 s + 1 ∑ m = 1 n − 1 σ 2 k − 2 s − 1 ( m ) σ 2 s + 1 ( n − m ) = 2 k + 3 4 k + 2 σ 2 k + 1 ( n ) + k 6 − n σ 2 k − 1 ( n ) + 1 2 k + 1 ∑ j = 2 k 2 k + 1 2 j B 2 j σ 2 k + 1 − 2 j ( n ) ,

where B j is the jth Bernoulli number.

A slightly different version of the binomial convolution sum of σ r ( n ) was evaluated by Kim and Bayad. Note that this slight change gives rise to a simpler expression on the right hand side.

Theorem 1.2

[3] For every k , n ∈ N we have

∑ s = 0 k − 1 2 k 2 s + 1 ∑ m = 1 n σ 2 k − 2 s − 1 ( 2 m − 1 ) σ 2 s + 1 ( 2 n − 2 m + 1 ) = 1 4 σ 2 k + 1 ∗ ( 2 n ) ,

where σ r ∗ ( n ) ≔ ∑ d ∣ n 2 ∤ n / d d r = σ r ( n ) − σ r ( n / 2 ) .

In the same article, they also provided several results about the binomial convolution sums of σ r ∗ ( n ) . Here, we introduce just two of them for the sake of simplicity.

Theorem 1.3

[3] For each k , n ∈ N we obtain

( i ) ∑ s = 0 k − 1 2 k 2 s + 1 ∑ m = 1 n − 1 σ 2 k − 2 s − 1 ∗ ( m ) σ 2 s + 1 ∗ ( n − m ) = 1 2 σ 2 k + 1 ∗ ( n ) − n 2 σ 2 k − 1 ∗ ( n ) ,

( i i ) ∑ s = 0 k − 1 2 k 2 s + 1 ∑ m = 1 2 n − 1 ( − 1 ) m + 1 σ 2 k − 2 s − 1 ∗ ( m ) σ 2 s + 1 ∗ ( 2 n − m ) = n σ 2 k − 1 ∗ ( 2 n ) .

After that, Kim and Park succeeded in evaluating the trinomial and quadrinomial convolution sums of the same divisor function as follows.

Theorem 1.4

[4] Let n ≥ 4 be an even integer and let k ∈ N . Then we have

( i ) ∑ a + b + c = 2 k + 1 a , b , c o d d a 2 k + 1 a , b , c ∑ m 1 + m 2 + m 3 = n m 3 even ( − 1 ) m 1 + 1 σ a ∗ ( m 1 ) σ b ∗ ( m 2 ) σ c ∗ ( m 3 ) = ( 2 k − 1 ) n 32 ( σ 2 k + 1 ∗ ( n ) − 2 n σ 2 k − 1 ∗ ( n ) ) ,

( i i ) ∑ a + b + c + d = 2 k a , b , c , d o d d a + b c + d + 1 2 k a , b , c , d ∑ m 1 + m 2 + m 3 + m 4 = n m 3 , m 4 o d d ( − 1 ) m 1 + 1 σ a ∗ ( m 1 ) σ b ∗ ( m 2 ) σ c ∗ ( m 3 ) σ d ∗ ( m 4 ) = 1 64 n σ 2 k + 1 ∗ ( n ) − 2 n 2 σ 2 k − 1 ∗ ( n ) − 64 ∑ m < n / 2 ( n / 2 − m ) σ 1 ∗ ( m ) σ 2 k − 1 ∗ ( n − 2 m ) .

Kim and Bayad [5] dealt with a similar divisor function

S ( n ) ≔ ∑ d ∣ n 2 ∤ d d

and introduced the orders, m -gonal shape number, type, and convexity derived from S ( n ) .

Let us consider another divisor function as follows [6]:

σ r ♯ ( n ) = ∑ d ∣ n n d ≡ 1 ( 4 ) d r − ( − 1 ) r ∑ d ∣ n n d ≡ − 1 ( 4 ) d r , σ r ♭ ( n ) = ∑ d ∣ n d ≡ 1 ( 4 ) d r − ( − 1 ) r ∑ d ∣ n d ≡ − 1 ( 4 ) d r .

Note that if r is even, then

σ r ♯ ( n ) = ∑ d ∣ n − 4 n / d d r and σ r ♭ ( n ) = ∑ d ∣ n − 4 d d r ,

where

− 4 d = 1 , if d ≡ 1 ( mod 4 ) − 1 , if d ≡ − 1 ( mod 4 ) 0 , otherwise ,

is a Kronecker symbol.

These divisor functions are related to the coefficients of (quasi) modular forms. In fact, it is well known that they appear as coefficients of Fourier series of modular forms for Γ 0 ( 16 ) (see Section 4.5 in [7]). Recently, Aygin and Hong obtained the results for convolution sums of σ r ♯ ( n ) and σ r ♭ ( n ) twisted by Dirichlet characters [8]. Also, σ r ♯ ( n ) = σ r ∗ ( n ) if r is odd, so all the above results about σ r ∗ ( n ) can be interpreted as the results about σ r ♯ ( n ) with “odd” indices. Kim, Bayad, and Park also gave the following result about the binomial convolution sum of divisor function σ r ♯ ( n ) as follows.

Theorem 1.5

[6] For each k , n ∈ N we obtain

∑ s = 0 k 2 k 2 s ∑ m = 1 n − 1 σ 2 k − 2 s ♯ ( m ) σ 2 s ♯ ( n − m ) = 1 2 σ 2 k + 1 ♯ ( n ) − 1 2 σ 2 k ♯ ( n ) .

Inspired by this work, the author together with Cho and Kim could evaluate the following “multinomial” convolution sum of σ r ♯ ( n ) with “even” indices.

Theorem 1.6

[9] For every t , k , n ∈ N we have

∑ ( s 1 , s 2 , … , s t ) ∈ N 0 t s 1 + s 2 + ⋯ + s t = k 2 k 2 s 1 , 2 s 2 , ⋯ , 2 s t ∑ M ( − 1 ) α σ 2 s 1 ♯ ( m 1 ) σ 2 s 2 ♯ ( m 2 ) ⋯ σ 2 s t ♯ ( m t ) = ( − 1 ) t − 1 2 ( 2 k − 1 ) ( t − 1 ) σ 2 k ♯ ( n ) ,

where

M = { ( m 1 , … , m t ) ∈ N t ∣ m 1 + ⋯ + m t = 2 t − 1 n , m 1 + ⋯ + m i ≡ 0 ( mod 2 i − 1 ) for all i }

and

α = m 2 + m 3 2 + ⋯ + m t 2 t − 2 .

Ramanujan [10] computed the following convolution sum:

∑ m = 0 n σ r ( m ) σ s ( n − m ) ,

where r and s are odd and r + s = 2 , 4 , 6 , 8 , 12 . Note that the summation begins at m = 0 and ends at m = n . For m = 0 or n , σ r ( 0 ) is defined to be 1 2 ζ ( − r ) , where ζ ( s ) is the Riemann zeta function. Recently, Bayad and Hajli [11] studied about the multidimensional zeta function and proved that the multidimensional Appell polynomials are special values at the nonpositive integers of these zeta functions.

In the present article, we generalize some of the above results including Theorem 1.6. For this purpose, we define the generalized divisor function σ r ♯ ( n ) as for positive integers N ≥ 3 , a nonnegative integer r , and 1 ≤ i ≤ N − 1

σ r ♯ ( n ; i , N ) = ∑ d ∣ n n d ≡ i ( N ) d r − ( − 1 ) r ∑ d ∣ n n d ≡ − i ( N ) d r .

If there does not exist a positive divisor d of n such that n d ≡ i ( mod N ) , then ∑ d ∣ n n d ≡ i ( N ) d r is assumed to be 0. It is obvious that σ r ♯ ( n ; 1 , 4 ) = σ r ♯ ( n ) and σ r ♯ ( n ; − i , N ) = ( − 1 ) r + 1 σ r ♯ ( n ; i , N ) . Now we state our main theorem.

Theorem 1.7

For every t , k , N ∈ N such that t ≥ 2 , 4 ∣ N , and N / 4 is odd, we have

∑ ( s 1 , … , s t ) ∈ N 0 t s 1 + ⋯ + s t = k 2 k 2 s 1 , ⋯ , 2 s t ∑ M ( − 1 ) α σ 2 s 1 ♯ ( m 1 ; N / 4 , N ) ⋯ σ 2 s t ♯ ( m t ; N / 4 , N ) = ( − 1 ) t − 1 2 ( 2 k − 1 ) ( t − 1 ) σ 2 k ♯ ( n ; N / 4 , N ) ,

where

M = { ( m 1 , … , m t ) ∈ N t ∣ m 1 + ⋯ + m t = 2 t − 1 n , m 1 + ⋯ + m i ≡ 0 ( mod 2 i − 1 ) for all i ≤ t }

and

α = m 2 + m 3 2 + ⋯ + m t 2 t − 2 .

Theorem 1.6 was proved by the use of Theorem 1.5. So, in order to prove Theorem 1.7, we need to generalize Theorem 1.5 to the case of σ r ♯ ( n ; N / 4 , N ) and it will be done in Corollary 2.4 ( i ) . To obtain this corollary, we derive a new result (Theorem 2.2) about the binomial convolution sum of σ r ♯ ( n ; i , N ) and then generalize Theorems 1.3(i) and 1.5 as its corollary (see Corollary 2.4). We comment that the technique used in the proof of Corollary 2.4 ( i ) is different from the one used in the proof of Theorem 1.5 given in [6]. Using Corollary 2.4 ( i ) we can obtain Proposition 2.6, which can be thought of as an even-indexed version of Theorem 1.3 ( i i ) . It can also be viewed as a special case of our main theorem for t = 2 . In Section 3, we evaluate three trinomial convolution sums of σ r ♯ ( n ; N / 4 , N ) and finally present the proof of Theorem 1.7.

2 Binomial convolution sums of σ r ♯

In combination with Cho and Kim, the author obtained a result about the binomial convolution sum of σ r ♯ ( n ; i , N ) .

Proposition 2.1

[9] Suppose N ≥ 3 and 1 ≤ i ≤ N − 1 . For all k , n ∈ N we have

∑ s = 0 2 k 2 k s ∑ m = 1 n − 1 σ 2 k − s ♯ ( m ; i , N ) σ s ♯ ( n − m ; i , N ) = σ 2 k + 1 ♯ ( n ; i , N ) − 1 − 2 i N σ 2 k ♯ ( n ; i , N ) − 2 n N σ 2 k − 1 ♯ ( n ; i , N ) .

To prove our main theorem we need the following result.

Theorem 2.2

Let N ≥ 4 be an even integer. For any k , n , i ∈ N with 0 < i < N / 2 , we have

∑ s = 0 2 k 2 k s ∑ m = 1 n − 1 σ 2 k − s ♯ ( m ; i , N ) σ s ♯ ( n − m ; N / 2 + i , N ) = − n N σ 2 k − 1 ♯ ( n ; i , N ) − n N σ 2 k − 1 ♯ ( n ; N / 2 + i , N ) + i N σ 2 k ♯ ( n ; i , N ) + i N − 1 2 σ 2 k ♯ ( n ; N / 2 + i , N ) .

We recall the identity of Huard, Ou, Spearman, and Williams to prove Theorem 2.2.

Theorem 2.3

([12], Theorem 1) Let f : Z 4 → C be a function such that

f ( a , b , x , y ) − f ( x , y , a , b ) = f ( − a , − b , x , y ) − f ( x , y , − a , − b )

for all a , b , x , y ∈ Z . Let n ∈ N . Then

∑ ( a , b , x , y ) ∈ N 4 a x + b y = n ( f ( a , b , x , − y ) − f ( a , − b , x , y ) + f ( a , a − b , x + y , y ) − f ( a , a + b , y − x , y ) + f ( b − a , b , x , x + y ) − f ( a + b , b , x , x − y ) ) = ∑ d ∣ n ∑ x ∈ N x < d ( f ( 0 , n / d , x , d ) + f ( n / d , 0 , d , x ) + f ( n / d , n / d , d − x , − x ) − f ( x , x − d , n / d , n / d ) − f ( x , d , 0 , n / d ) − f ( d , x , n / d , 0 ) ) .

Proof of Theorem 2.2

Let

F i , N ( a ) = 1 if a ≡ i ( mod N ) 0 otherwise , F N ( n ) = F 0 , N ( n ) .

We take f ( a , b , x , y ) = ( F i , N ( a ) + F − i , N ( a ) ) F N 2 , N ( a − b ) ( x − y ) 2 k ( k ∈ N ) in Theorem 2.3. Since F i , N ( a ) = F − i , N ( − a ) and F N 2 , N ( a ) = F N 2 , N ( − a ) for a ∈ Z and even N ≥ 2 , we obtain

f ( a , b , x , y ) − f ( x , y , a , b ) = ( F i , N ( a ) + F − i , N ( a ) ) F N 2 , N ( a − b ) ( x − y ) 2 k − ( F i , N ( x ) + F − i , N ( x ) ) F N 2 , N ( x − y ) ( a − b ) 2 k = ( F − i , N ( − a ) + F i , N ( − a ) ) F N 2 , N ( − a + b ) ( x − y ) 2 k − ( F i , N ( x ) + F − i , N ( x ) ) F N 2 , N ( x − y ) ( − a + b ) 2 k = f ( − a , − b , x , y ) − f ( x , y , − a , − b ) .

Then the left hand side is equal to

∑ ( a , b , x , y ) ∈ N 4 a x + b y = n ( ( F i , N ( a ) + F − i , N ( a ) ) F N 2 , N ( a − b ) ( x + y ) 2 k − ( F i , N ( a ) + F − i , N ( a ) ) F N 2 , N ( a + b ) ( x − y ) 2 k ) = ∑ ( a , b , x , y ) ∈ N 4 a x + b y = n ( F i , N ( a ) + F − i , N ( a ) ) F N 2 , N ( a − b ) ∑ s = 0 2 k 2 k s x 2 k − s y s − ( F i , N ( a ) + F − i , N ( a ) ) F N 2 , N ( a + b ) ∑ s = 0 2 k 2 k s ( − 1 ) s x 2 k − s y s = ∑ s = 0 2 k 2 k s ∑ m = 1 n − 1 ∑ x ∣ m m x ≡ i ( N ) x 2 k − s ∑ y ∣ n − m n − m y ≡ N 2 + i ( N ) y s + ∑ x ∣ m m x ≡ − i ( N ) x 2 k − s ∑ y ∣ n − m n − m y ≡ N 2 − i ( N ) y s − ( − 1 ) s ∑ x ∣ m m x ≡ i ( N ) x 2 k − s ∑ y ∣ n − m n − m y ≡ N 2 − i ( N ) y s + ∑ x ∣ m m x ≡ − i ( N ) x 2 k − s ∑ y ∣ n − m n − m y ≡ N 2 + i ( N ) y s = ∑ s = 0 2 k 2 k s ∑ m = 1 n − 1 ∑ x ∣ m m x ≡ i ( N ) x 2 k − s − ( − 1 ) s ∑ x ∣ m m x ≡ − i ( N ) x 2 k − s ∑ y ∣ n − m n − m y ≡ N 2 + i ( N ) y s − ( − 1 ) s ∑ y ∣ n − m n − m y ≡ N 2 − i ( N ) y s = ∑ s = 0 2 k 2 k s ∑ m = 1 n − 1 σ 2 k − s ♯ ( m ; i , N ) σ s ♯ ( n − m ; N / 2 + i , N ) .

The right hand side is − ( U 1 + U 2 ) , where

U 1 = ∑ d ∣ n ∑ x = 1 d − 1 ( F i , N ( x ) + F − i , N ( x ) ) F N 2 , N ( x − d ) ( n / d ) 2 k , U 2 = ∑ d ∣ n ∑ x = 1 d − 1 ( F i , N ( d ) + F − i , N ( d ) ) F N 2 , N ( d − x ) ( n / d ) 2 k .

Replacing x by d − x , U 1 is equal to

∑ d ∣ n ∑ x = 1 d − 1 ( F i , N ( d − x ) + F − i , N ( d − x ) ) F N 2 , N ( x ) ( n / d ) 2 k = ∑ d ∣ n ( n / d ) 2 k F N 2 + i , N ( d ) + F N 2 − i , N ( d ) ∑ x = 1 d − 1 F N 2 , N ( x ) = ∑ d ∣ n ( n / d ) 2 k F N 2 + i , N ( d ) + F N 2 − i , N ( d ) ∑ x = 1 d − 1 F N 2 ( x ) − F N ( x ) = ∑ d ∣ n ( n / d ) 2 k F N 2 + i , N ( d ) + F N 2 − i , N ( d ) 2 ( d − 1 ) N − d − 1 N = U 1 , 1 − U 1 , 2 ,

where

U 1 , 1 = ∑ d ∣ n d ≡ N 2 + i ( N ) ( n / d ) 2 k 2 ( d − 1 ) N + ∑ d ∣ n d ≡ N 2 − i ( N ) ( n / d ) 2 k 2 ( d − 1 ) N , U 1 , 2 = ∑ d ∣ n d ≡ N 2 + i ( N ) ( n / d ) 2 k d − 1 N + ∑ d ∣ n d ≡ N 2 − i ( N ) ( n / d ) 2 k d − 1 N .

Then U 1 , 1 is

∑ d ∣ n n d ≡ N 2 + i ( N ) d 2 k 2 ( n / d − 1 ) N + ∑ d ∣ n n d ≡ N 2 − i ( N ) d 2 k 2 ( n / d − 1 ) N = ∑ d ∣ n n d ≡ N 2 + i ( N ) d 2 k 2 n / d − 2 i N + ∑ d ∣ n n d ≡ N 2 − i ( N ) d 2 k 2 n / d − N + 2 i N = 2 n N ∑ d ∣ n n d ≡ N 2 + i ( N ) d 2 k − 1 + ∑ d ∣ n n d ≡ N 2 − i ( N ) d 2 k − 1 − 2 i N ∑ d ∣ n n d ≡ N 2 + i ( N ) d 2 k − ∑ d ∣ n n d ≡ N 2 − i ( N ) d 2 k − ∑ d ∣ n n d ≡ N 2 − i ( N ) d 2 k = 2 n N σ 2 k − 1 ♯ ( n ; N / 2 + i , N ) − 2 i N σ 2 k ♯ ( n ; N / 2 + i , N ) − ∑ d ∣ n n d ≡ N 2 − i ( N ) d 2 k .

Also, U 1 , 2 equals

∑ d ∣ n n d ≡ N 2 + i ( N ) d 2 k n / d − 1 N + ∑ d ∣ n n d ≡ N 2 − i ( N ) d 2 k n / d − 1 N = ∑ d ∣ n n d ≡ N 2 + i ( N ) d 2 k n / d − N / 2 − i N + ∑ d ∣ n n d ≡ N 2 − i ( N ) d 2 k n / d − N / 2 + i N = ∑ d ∣ n n d ≡ N 2 + i ( N ) d 2 k n / d − N / 2 − i N + ∑ d ∣ n n d ≡ N 2 − i ( N ) d 2 k n / d + N / 2 + i N − 1

= n N ∑ d ∣ n n d ≡ N 2 + i ( N ) d 2 k − 1 + ∑ d ∣ n n d ≡ N 2 − i ( N ) d 2 k − 1 − 1 2 + i N ∑ d ∣ n n d ≡ N 2 + i ( N ) d 2 k − ∑ d ∣ n n d ≡ N 2 − i ( N ) d 2 k − ∑ d ∣ n n d ≡ N 2 − i ( N ) d 2 k = n N σ 2 k − 1 ♯ ( n ; N / 2 + i , N ) − 1 2 + i N σ 2 k ♯ ( n ; N / 2 + i , N ) − ∑ d ∣ n n d ≡ N 2 − i ( N ) d 2 k .

Hence, we obtain

U 1 = U 1 , 1 − U 1 , 2 = n N σ 2 k − 1 ♯ ( n ; N / 2 + i , N ) − i N − 1 2 σ 2 k ♯ ( n ; N / 2 + i , N ) .

Next, replacing x by d − x , we compute U 2 as follows:

∑ d ∣ n ∑ x = 1 d − 1 ( F i , N ( d ) + F − i , N ( d ) ) F N 2 , N ( x ) ( n / d ) 2 k = ∑ d ∣ n ∑ x = 1 d − 1 ( F i , N ( d ) + F − i , N ( d ) ) ( n / d ) 2 k 2 ( d − 1 ) N − d − 1 N = ∑ d ∣ n d ≡ i ( N ) ( n / d ) 2 k 2 ( d − 1 ) N − d − 1 N + ∑ d ∣ n d ≡ − i ( N ) ( n / d ) 2 k 2 ( d − 1 ) N − d − 1 N .

We define U 2 , 1 and U 2 , 2 as

U 2 , 1 = ∑ d ∣ n d ≡ i ( N ) ( n / d ) 2 k 2 ( d − 1 ) N + ∑ d ∣ n d ≡ − i ( N ) ( n / d ) 2 k 2 ( d − 1 ) N , U 2 , 2 = ∑ d ∣ n d ≡ i ( N ) ( n / d ) 2 k d − 1 N + ∑ d ∣ n d ≡ − i ( N ) ( n / d ) 2 k d − 1 N

and we evaluate U 2 , 1 as follows:

∑ d ∣ n n d ≡ i ( N ) d 2 k 2 ( n / d − 1 ) N + ∑ d ∣ n n d ≡ − i ( N ) d 2 k 2 ( n / d − 1 ) N = ∑ d ∣ n n d ≡ i ( N ) d 2 k 2 ( n / d − i ) N + ∑ d ∣ n n d ≡ − i ( N ) d 2 k 2 ( n / d + i ) N − 1 = 2 n N ∑ d ∣ n n d ≡ i ( N ) d 2 k − 1 + ∑ d ∣ n n d ≡ − i ( N ) d 2 k − 1 − 2 i N ∑ d ∣ n n d ≡ i ( N ) d 2 k − ∑ d ∣ n n d ≡ − i ( N ) d 2 k − ∑ d ∣ n n d ≡ − i ( N ) d 2 k = 2 n N σ 2 k − 1 ♯ ( n ; i , N ) − 2 i N σ 2 k ♯ ( n ; i , N ) − ∑ d ∣ n n d ≡ − i ( N ) d 2 k .

Finally, in the same way we can obtain that

U 2 , 2 = ∑ d ∣ n n d ≡ i ( N ) d 2 k n / d − 1 N + ∑ d ∣ n n d ≡ − i ( N ) d 2 k n / d − 1 N = n N σ 2 k − 1 ♯ ( n ; i , N ) − i N σ 2 k ♯ ( n ; i , N ) − ∑ d ∣ n n d ≡ − i ( N ) d 2 k .

Hence, we see that

U 2 = U 2 , 1 − U 2 , 2 = n N σ 2 k − 1 ♯ ( n ; i , N ) − i N σ 2 k ♯ ( n ; i , N ) .

Finally, we conclude that − ( U 1 + U 2 ) is equal to

− n N σ 2 k − 1 ♯ ( n ; i , N ) − n N σ 2 k − 1 ♯ ( n , N / 2 + i , N ) + i N σ 2 k ♯ ( n ; i , N ) + i N − 1 2 σ 2 k ♯ ( n ; N / 2 + i , N ) . □

Corollary 2.4

For any k , n , N ∈ N with 4 ∣ N , we have

( i ) ∑ s = 0 k 2 k 2 s ∑ m = 1 n − 1 σ 2 k − 2 s ♯ ( m ; N / 4 , N ) σ 2 s ♯ ( n − m ; N / 4 , N ) = 1 2 σ 2 k + 1 ♯ ( n ; N / 4 , N ) − 1 2 σ 2 k ♯ ( n ; N / 4 , N ) , ( i i ) ∑ s = 0 k − 1 2 k 2 s + 1 ∑ m = 1 n − 1 σ 2 k − 2 s − 1 ♯ ( m ; N / 4 , N ) σ 2 s + 1 ♯ ( n − m ; N / 4 , N ) = 1 2 σ 2 k + 1 ♯ ( n ; N / 4 , N ) − 2 n N σ 2 k − 1 ♯ ( n ; N / 4 , N ) .

Proof

By Proposition 2.1 and Theorem 2.2, we obtain that

∑ s = 0 2 k 2 k s ∑ m = 1 n − 1 σ 2 k − s ♯ ( m ; N / 4 , N ) σ s ♯ ( n − m ; N / 4 , N ) = σ 2 k + 1 ♯ ( n ; N / 4 , N ) − 1 2 σ 2 k ♯ ( n ; N / 4 , N ) − 2 n N σ 2 k − 1 ♯ ( n ; N / 4 , N ) ,

and

∑ s = 0 2 k 2 k s ∑ m = 1 n − 1 ( − 1 ) s σ 2 k − s ♯ ( m ; N / 4 , N ) σ s ♯ ( n − m ; N / 4 , N ) = 2 n N σ 2 k − 1 ♯ ( n ; N / 4 , N ) − 1 2 σ 2 k ♯ ( n ; N / 4 , N ) .

Adding these identities we obtain the identity (i) because σ r ♯ ( n ; i , N ) = − ( − 1 ) r σ r ♯ ( n ; − i , N ) . Similarly, we easily deduce (ii).□

Lemma 2.5

Let k , n , N ∈ N and assume that N / 4 is an odd integer. Then

σ k ♯ ( 2 n ; N / 4 , N ) = 2 k σ k ♯ ( n ; N / 4 , N ) .

Proof

If d ∣ 2 n and 2 n d ≡ N 4 ( mod N ) ( resp., 2 n d ≡ − N 4 ( mod N ) ) , then d must be even. Put d ′ = d 2 . Then d ′ ∣ n and n d ′ ≡ N 4 ( mod N ) ( resp., n d ′ ≡ − N 4 ( mod N ) ) . Hence,

σ k ♯ ( 2 n ; N / 4 , N ) = ∑ d ∣ 2 n 2 n d ≡ N 4 ( N ) d k − ( − 1 ) k ∑ d ∣ 2 n 2 n d ≡ − N 4 ( N ) d k = ∑ d ∣ n n d ≡ N 4 ( N ) ( 2 d ) k − ( − 1 ) k ∑ d ∣ n n d ≡ − N 4 ( N ) ( 2 d ) k = 2 k σ k ♯ ( n ; N / 4 , N ) . □

Moreover, we can obtain another result about the binomial convolution sum, which turns out to be a special case of our main theorem.

Proposition 2.6

Let k , n , N be positive integers with N / 4 odd. Then

( i ) ∑ s = 0 k 2 k 2 s ∑ m = 1 2 n − 1 ( − 1 ) m σ 2 k − 2 s ♯ ( m ; N / 4 , N ) σ 2 s ♯ ( 2 n − m ; N / 4 , N ) = − 2 2 k − 1 σ 2 k ♯ ( n ; N / 4 , N ) , ( i i ) ∑ s = 0 k − 1 2 k 2 s + 1 ∑ m = 1 2 n − 1 ( − 1 ) m σ 2 k − 2 s − 1 ♯ ( m ; N / 4 , N ) σ 2 s + 1 ♯ ( 2 n − m ; N / 4 , N ) = 2 2 k + 2 n N σ 2 k − 1 ♯ ( n ; N / 4 , N ) .

Proof

(i) We have

∑ s = 0 k 2 k 2 s ∑ m = 1 2 n − 1 ( − 1 ) m σ 2 k − 2 s ♯ ( m ; N / 4 , N ) σ 2 s ♯ ( 2 n − m ; N / 4 , N ) = 2 ∑ s = 0 k 2 k 2 s ∑ m = 1 m even 2 n − 1 σ 2 k − 2 s ♯ ( m ; N / 4 , N ) σ 2 s ♯ ( 2 n − m ; N / 4 , N ) − ∑ s = 0 k 2 k 2 s ∑ m = 1 2 n − 1 σ 2 k − 2 s ♯ ( m ; N / 4 , N ) σ 2 s ♯ ( 2 n − m ; N / 4 , N ) = 2 ∑ s = 0 k 2 k 2 s ∑ m = 1 n − 1 σ 2 k − 2 s ♯ ( 2 m ; N / 4 , N ) σ 2 s ♯ ( 2 n − 2 m ; N / 4 , N ) − ∑ s = 0 k 2 k 2 s ∑ m = 1 2 n − 1 σ 2 k − 2 s ♯ ( m ; N / 4 , N ) σ 2 s ♯ ( 2 n − m ; N / 4 , N ) .

From (i) of Corollary 2.4 and Lemma 2.5, we easily deduce that this is equal to

2 2 k + 1 ∑ s = 0 k 2 k 2 s ∑ m = 1 n − 1 σ 2 k − 2 s ♯ ( m ; N / 4 , N ) σ 2 s ♯ ( n − m ; N / 4 , N ) − ∑ s = 0 k 2 k 2 s ∑ m = 1 2 n − 1 σ 2 k − 2 s ♯ ( m ; N / 4 , N ) σ 2 s ♯ ( 2 n − m ; N / 4 , N ) = 2 2 k ( σ 2 k + 1 ♯ ( n ; N / 4 , N ) − σ 2 k ♯ ( n ; N / 4 , N ) ) − 1 2 ( σ 2 k + 1 ♯ ( 2 n ; N / 4 , N ) − σ 2 k ♯ ( 2 n ; N / 4 , N ) ) = − 2 2 k − 1 σ 2 k ♯ ( n ; N / 4 , N ) .

(ii) From (ii) of Corollary 2.4 and Lemma 2.5, we obtain

∑ s = 0 k − 1 2 k 2 s + 1 ∑ m = 1 2 n − 1 ( − 1 ) m σ 2 k − 2 s − 1 ♯ ( m ; N / 4 , N ) σ 2 s + 1 ♯ ( 2 n − m ; N / 4 , N ) = 2 ∑ s = 0 k − 1 2 k 2 s + 1 ∑ m = 1 m even 2 n − 1 σ 2 k − 2 s − 1 ♯ ( m ; N / 4 , N ) σ 2 s + 1 ♯ ( 2 n − m ; N / 4 , N ) − ∑ s = 0 k − 1 2 k 2 s + 1 ∑ m = 1 2 n − 1 σ 2 k − 2 s − 1 ♯ ( m ; N / 4 , N ) σ 2 s + 1 ♯ ( 2 n − m ; N / 4 , N ) = 2 ∑ s = 0 k − 1 2 k 2 s + 1 ∑ m = 1 n − 1 σ 2 k − 2 s − 1 ♯ ( 2 m ; N / 4 , N ) σ 2 s + 1 ♯ ( 2 n − 2 m ; N / 4 , N ) − ∑ s = 0 k − 1 2 k 2 s + 1 ∑ m = 1 2 n − 1 σ 2 k − 2 s − 1 ♯ ( m ; N / 4 , N ) σ 2 s + 1 ♯ ( 2 n − m ; N / 4 , N ) = 2 2 k + 1 ∑ s = 0 k − 1 2 k 2 s + 1 ∑ m = 1 n − 1 σ 2 k − 2 s − 1 ♯ ( m ; N / 4 , N ) σ 2 s + 1 ♯ ( n − m ; N / 4 , N ) − ∑ s = 0 k − 1 2 k 2 s + 1 ∑ m = 1 2 n − 1 σ 2 k − 2 s − 1 ♯ ( m ; N / 4 , N ) σ 2 s + 1 ♯ ( 2 n − m ; N / 4 , N ) = 2 2 k + 1 1 2 σ 2 k + 1 ♯ ( n ; N / 4 , N ) − 2 n N σ 2 k − 1 ♯ ( n ; N / 4 , n ) − 1 2 σ 2 k + 1 ♯ ( 2 n ; N / 4 , N ) − 4 n N σ 2 k − 1 ♯ ( 2 n ; N / 4 , N ) = 2 2 k + 2 n N σ 2 k − 1 ♯ ( n ; N / 4 , N ) . □

3 Applications of Theorem 2.2 to trinomial and multinomial convolution sums of σ r ♯

We present three identities Proposition 3.1 ( i ) – ( i i i ) about trinomial convolution sums of σ r ♯ ( n ; N / 4 , N ) and Part (ii) will be generalized to the case of multinomial convolution sum.

Proposition 3.1

Let k , n , N be positive integers with N / 4 odd. Then we have

( i ) ∑ ( a , b , c ) ∈ N 0 3 a + b + c = k 2 k 2 a , 2 b , 2 c ∑ ( m 1 , m 2 , m 3 ) ∈ N 3 m 1 + m 2 + m 3 = 2 n m 3 even ( − 1 ) m 1 σ 2 a ♯ ( m 1 ; N / 4 , N ) σ 2 b ♯ ( m 2 ; N / 4 , N ) σ 2 c ♯ ( m 3 ; N / 4 , N ) = − 2 2 k − 2 ( σ 2 k + 1 ♯ ( n ; N / 4 , N ) − σ 2 k ♯ ( n ; N / 4 , N ) ) , ( i i ) ∑ ( a , b , c ) ∈ N 0 3 a + b + c = k 2 k 2 a , 2 b , 2 c ∑ ( m 1 , m 2 , m 3 ) ∈ N 3 m 1 + m 2 + m 3 = 4 n m 3 even ( − 1 ) m 2 + m 3 2 σ 2 a ♯ ( m 1 ; N / 4 , N ) σ 2 b ♯ ( m 2 ; N / 4 , N ) σ 2 c ♯ ( m 3 ; N / 4 , N ) = 2 4 k − 2 σ 2 k ♯ ( n ; N / 4 , N ) , ( i i i ) ∑ ( a , b , c ) ∈ N 0 3 a + b + c = k 2 k 2 a , 2 b , 2 c ∑ ( m 1 , m 2 , m 3 ) ∈ N 3 m 1 + m 2 + m 3 = 4 n m 3 ≡ 2 ( 4 ) ( − 1 ) m 1 σ 2 a ♯ ( m 1 ; N / 4 , N ) σ 2 b ♯ ( m 2 ; N / 4 , N ) σ 2 c ♯ ( m 3 ; N / 4 , N ) = − 2 4 k − 2 σ 2 k + 1 ♯ ( n ; N / 4 , N ) .

Proof of (i)

Note that

∑ a + b + c = k 2 k 2 a , 2 b , 2 c ∑ ( m 1 , m 2 , m 3 ) ∈ N 3 m 1 + m 2 + m 3 = 2 n m 3 even ( − 1 ) m 1 σ 2 a ♯ ( m 1 ; N / 4 , N ) σ 2 b ♯ ( m 2 ; N / 4 , N ) σ 2 c ♯ ( m 3 ; N / 4 , N ) = ∑ a + b + c = k ( 2 k ) ! ( 2 a + 2 b ) ! ( 2 c ) ! ( 2 a + 2 b ) ! ( 2 a ) ! ( 2 b ) ! ∑ ( m 1 , m 2 , m 3 ) ∈ N 3 m 1 + m 2 + m 3 = 2 n m 3 even ( − 1 ) m 1 σ 2 a ♯ ( m 1 ; N / 4 , N ) σ 2 b ♯ ( m 2 ; N / 4 , N ) σ 2 c ♯ ( m 3 ; N / 4 , N ) .

We set l = a + b and 2 t = m 1 + m 2 . Then we have

∑ l = 0 k 2 k 2 l ∑ t = 1 n − 1 ∑ a = 0 l 2 l 2 a ∑ m 1 = 1 2 t − 1 ( − 1 ) m 1 σ 2 a ♯ ( m 1 ; N / 4 , N ) σ 2 l − 2 a ♯ ( 2 t − m 1 ; N / 4 , N ) σ 2 k − 2 l ♯ ( 2 n − 2 t ; N / 4 , N ) .

By (i) of Corollary 2.4 and Lemma 2.5, we obtain that

− 1 2 ∑ l = 0 k 2 k 2 l ∑ t = 1 n − 1 σ 2 l ♯ ( 2 t ; N / 4 , N ) σ 2 k − 2 l ♯ ( 2 n − 2 t ; N / 4 , N ) = − 2 2 k − 1 ∑ l = 0 k 2 k 2 l ∑ t = 1 n − 1 σ 2 l ♯ ( t ; N / 4 , N ) σ 2 k − 2 l ♯ ( n − t ; N / 4 , N ) = − 2 2 k − 2 ( σ 2 k + 1 ♯ ( n ; N / 4 , N ) − σ 2 k ♯ ( n ; N / 4 , N ) ) . □

Proof of (ii)

Putting l = a + b and 2 t = m 1 + m 2 , we have

∑ a + b + c = k 2 k 2 a , 2 b , 2 c ∑ ( m 1 , m 2 , m 3 ) ∈ N 3 m 1 + m 2 + m 3 = 4 n m 3 even ( − 1 ) m 1 + m 3 2 σ 2 a ♯ ( m 1 ; N / 4 , N ) σ 2 b ♯ ( m 2 ; N / 4 , N ) σ 2 c ♯ ( m 3 ; N / 4 , N ) = ∑ l = 0 k 2 k 2 l ∑ t = 1 2 n − 1 ( − 1 ) t ∑ a = 0 l 2 l 2 a ∑ m 1 = 1 2 t − 1 ( − 1 ) m 1 σ 2 a ♯ ( m 1 ; N / 4 , N ) σ 2 l − 2 a ♯ ( 2 t − m 1 ; N / 4 , N ) σ 2 k − 2 l ♯ ( 4 n − 2 t ; N / 4 , N ) .

From Lemma 2.5 and Proposition 2.6, we immediately infer

− 1 2 ∑ l = 0 k 2 k 2 l ∑ t = 1 2 n − 1 ( − 1 ) t σ 2 l ♯ ( 2 t ; N / 4 , N ) σ 2 k − 2 l ♯ ( 4 n − 2 t ; N / 4 , N ) = − 2 2 k − 1 ∑ l = 0 k 2 k 2 l ∑ t = 1 2 n − 1 ( − 1 ) t σ 2 l ♯ ( t ; N / 4 , N ) σ 2 k − 2 l ♯ ( 2 n − t ; N / 4 , N ) = 2 4 k − 2 σ 2 k ♯ ( n ; N / 4 , N ) . □

Proof of (iii)

Replace 2 n by 4 n in (i) of Proposition 3.1. Then we have

∑ ( a , b , c ) ∈ N 0 3 a + b + c = k 2 k 2 a , 2 b , 2 c ∑ ( m 1 , m 2 , m 3 ) ∈ N 3 m 1 + m 2 + m 3 = 4 n m 3 even ( − 1 ) m 1 σ 2 a ♯ ( m 1 ; N / 4 , N ) σ 2 b ♯ ( m 2 ; N / 4 , N ) σ 2 c ♯ ( m 3 ; N / 4 , N ) = − 2 2 k − 2 ( σ 2 k + 1 ♯ ( 2 n ; N / 4 , N ) − σ 2 k ♯ ( 2 n ; N / 4 , N ) ) .

The (iii) in Proposition 3.1 is correct if subtract (ii) in Proposition 3.1 from the above term.□

We now generalize Proposition 3.1(ii) to the case of multinomial convolution sum as follows.

Proof of Theorem 1.7

We proceed by induction on t . The assertion for the case t = 2 is immediate due to Proposition 2.6. Suppose that the assertion is true for some t − 1 ≥ 2 . Namely, we assume that for all l , m , N ∈ N with N / 4 odd, we have

∑ s 1 + ⋯ + s t − 1 = l 2 l 2 s 1 , … , 2 s t − 1 × ∑ M ′ ( − 1 ) α ′ σ 2 s 1 ♯ ( m 1 ; N / 4 , N ) ⋯ σ 2 s t ♯ ( m t − 1 ; N / 4 , N ) = ( − 1 ) t 2 ( 2 l − 1 ) ( t − 2 ) σ 2 l ♯ ( m ) ,

where α ′ = m 2 + m 3 2 + ⋯ + m t − 1 2 t − 3 and

M ′ = { ( m 1 , … , m t − 1 ) ∈ N t ∣ m 1 + ⋯ + m t − 1 = 2 t − 1 m , m 1 + ⋯ + m i ≡ 0 ( mod 2 i − 1 ) for all 2 ≤ i ≤ t − 1 } .

Fix l = s 1 + ⋯ + s t − 1 and m = m 1 + ⋯ + m t − 1 2 t − 2 . Applying Lemma 2.5 and Proposition 2.6, we obtain

∑ s 1 + ⋯ + s t = k 2 k 2 s 1 , … , 2 s t ∑ M ( − 1 ) α σ 2 s 1 ♯ ( m 1 ; N / 4 , N ) ⋯ σ 2 s t ♯ ( m t ; N / 4 , N ) = ∑ s 1 + ⋯ + s t = k ( 2 k ) ! ( 2 s 1 + ⋯ + 2 s t − 1 ) ! ( 2 s t ) ! ( 2 s 1 + ⋯ + 2 s t − 1 ) ! ( 2 s 1 ) ! ⋯ ( 2 s t − 1 ) ! ∑ M ( − 1 ) α σ 2 s 1 ♯ ( m 1 ; N / 4 , N ) ⋯ σ 2 s t ♯ ( m t ; N / 4 , N ) = ∑ l = 0 k 2 k 2 l ∑ m = 1 2 n − 1 ( − 1 ) m σ 2 k − 2 l ♯ ( 2 t − 2 ( 2 n − m ) ; N / 4 , N ) ∑ s 1 + ⋯ + s t − 1 = l 2 l 2 s 1 , … , 2 s t − 1 ∑ M ′ ( − 1 ) α ′ σ 2 s 1 ♯ ( m 1 ; N / 4 , N ) ⋯ × σ 2 s t − 1 ♯ ( m t − 1 ; N / 4 , N ) = ∑ l = 0 k 2 k 2 l ∑ m = 1 2 n − 1 ( − 1 ) m ( − 2 2 l − 1 ) t − 2 σ 2 l ♯ ( m ; N / 4 , N ) σ 2 k − 2 l ♯ ( 2 t − 1 n − 2 t − 2 m ; N / 4 , N ) = ( − 2 2 k − 1 ) t − 2 ∑ l = 0 k 2 k 2 l ∑ m = 1 2 n − 1 ( − 1 ) m σ 2 l ♯ ( m ; N / 4 , N ) σ 2 k − 2 l ♯ ( 2 n − m ; N / 4 , N ) = ( − 2 2 k − 1 ) t − 1 σ 2 k ♯ ( n ; N / 4 , N ) . □

Acknowledgements

The author wishes to thank the anonymous referee for their useful comments and would like to thank Bumkyu Cho for suggesting an idea of this paper.

  1. Funding information: This work was supported by the National Research Foundation of Korea (NRF-2018R1A2B6001645 and NRF-2019R1C1C1010211).

  2. Conflict of interest: The author states no conflict of interest.

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Received: 2021-08-27
Revised: 2022-03-23
Accepted: 2022-04-08
Published Online: 2022-06-22

© 2022 Ho Park, published by De Gruyter

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

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https://doi.org/10.1515/math-2022-0038
Keywords for this article
divisor functions; convolution sums
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  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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