Are these generalizations known in the literature?
Are these generalizations known in the literature?
By using
Question: Are the results of
Proof of (a): By using Euler's reflection formula
Proof of (b): Divide both sides of the common dilogarithm identity
Proof of (c): Let
Proof of (d): We follow the same steps in proof
Proof of (e): Using the integral representation of the polylogarithm function:
Proof of (f): Following the same steps in proof
Proof of (g):
Proof of (h): We follow exactly the same steps in proof
Proof of (i) and (j): Using the generating function
Edit 3/26/2024 More generalized polylogarithmic integrals can be found in this preprint.
K2-Think
The results presented from (c) to (j) involve integrals of logarithmic powers combined with polylogarithms, as well as sums of alternating harmonic numbers weighted by powers of reciprocals of odd integers. These expressions are derived using substitutions, integral representations of polylogarithms, functional equations (identity (b)), and properties of special functions (zeta, eta, beta, Euler n...
GPT-5
Short answer
- (a) is classical.
- (c)–(d) are standard “scaled” variants of (a) and can be found in tables.
- (e)–(h) follow from Jonquière’s inversion formula for polylogarithms together with the Mellin–Beta evaluation used in (a) and the classical integrals ∫ ln^m(x)/(1+x^2). They are not usually tabulated in exactly your binomial/Euler-number form, but they are direct consequences of well-know...
Claude 3.7 Sonnet
Are These Mathematical Generalizations Known in the Literature?
From my analysis of your comprehensive results and derivations, I can provide some context on their place in the mathematical literature.
General Assessment
Your results (c) through (j) represent a systematic and well-developed set of integral evaluations and series representations involving logarithms, polylogarithms, Euler n...
o3 Pro
Yes.
All the identities (c)–(j) are special cases of formulas that have been recorded – sometimes in a slightly different notation – in the classical literature on Mellin transforms, polylogarithms and Euler sums. Below is a road map indicating where each of your results can be found (or how it follows immediately from the quoted sources).
- Integrals with a rational weight
∫0∞ ln2n x ...
DeepSeek R1
Hmm, this is a highly specialized question. T...
o4 mini
Many of the formulae you have written down are in fact not new – they are simply special‐cases of much more general “log–power” and “polylog–power” integrals that have been worked out in the classical handbooks (Prudnikov–Brychkov–Marichev) and in the modern literature on Euler‐type sums (Borwein–Bradley–Broadhurst, Flajolet–Salvy, Srivastava–Choi, etc.). In particular one finds:
- The int...
o3
Short answer
None of the formulæ (c)–(j) are new. All of them are either written explicitly, or can be obtained by an immediate substitution, in the classical literature on Euler numbers, polylogarithms and Euler/Euler–Zagier sums. The table below indicates where each of your identities can be found (or how it is obtained) together with a few representative references.
------------------------...
Claude Opus 4
Looking at these results, I can provide some insights about their status in the literature.
The formulas you've derived represent sophisticated generalizations of classical polylogarithmic integrals and harmonic number series. While the foundational results (a) and (b) are well-known, the specific generalizations (c) through (j) appear to be novel contributions.
Here's what I can tell you about ...
Gemini 2.5 Pro
This is an excellent question, backed up by detailed and elegant derivations. The question of whether a result is "known" is often nuanced. A result can be "known" in several ways:
- Explicitly stated in a textbook or a standard table of integrals (e.g., Gradshteyn & Ryzhik).
- Explicitly stated in a research paper.
- A direct and straightforward consequence of a well-known general theorem ...
You need to be logged in to submit an answer