scholarly journals A study of approximate normal distribution derived from combinatoric convolution sums of divisor functions

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1811-1821
Author(s):  
Daeyeoul Kim ◽  
Kwangchul Lee ◽  
Gyeong-Sig Seo
Keyword(s):  
Normal Distribution ◽  
Legendre Polynomials ◽  
Bernoulli Polynomials ◽  
Convolution Sums ◽  
Divisor Functions

In this paper, we consider the relations between Bernoulli polynomials, Legendre polynomials and combinatoric convolution sums of divisor functions. In addition, we give examples of approximate normal distribution derived from combinatoric convolution sums of divisor functions.

Evaluating binomial convolution sums of divisor functions in terms of Euler and Bernoulli polynomials

2018 ◽  
Vol 14 (02) ◽  
pp. 509-525 ◽  
Author(s):  
Bumkyu Cho ◽  
Ho Park
Keyword(s):  
Recent Result ◽  
Bernoulli Polynomials ◽  
Convolution Sums ◽  
Divisor Functions ◽  
Binomial Convolution ◽  
Special Case

In this paper, we provide two identities about binomial convolution sums of [Formula: see text] with [Formula: see text], which are expressed in terms of Euler and Bernoulli polynomials. A recent result of Kim, Bayad and Park turns out to be a special case of one of the two identities when [Formula: see text].

scholarly journals A PRODUCT FORMULA FOR COMBINATORIC CONVOLUTION SUMS OF ODD DIVISOR FUNCTIONS

2016 ◽  
Vol 38 (2) ◽  
pp. 243-257
Author(s):  
Kwangchul Lee ◽  
Daeyeoul Kim ◽  
Gyeong-Sig Seo
Keyword(s):  
Product Formula ◽  
Convolution Sums ◽  
Divisor Functions

scholarly journals Old and New Identities for Bernoulli Polynomials via Fourier Series

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Luis M. Navas ◽  
Francisco J. Ruiz ◽  
Juan L. Varona
Keyword(s):  
Fourier Series ◽  
Linear Combination ◽  
Legendre Polynomials ◽  
Fourier Coefficients ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Gegenbauer Polynomials ◽  
Linear Combinations ◽  
The Given

The Bernoulli polynomialsBkrestricted to[0,1)and extended by periodicity haventh sine and cosine Fourier coefficients of the formCk/nk. In general, the Fourier coefficients of any polynomial restricted to[0,1)are linear combinations of terms of the form1/nk. If we can make this linear combination explicit for a specific family of polynomials, then by uniqueness of Fourier series, we get a relation between the given family and the Bernoulli polynomials. Using this idea, we give new and simpler proofs of some known identities involving Bernoulli, Euler, and Legendre polynomials. The method can also be applied to certain families of Gegenbauer polynomials. As a result, we obtain new identities for Bernoulli polynomials and Bernoulli numbers.

scholarly journals CONVOLUTION SUMS ARISING FROM DIVISOR FUNCTIONS

2013 ◽  
Vol 50 (2) ◽  
pp. 331-360 ◽  
Author(s):  
Aeran Kim ◽  
Daeyeoul Kim ◽  
Li Yan
Keyword(s):  
Convolution Sums ◽  
Divisor Functions
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