Bernoulli Identities and Combinatoric Convolution Sums with Odd Divisor Functions

Kim, Daeyeoul; Park, Yoon Kyung

doi:10.1155/2014/890973

Cited by 4 publications

(3 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The aim of this article is to study two combinatoric convolution sums of the analogous type (1.5) and (1.6) in [13]. Using these new formulas and addition theorem of Bernoulli or Euler polynomials, we derive the explicit formulas for the third and fourth-order convolution sums of divisor functions.…”

Section: Resultsmentioning

confidence: 99%

Certain Combinatoric Convolution Sums and Their Relations to Bernoulli and Euler Polynomials

Kim¹,

Bayad²,

Ikikardes³

2015

Journal of the Korean Mathematical Society

Self Cite

View full text Add to dashboard Cite

Abstract. In this paper, we give relationship between Bernoulli-Euler polynomials and convolution sums of divisor functions. First, we establish two explicit formulas for certain combinatoric convolution sums of divisor functions derived from Bernoulli and Euler polynomials. Second, as applications, we show five identities concerning the third and fourth-order convolution sums of divisor functions expressed by their divisor functions and linear combination of Bernoulli or Euler polynomials.

show abstract

Section: Resultsmentioning

confidence: 99%

Certain Combinatoric Convolution Sums and Their Relations to Bernoulli and Euler Polynomials

Kim¹,

Bayad²,

Ikikardes³

2015

Journal of the Korean Mathematical Society

Self Cite

View full text Add to dashboard Cite

show abstract

“…While this result is not directly related to the methods presented in this paper, we give it a short proof for the sake of completeness, and to illustrate a different way of obtaining divisor function congruences, starting from certain convolution identities. We start from the following result of D. Kim [34,35,36]:…”

Section: Related Congruences and Conjectures For Divisor Functionsmentioning

confidence: 99%

Computer-assisted proofs of congruences for multipartitions and divisor function convolutions, based on methods of differential algebra

Pascadi

2021

Ramanujan J

View full text Add to dashboard Cite

This paper provides algebraic proofs for several types of congruences involving the multipartition function and self-convolutions of the divisor function. Our computations use methods of Differential Algebra in Z qZ, implemented in a couple of MAPLE programs available as ancillary files on arXiv [37,38].The first results of the paper are Ramanujan-type congruences of the form p * k (qn + r) ≡q 0 and σ * k (qn + r) ≡q 0, where p(n) and σ(n) are the partition and divisor functions, q > 3 is prime, and * k denotes k th -order self-convolution. We prove all the valid congruences of this form for q ∈ {5, 7, 11}, including the three Ramanujan congruences [1], and a nontrivial one for q = 17. All such multipartition congruences have already been settled in principle up to a numerical verification due to D. Eichhorn and K. Ono [2] via modular forms, but our proofs are purely algebraic. On the other hand, the majority of the divisor function congruences are new results.We then proceed to search for more general congruences modulo small primes, concerning linear combinations of σ * k (pn + r) for different values of k, as well as weighted convolutions of p(n) and σ(n) with polynomial weights. The paper ends with a few corollaries and extensions for the divisor function congruences, including proofs for three conjectures of N. C. Bonciocat.

show abstract

“…Glaisher [3], H. Hahn [4], J.G. Huard et al [5], D. Kim et al [8], G. Melfi [12] and K.S. Williams [13].…”

Section: Introductionmentioning

confidence: 99%

Triple and Fifth Product of Divisor Functions and Tree Model

Kim¹,

Cheong²,

Park³

2016

Journal of applied mathematics & informatics

Self Cite

View full text Add to dashboard Cite

show abstract

Bernoulli Identities and Combinatoric Convolution Sums with Odd Divisor Functions

Abstract: We study the combinatoric convolution sums involving odd divisor functions, their relations to Bernoulli numbers, and some interesting applications.

Cited by 4 publications

References 6 publications

Certain Combinatoric Convolution Sums and Their Relations to Bernoulli and Euler Polynomials

Certain Combinatoric Convolution Sums and Their Relations to Bernoulli and Euler Polynomials

Computer-assisted proofs of congruences for multipartitions and divisor function convolutions, based on methods of differential algebra

Triple and Fifth Product of Divisor Functions and Tree Model

Contact Info

Product

Resources

About