Harmonic Number Identities Via Euler’s Transform
نویسنده
چکیده
We evaluate several binomial transforms by using Euler's transform for power series. In this way we obtain various binomial identities involving power sums with harmonic numbers.
منابع مشابه
Course MA2C02, Hilary Term 2011 Section 7: Trigonometric Identities, Complex Exponentials and Periodic Sequences
7 Trigonometric Identities, Complex Exponentials and Periodic Sequences 26 7.1 Basic Trigonometric Identities . . . . . . . . . . . . . . . . . . 26 7.2 Basic Trigonometric Integrals . . . . . . . . . . . . . . . . . . 28 7.3 Basic Properties of Complex Numbers . . . . . . . . . . . . . 29 7.4 Complex Numbers and Trigonometrical Identities . . . . . . . 31 7.5 The Exponential of a Complex Numbe...
متن کاملCourse Ma2c03, Hilary Term 2014 Section 7: Periodic Series and Functions
7 Periodic Series and Functions 128 7.1 Basic Trigonometric Identities . . . . . . . . . . . . . . . . . . 128 7.2 Basic Trigonometric Integrals . . . . . . . . . . . . . . . . . . 130 7.3 Basic Properties of Complex Numbers . . . . . . . . . . . . . 131 7.4 Complex Numbers and Trigonometrical Identities . . . . . . . 133 7.5 The Exponential of a Complex Number . . . . . . . . . . . . . 133 7.6...
متن کاملOn the Fourier Transform of the Greatest Common Divisor
The discrete Fourier transform of the greatest common divisor � id[a](m) = m � k=1 gcd(k,m)α m , with αm a primitive m-th root of unity, is a multiplicative function that generalizes both the gcd-sum function and Euler’s totient function. On the one hand it is the Dirichlet convolution of the identity with Ramanujan’s sum, � id[a] = id ∗ c•(a), and on the other hand it can be written as a gener...
متن کاملIntegrals of Fractional Parts and Some New Identities on Bernoulli Numbers
In this paper, we calculate the values of the integrals ∫ 1 0 { 1 x}dx, ∫ ∫ 0≤x,y≤1 { 1 x+y}dxdy, ∫ ∫ ∫ 0≤x,y,z≤1 { 1 x+y+z}dxdydz and ∫ 1 0 { 1 x}{ 1 1−x}dx, where m and n are positive integers and {u} is the fractional part of u, and express their values in terms of Euler’s constant and Riemann-Zeta function. We also obtain a set of identities involving the Bernoulli and Harmonic numbers.
متن کاملq-HYPERGEOMETRIC PROOFS OF POLYNOMIAL ANALOGUES OF THE TRIPLE PRODUCT IDENTITY, LEBESGUE’S IDENTITY AND EULER’S PENTAGONAL NUMBER THEOREM
X iv :m at h/ 02 03 22 9v 1 [ m at h. C O ] 2 2 M ar 2 00 2 2000]Primary 05A19, 33D15 q-HYPERGEOMETRIC PROOFS OF POLYNOMIAL ANALOGUES OF THE TRIPLE PRODUCT IDENTITY, LEBESGUE’S IDENTITY AND EULER’S PENTAGONAL NUMBER THEOREM S. OLE WARNAAR Abstract. We present alternative, q-hypergeometric proofs of some polynomial analogues of classical q-series identities recently discovered by Alladi and Berk...
متن کامل