相关论文: On a multiple harmonic power series
We evaluate binomial series with harmonic number coefficients, providing recursion relations, integral representations, and several examples. The results are of interest to analytic number theory, the analysis of algorithms, and…
Two types of finite series of products of harmonic numbers involving nonnegative integer powers are evaluated, also yielding two other important harmonic number identities. The recursion formulas for these sums are derived, which are easily…
We show that the compositions of positive integers may be interpreted in terms of powers of some power series, over arbitrary commutative ring. As consequences, several closed formulas for the compositions as well as for the generalized…
We prove the cyclic sum formulas for certain two-parameter multiple series. These are new and non-trivial generalizations of the cyclic sum formulas for multiple zeta values and multiple zeta-star values.
In a previous paper, we saw how to create formulae for the sum of the terms of a harmonic progression of order $k$, $HP_k(n)$, with integer parameters, $a$ and $b$. In this new paper we make those formulae more general by lifting the…
Further formulas are presented involving quantum mechanics, thermodynamics, and integrable systems. Modifications of dispersionless theory are developed.
The first part of this note is a short introduction on continued fraction expansions for certain algebraic power series. In the last part, as an illustration, we present a family of algebraic continued fractions of degree 4, including a toy…
This paper presents new formulae for the harmonic numbers of order $k$, $H_{k}(n)$, and for the partial sums of two Fourier series associated with them, denoted here by $C^m_{k}(n)$ and $S^m_{k}(n)$. I believe this new formula for…
A new set of formulas for primes is presented. These formulas are more efficient and grow much slower than the two known formulas of Mills and Wright. 3 new formulas are explained.
We present several sequences involving harmonic numbers and the central binomial coefficients. The calculational technique is consists of a special summation method that allows, based on proper two-valued integer functions, to calculate…
We obtain some results about the repeated exponentiation modulo a prime power from the viewpoint of arithmetic dynamical systems. Especially, we extend two asymptotic formulas about periodic points and tails in the case of modulo a prime to…
In this paper, we give explicit evaluation for some infinite series involving generalized (alternating) harmonic numbers. In addition, some formulas for generalized (alternating) harmonic numbers will also be derived.
The paper provides a necessary and sufficient condition for the composition of multivariable formal power series and present the Generalized Chain Rule for formal power series of multiple variables.
We develop new closed form representations of sums of (n + {\alpha})th shifted harmonic numbers and reciprocal binomial coefficients in terms of {\alpha}th shifted harmonic numbers. Some interesting new consequences and illustrative…
Efficient algorithms are known for many operations on truncated power series (multiplication, powering, exponential, ...). Composition is a more complex task. We isolate a large class of power series for which composition can be performed…
We continue our solution of the inverse problem started by the first author in [Int. J. Mod. Phys. A 35, xxxx (2020), in production]. Additional potential functions for exactly solvable problems that correspond to the same energy spectrum…
This paper presents formulae for the sum of the terms of a harmonic progression of order $k$ with integer parameters, $\mathrm{HP}_k(n)$, and for the partial sums of its two associated Fourier series, $C^z_{k}(a,b,n)$ and $S^z_{k}(a,b,n)$.…
Several combinatorial identities are presented, involving Stirling functions of the second kind with a complex variable. The identities involve also Stirling numbers of the first kind, binomial coefficients and harmonic numbers.
We construct the generating function for products of inverse central binomial coefficients with harmonic numbers.
We prove an irreducibility criterion for polynomials with power series coefficients generalizing previous known results concerning quasi-ordinary polynomials.