In summary, the main purpose of this paper is to derive the formulas in Sections 4 and 5. Section 4 hold n-fold iterated integral representations of some special functions (where n is a positive integer), though somewhat dense, all the material up to this and including Section 3 is just advanced Calc 3 level material; Sections 4&5 are the analysis content, section 5 contains fractional integrals as analytic continuations of the previous section's formulas on the variable n which gets continued to a complex-valued parameter.
benorin
Homework Helper
Introduction
This bit is what new thing you can learn reading this:) As for original content, I only have hope that the method of using the sets
$$C_N^n: = \left\{ { \vec x \in {\mathbb{R}^n}|{x_i} \ge 0\forall i,\sum\limits_{k = 1}^n {x_k^{2N}} < n – 1 } \right\}$$
and Dirichlet integrals to evaluate certain integrals of the type
$$\mathop {\lim }\limits_{N \to \infty } \int\limits_{C_N^n} {f\left( {\vec x} \right)d\mu } = \int\limits_{{{\left[ {0,1} \right]}^n}} {f\left( {\vec x} \right)d\mu }$$
might be original material as I have never seen it my reading.
Summary
The main purpose of this paper is to derive the formulas in Sections 4 and 5. Section 4 hold n-fold iterated integral representations of some special functions (where n is a positive integer), though somewhat dense, all the material up to this and including Section 3 is just advanced Calc 3 level material; Sections 4&5 are the analysis content, section 5 contains fractional integrals as analytic...

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Greg Bernhardt
Do you have a specific question? ##\lim_{N\to \infty} C_N^n=[0,1)## is straightforward.

Yes, it is, I had used that particular definition of ##C_N^n## with the ##\leq n-1## (opposed to ##\leq n##) to accommodate and singular point of an a few integrands in section 4, whereas the latter would converge to ##\left[ 0,1\right]##. These two definitions cover all of the integrals in the text.

I updated the Insight to include more exercises in sections 4 and 5 and added a few answers to section 1 at the end.

A few days ago I added the rest of the solutions to the exercises in section 1 (at the end of the document).

I just wanted to say that God gave me every bit of skills, inspiration, people who helped me flesh this note out over the years I worked on it. Some people that helped me after having worked on this note throughout college and finally having had my analysis prof Akemann from UCSB read what was the beginnings of this note, of which he said what I was working with wasn’t well defined, and some years past. A couple of years ago God told me the finish my paper (I learned the other day that this work is called a note, because papers are published-I had been calling it my paper for quite a long time tho) so I came to PF and I got help from @fresh_42 and @FactChecker with and actually quite a lot of other math people here on PF helped (that would be a long list but I will just say search threads started by me and containing the word paper and from these search results you can see all the help I got, also there’s an Insights and Blog dev sub forum which is hidden by default with a few more threads with several advisors who also helped me).

Sorry this turned into kind of long winded credit where credit is due. I do feel that God gets the credit here, having worked all things to this end.

I’m currently rewriting this note, and seek a co-author if any of you are interested? I will say that I’m neural diverse, and if you contact me being interested in being a co-author I will tell you point-blank what my issues are, I just don’t feel comfortable doing so here publicly as I’m not knowing how these types of issues are handled in math circles yet.

Sorry for the essay.

-Ben

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jedishrfu and Greg Bernhardt
Update: I added generous hints to Section 5 exercise 4) parts a) thru f), namely the desired fractional integral is stated: still remains to be shown that the fractional integral is equal to the series definition.

The overhaul is through it's main phase, though I'm still going to post some more Answers to Exercises - Section 5. If previously you glanced at this Insight Article and found it rather too lengthy for you taste, please give it another go being as I have streamlined this work with the recommendation that you simply scan through sections 1 & 2. Sections 3 & 4 however are important for the proofs, and section 5 is the main results in terms of fractional integral representations of the Lerch Transcendent family of special functions.

Looking for co-author of this note that it might be elevated to the status of a paper. If you are interested, please say so here in the comments or send me a private message.

I have just this minute added complete solutions to Exercises in Section 4, I do now believe there's only one problem whom does not have a solution provided now. Not bad!

jedishrfu

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