How Can Dirichlet Integrals Be Generalized Using Special Functions?

[benorin]

So I'm writing a paper involving Gamma and Beta functions, Dirichlet integrals, Generalized Hypergeometric functions, and a small grab-bag of other miscellaneous special functions. There's this grouping of results that I am most certianly going to generalize; in fact I feel rather like quoting Gauss, whom said "My results, I've had them for some time. Problem is, I don't yet know how I am to arrive at them."
Blah, blah, blah... what I've proved is in the attached PDF.

Have at it. Any conjectures?

 

[amcavoy]

So I'm writing a paper involving Gamma and Beta functions, Dirichlet integrals, Generalized Hypergeometric functions, and a small grab-bag of other miscellaneous special functions. There's this grouping of results that I am most certianly going to generalize; in fact I feel rather like quoting Gauss, whom said "My results, I've had them for some time. Problem is, I don't yet know how I am to arrive at them."
Blah, blah, blah... what I've proved is in the attached PDF.
Have at it. Any conjectures?

Sorry to go off-topic, but was this written in MS Word?

 

[benorin]

Yes, the paper was written with MS word/MathType combo => print to PDF.

 

[bomba923]

Ahh...I always use MathType too
(:Love:)

 

[benorin]

C'mon, don't make me nag

No way! Someone's got to post something . I really want some feedback on this one. C'mon, don't make me nag.

 

[slayerchange]

Hi, at first look , u don't need to write p and q near F , because they are indexed via a's and b's inside of function. :) ,

 

[matt grime]

What do you want us to say? It's one page of formula without any proof that they are even remotely correct. THere are no explanations, no introductory words, and no indication of if the work is even original or not. I will remain sceptical of their worth until you correct that.

 

[benorin]

Here's it is, my work-in-progress

Do bear in mind I wrote this paper after having taken only the standard undergard calc sequence, and I haven't updated the proof techniques to match the rigors of analysis.

Regarding the content of my paper: I have come-up with a good portion of it's content of my own work, my results are--sadly--not original, save perhaps one thing, a description of the hypercube for use with Dirichlet Intergrals: but it's broken. Need to re-write using perhaps nets or filters or lim sup/inf convergence of sequences of sets instead of ordinary limits of integer indexed families of sets... blah, blah, blah: I'm sorry.

The intro to the gamma function section consists of proofs I came up, and which are original to my knowledge, and it is an unorthodox procession of theorems and links between standard definitions of it; the inductive proof of the generalized Dirichlet Integrals is distinctive, somewhat original, yet lacking.

Also attached is a proof for the Lerch Transcendent which differs from that given in the paper, and, if you don't have time to read the former, somewhat lengthy rendition.

Do be kind.

 

[benorin]

So, any comments?

 

[z-component]

Might I ask which PDF printer you used?

 

[benorin]

Adobe Pro PDF printer

 

[dextercioby]

So why would you need such a paper?

Daniel.

 

[benorin]

If I may tell the story...

If I may tell the story...

I started out by asking this question: "What is \frac{d}{dn}\left(n!\right) ?"

Okay, so you can't differentiate discrete functions. Bummer. Hey, wait... Euler was here... the Gamma function? do tell. Neat integral. Analytic continuation, what's that? I see: nice.

After exhausting the understandable potions of my less than extensive library on the subject I was gifted this text: Solved Problems: Gamma and Beta functions, Legendre polynomials, and Bessel functions, by Farrell & Ross. Therein was presented the 2-d and 3-d Dirichlet Integrals, with proofs, which I then promptly generalized to n-d (the proof of that took me about a year.) This spurred my long-lasting love affair Gamma function, and the rest follows.