Sums of Powers, Bernoulli Numbers, and the Gamma Function

1985 ◽  
pp. 150-180
Author(s):  
Bruce C. Berndt
Keyword(s):  
Gamma Function ◽  
Bernoulli Numbers ◽  
Sums Of Powers

scholarly journals Obtaining Easily Sums of Powers on Arithmetic Progressions and Properties of Bernoulli Polynomials by Operator Calculus

2017 ◽  
Vol 9 (5) ◽  
pp. 73
Author(s):  
Do Tan Si
Keyword(s):  
Differential Operator ◽  
Generating Function ◽  
Arithmetic Progression ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Arithmetic Progressions ◽  
Operator Calculus ◽  
Sums Of Powers ◽  
The Way

We show that a sum of powers on an arithmetic progression is the transform of a monomial by a differential operator and that its generating function is simply related to that of the Bernoulli polynomials from which consequently it may be calculated. Besides, we show that it is obtainable also from the sums of powers of integers, i.e. from the Bernoulli numbers which in turn may be calculated by a simple algorithm.By the way, for didactic purpose, operator calculus is utilized for proving in a concise manner the main properties of the Bernoulli polynomials. 

Sums of Powers of Integers and Bernoulli Numbers

1970 ◽  
Vol 54 (389) ◽  
pp. 272 ◽  
Author(s):  
Howard Sherwood
Keyword(s):  
Bernoulli Numbers ◽  
Sums Of Powers

scholarly journals Fibonacci Series from Power Series

2020 ◽  
Author(s):  
Kunle Adegoke
Keyword(s):  
Power Series ◽  
Infinite Series ◽  
Gamma Function ◽  
Bernoulli Numbers ◽  
Trigonometric Functions ◽  
Euler Numbers ◽  
Lucas Numbers ◽  
Fibonacci Series ◽  
Digamma Function ◽  
Inverse Trigonometric Functions

We show how every power series gives rise to a Fibonacci series and a companion series involving Lucas numbers. For illustrative purposes, Fibonacci series arising from trigonometric functions, inverse trigonometric functions, the gamma function and the digamma function are derived. Infinite series involving Fibonacci and Bernoulli numbers and Fibonacci and Euler numbers are also obtained.

scholarly journals Some inequalities arising from analytic summability of functions

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 3223-3230
Author(s):  
Sh. Saadat ◽  
M.H. Hooshmand
Keyword(s):  
Functional Equation ◽  
Holomorphic Function ◽  
Upper Bound ◽  
Bernoulli Polynomials ◽  
Upper Bounds ◽  
Bernoulli Numbers ◽  
Trigonometric Functions ◽  
Natural Numbers ◽  
Sums Of Powers ◽  
The Difference

Analytic summability of functions was introduced by the second author in 2016. He utilized Bernoulli numbers and polynomials for a holomorphic function to construct analytic summability. The analytic summand function f? (if exists) satisfies the difference functional equation f?(z) = f (z) + f?(z-1). Moreover, some upper bounds for f? and several inequalities between f and f? were presented by him. In this paper, by using Alzer?s improved upper bound for Bernoulli numbers, we improve those upper bounds and obtain some inequalities and new upper bounds. As some applications of the topic, we obtain several upper bounds for Bernoulli polynomials, sums of powers of natural numbers, (e.g., 1p+2p+3p+...+rp ? 2p! ?p+1 (e?r-1)) and several inequalities for exponential, hyperbolic and trigonometric functions.

scholarly journals Sums of Powers of Integers and Bernoulli Numbers Clarified

2017 ◽  
Vol 9 (2) ◽  
pp. 12
Author(s):  
DO TAN SI
Keyword(s):  
Bernoulli Numbers ◽  
Inverse Matrix ◽  
Simple Method ◽  
Pascal Triangle ◽  
Sums Of Powers ◽  
As If

This work exposes a very simple method for calculating at the same time the sums of powers of the first integers S_m(n) and the Bernoulli numbers B_m. This is possible thank only to the relation S_m(x+1)-S_m(x)= x^m and the Pascal formula concerning S_m(n) which may be explained as if the vector  n^2-n, n^3-n,...,n^(m+1)-n  is the transform of the vector S_1(n), S_2(n),...,S_m(n)  by a matrix P built from the Pascal triangle. Very useful relations between the sums S_m(n), the Bernoulli numbers B_m and elements of the inverse matrix of P are deduced, leading straightforwardly to known and new properties of them.

scholarly journals Fibonacci series from power series

2021 ◽  
Vol 27 (3) ◽  
pp. 44-62
Author(s):  
Kunle Adegoke ◽  
Keyword(s):  
Power Series ◽  
Infinite Series ◽  
Gamma Function ◽  
Bernoulli Numbers ◽  
Trigonometric Functions ◽  
Euler Numbers ◽  
Lucas Numbers ◽  
Fibonacci Series ◽  
Digamma Function

We show how every power series gives rise to a Fibonacci series and a companion series involving Lucas numbers. For illustrative purposes, Fibonacci series arising from trigonometric functions, the gamma function and the digamma function are derived. Infinite series involving Fibonacci and Bernoulli numbers and Fibonacci and Euler numbers are also obtained.

ON THE INCOMPLETE GAMMA FUNCTION AND ITS NEUTRIX CONVOLUTION FOR NEGATIVE INTEGERS

2019 ◽  
Vol 10 (1) ◽  
pp. 30-51
Author(s):  
Mongkolsery Lin ◽  
◽  
Brian Fisher ◽  
Somsak Orankitjaroen ◽  
◽  
...  
Keyword(s):  
Gamma Function ◽  
Incomplete Gamma Function
Sign in / Sign up

Export Citation Format

Share Document