scholarly journals Some combinatorial series identities associated with the Digamma function and harmonic numbers

2000 ◽  
Vol 13 (3) ◽  
pp. 101-106 ◽  
Author(s):  
Tsu-Chen Wu ◽  
Shih-Tong Tu ◽  
H.M. Srivastava
Keyword(s):  
Harmonic Numbers ◽  
Digamma Function

scholarly journals A New Class of Symmetric Beta Type Distributions Constructed by Means of Symmetric Bernstein Type Basis Functions

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 779 ◽  
Author(s):  
Fusun Yalcin ◽  
Yilmaz Simsek
Keyword(s):  
Bernoulli Polynomials ◽  
Random Variable ◽  
Stirling Numbers ◽  
Moment Generating Function ◽  
Basis Functions ◽  
Integral Representations ◽  
Harmonic Numbers ◽  
Bernstein Type ◽  
Digamma Function ◽  
New Class

The main aim of this paper is to define and investigate a new class of symmetric beta type distributions with the help of the symmetric Bernstein-type basis functions. We give symmetry property of these distributions and the Bernstein-type basis functions. Using the Bernstein-type basis functions and binomial series, we give some series and integral representations including moment generating function for these distributions. Using generating functions and their functional equations, we also give many new identities related to the moments, the polygamma function, the digamma function, the harmonic numbers, the Stirling numbers, generalized harmonic numbers, the Lah numbers, the Bernstein-type basis functions, the array polynomials, and the Apostol–Bernoulli polynomials. Moreover, some numerical values of the expected values for the logarithm of random variable are given.

Some evaluations of infinite series involving parametric harmonic numbers

2019 ◽  
Vol 15 (07) ◽  
pp. 1531-1546
Author(s):  
Ce Xu
Keyword(s):  
Infinite Series ◽  
Double Series ◽  
Asymptotic Formulas ◽  
Residue Theorem ◽  
Harmonic Numbers ◽  
Digamma Function ◽  
Special Cases

In this paper, we deduce the asymptotic formulas of parametric digamma function [Formula: see text] at the integers and poles. Then using these identities and residue theorem, we establish a large number of formulas of double series involving parametric harmonic numbers. Some illustrative special cases as well as immediate consequences of the main results are also considered.

scholarly journals New fractional approaches for n-polynomial P-convexity with applications in special function theory

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shu-Bo Chen ◽  
Saima Rashid ◽  
Muhammad Aslam Noor ◽  
Zakia Hammouch ◽  
Yu-Ming Chu
Keyword(s):  
Fractional Calculus ◽  
Convex Functions ◽  
Special Functions ◽  
Real Life ◽  
Fractional Integrals ◽  
Digamma Function ◽  
Novel Approach ◽  
Inequality Theory ◽  
New Strategy ◽  
Significant Mechanism

Abstract Inequality theory provides a significant mechanism for managing symmetrical aspects in real-life circumstances. The renowned distinguishing feature of integral inequalities and fractional calculus has a solid possibility to regulate continuous issues with high proficiency. This manuscript contributes to a captivating association of fractional calculus, special functions and convex functions. The authors develop a novel approach for investigating a new class of convex functions which is known as an n-polynomial $\mathcal{P}$ P -convex function. Meanwhile, considering two identities via generalized fractional integrals, provide several generalizations of the Hermite–Hadamard and Ostrowski type inequalities by employing the better approaches of Hölder and power-mean inequalities. By this new strategy, using the concept of n-polynomial $\mathcal{P}$ P -convexity we can evaluate several other classes of n-polynomial harmonically convex, n-polynomial convex, classical harmonically convex and classical convex functions as particular cases. In order to investigate the efficiency and supremacy of the suggested scheme regarding the fractional calculus, special functions and n-polynomial $\mathcal{P}$ P -convexity, we present two applications for the modified Bessel function and $\mathfrak{q}$ q -digamma function. Finally, these outcomes can evaluate the possible symmetric roles of the criterion that express the real phenomena of the problem.

On 3-adic Valuations of Generalized Harmonic Numbers

Integers ◽  
2012 ◽  
Vol 12 (2) ◽  
Author(s):  
Ken Kamano
Keyword(s):  
Harmonic Numbers ◽  
Generalized Harmonic Numbers

Abstract.We investigate 3-adic valuations of generalized harmonic numbers

scholarly journals Harmonic Numbers and Cubed Binomial Coefficients

2011 ◽  
Vol 2011 ◽  
pp. 1-14
Author(s):  
Anthony Sofo
Keyword(s):  
Closed Form ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Polygamma Functions ◽  
Form Representation ◽  
The Given

Euler related results on the sum of the ratio of harmonic numbers and cubed binomial coefficients are investigated in this paper. Integral and closed-form representation of sums are developed in terms of zeta and polygamma functions. The given representations are new.

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