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Keyword: Harmonic Sums (7)	31 Mar 2025	Run

articleNo Access
The analytic structure of the BFKL equation and reflection identities of harmonic sums at weight five
- Mohammad Joubat and
- Alexander Prygarin
International Journal of Modern Physics A20 Apr 2019
Preview Abstract
We analyze the structure of the eigenvalue of the color-singlet Balitsky–Fadin–Kuraev–Lipatov (BFKL) equation in N=4 SYM in terms of the meromorphic functions obtained by the analytic continuation of harmonic sums from positive even integer values of the argument to the complex plane. The meromorphic functions we discuss have pole singularities at negative integers and take finite values at all other points. We derive the reflection identities for harmonic sums at weight five decomposing a product of two harmonic sums with mixed pole structure into a linear combination of terms each having a pole at either negative or non-negative values of the argument. The pole decomposition demonstrates how the product of two simpler harmonic sums can build more complicated harmonic sums at higher weight. We list a minimal irreducible set of bilinear reflection identities at weight five which presents the main result of the paper. We show how the reflection identities can be used to restore the functional form of the next-to-leading eigenvalue of the color-singlet BFKL equation in N=4SYM, i.e. we argue that it is possible to restore the full functional form on the entire complex plane provided one has information how the function looks like on just two lines on the complex plane. Finally we discuss how nonlinear reflection identities can be constructed from our result with the use of well known quasi-shuffle relations for harmonic sums.
articleNo Access
Analytic continuation of harmonic sums near the integer values
- V. N. Velizhanin
International Journal of Modern Physics A30 Nov 2020
Preview Abstract
We present a simple method for analytic continuation of harmonic sums near negative and positive integer numbers. We provide a precomputed database for the exact expansion of harmonic sums over a small parameter near these integer numbers, along with MATHEMATICA code, which shows the application of the database for actual problems. We also provide the FORM code that was used to obtain the database mentioned above. The applications of the obtained database for the study of evolution equations in the quantum field theory are discussed.
articleNo Access
Analytic continuation of harmonic sums with purely imaginary indices near the integer values
- V. N. Velizhanin
International Journal of Modern Physics A10 Mar 2023
Preview Abstract
We present a simple algebraic method for the analytic continuation of harmonic sums with integer real or purely imaginary indices near negative and positive integers. We provide a MATHEMATICA code for exact expansion of harmonic sums in a small parameter near these integers. As an application, we consider the analytic continuation of the anomalous dimension of twist-1 operators in the ABJM model, which contains nested harmonic sums with purely imaginary indices. We found that in the BFKL-like limit the result has the same single-logarithmic behavior as in $𝒩 = 4$ SYM and QCD, however, we did not find a general expression for the “BFKL Pomeron” eigenvalue in this model. For the slope function, we found full agreement with the expansion of the known general result and give predictions for the first three perturbative terms in the expansion of the next-to-slope function. The proposed method of analytic continuation can also be used for other generalization of nested harmonic sums.
articleNo Access
A congruence involving harmonic sums modulo $p^{α} q^{β}$
- Tianxin Cai,
- Zhongyan Shen, and
- Lirui Jia
International Journal of Number Theory25 Apr 2017
Preview Abstract
In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$ ,
$Z (p^{r}) \equiv - 2 p^{r - 1} B_{p - 3} (mod p^{r}),$
where $Z (n) = \sum_{\binom{i + j + k = n}{i, j, k \in 𝒫_{n}}} \frac{1}{i j k}$ and $𝒫_{n}$ denote the set of positive integers which are prime to $n$ . In this paper, we obtain an unexpected congruence for distinct odd primes $p$ , $q$ and positive integers $α, β$ ,
$Z (p^{α} q^{β}) \equiv 2 (2 - q) (1 - \frac{1}{q^{3}}) p^{α - 1} q^{β - 1} B_{p - 3} (mod p^{α})$
and the necessary and sufficient condition for
$Z (p^{α} q^{β}) \equiv 0 (mod p^{α} q^{β}) .$
Finally, we raise a conjecture that for $n > 1$ and odd prime power $p^{α} ∥ n$ , $α \geq 1$ ,
$Z (n) \equiv \prod_{\binom{q | n}{q \neq p}} (1 - \frac{2}{q}) (1 - \frac{1}{q^{3}}) (- \frac{2 n}{p}) B_{p - 3} (mod p^{α}) .$
However, we fail to prove it even for $n$ with three distinct prime factors.
articleNo Access
On some combinatorial identities and harmonic sums
- Necdet Batir
International Journal of Number Theory01 Feb 2017
Preview Abstract
For any $m, n \in ℕ$ we first give new proofs for the following well-known combinatorial identities
$S_{n} (m) = \sum_{k = 1}^{n} (\binom{n}{k}) \frac{{(- 1)}^{k - 1}}{k^{m}} = \sum_{n \geq r_{1} \geq r_{2} \geq \dots \geq r_{m} \geq 1} \frac{1}{r_{1} r_{2} \dots r_{m}}$
and
$\sum_{k = 1}^{n} {(- 1)}^{n - k} (\binom{n}{k}) k^{n} = n!,$
and then we produce the generating function and an integral representation for $S_{n} (m)$ . Using them we evaluate many interesting finite and infinite harmonic sums in closed form. For example, we show that
$ζ (3) = \frac{1}{9} \sum_{n = 1}^{\infty} \frac{H_{n}^{3} + 3 H_{n} H_{n}^{(2)} + 2 H_{n}^{(3)}}{2^{n}},$
and
$ζ (5) = \frac{2}{45} \sum_{n = 1}^{\infty} \frac{H_{n}^{4} + 6 H_{n}^{2} H_{n}^{(2)} + 8 H_{n} H_{n}^{(3)} + 3 {(H_{n}^{(2)})}^{2} + 6 H_{n}^{(4)}}{n 2^{n}},$
where $H_{n}^{(i)}$ are generalized harmonic numbers defined below.
articleNo Access
Super congruence on harmonic sums
- Peng Yang and
- Tianxin Cai
International Journal of Number Theory16 Oct 2017
Preview Abstract
Wang and Cai established the following harmonic congruence for odd prime and positive integer $r$ :
$\sum_{\begin{array}{c} i + j + k = p^{r} \\ i, j, k \in 𝒫_{p} \end{array}} \frac{1}{i j k} \equiv - 2 p^{r - 1} B_{p - 3} (mod p^{r}),$
where $𝒫_{p}$ denote the set of positive integers which are prime to $n$ . In this paper, the authors generalize it into super congruence for odd prime $p$ and positive integer $r$ :
$\sum_{\begin{array}{c} i + j + k = p^{r} \\ i, j, k \in 𝒫_{p} \end{array}} \frac{1}{i^{α} j^{β} k^{γ}} \equiv τ (α, β, γ) B_{p - α - β - γ} p^{r - 1} (mod p^{r}),$
where
$τ (α, β, γ) = \frac{1}{α + β + γ} (\sum_{j = 1}^{β} {(- 1)}^{α + β - j} (\begin{matrix} α + β - j - 1 \\ α - 1 \end{matrix}) (\begin{matrix} α + β + γ \\ j \end{matrix})$
$+ \sum_{j = 1}^{α} {(- 1)}^{α + β - j} (\begin{matrix} α + β - j - 1 \\ β - 1 \end{matrix}) (\begin{matrix} α + β + γ \\ j \end{matrix}))$
$+ 2 {(- 1)}^{γ} (\begin{matrix} α + β \\ α \end{matrix}) δ_{p - 1, α + β + γ},$
and $α, β, γ$ are positive integers such that $α + β + γ \leq p .$
articleNo Access
Analytic continuation for multiple zeta values using symbolic representations
- Lin Jiu,
- Christophe Vignat, and
- Tanay Wakhare
International Journal of Number Theory25 Sep 2019
Preview Abstract
We introduce a symbolic representation of $r$ -fold harmonic sums at negative indices. This representation allows us to recover and extend some recent results by Duchamp et al., such as recurrence relations and generating functions for these sums. This approach is also applied to the study of the family of extended Bernoulli polynomials, which appear in the computation of harmonic sums at negative indices. It also allows us to reinterpret the Raabe analytic continuation of the multiple zeta function as both a constant term extension of Faulhaber’s formula, and as the result of a natural renormalization procedure for Faulhaber’s formula.