use inverse mellin transform (see Rainville, 1960).

Thank you, Abdus!

Dear Mykola,

find attached some my calculations, perhaps it will be of some help - hopefully I didn't skip some crucial steps inside.

Regards,

Tibor

Sorry, the attachment is coming!

TP

Thank you very much, Tibor!

I shall look closer at the sum over k.

My best regards,

Mykola

Dear Mykola,

I have derived another, integral form expressions for the real and imaginary part of 2F1. The file is closed.

Kind regards,

Tibor

The seocnd addend in (1) is superfluous! The corrected file is attached.

TP

Dear Tibor,

many thanks once again!

Best,

Mykola

Dear Tibor,

I have rederived your results from another side

and cheched with Mathematica.

Best wishes,

MSh

Dear Mykola,

actually we have now three different fashion derivations + the simulation. Is some your deeper question solved by these efforts?

Kind regards,

Tibor

Dear Tibor,

my question arose in the calculation of a Feynman integral.

To express it as an imaginary part of a hypergeometric function appeared

as one of several possibilities of its calculation.

Since this integral is a part of the integrand for further integration,

my desire is to get it in terms of named functions, or at least in a most

tractable form.

I would like to calculate it in a few other ways and compare the results.

Eventually, this could give some interesting relationships among different mathematical objects.

Besides of all that, the question seems to be interesting on its own right.

Moreover, I already learned something new from you and

I am grateful to you for your help.

Kind regards,

Mykola

Dear Mykola,

the pleasure was mine - thank you for the challenge. I agrre, the question was interesting "per se".

Best regards,

Tibor

Dear Mykola,

finally, some concrete result. Both real and imaginary part of 2F1 i have expressed via the Horn H_4 function (see part D of the PDF). However, it is a double-series as well, but I think that is is the floor. I don't know about some in-built calculation routine for Horn functions.

Kind regards,

Tibor

And the PDF!

T.

Dear Tibor,

thank you for your nice calculation!

Since one of the nominator parameters of the $_2F_1$ you consider

equals to $1$, your results actually cover a large subclass of Gauss hypergeometric function, the incomplete Beta function:

$$

B_z(a,b)=a^{-1}z^a\,_2F_1(a,1-b,a+1;z)=a^{-1}z^a(1-z)^{b-1}\,_2F_1\Big(1-b,1,a+1;\frac{z}{z-1}\Big).

$$

The first equality is the entry 6.6.8, p.263 of \cite{AS}, and the second is the result of a linear transformation from \cite{PBM3}, 7.3.1.3.

In fact, all generalized inverse trigonometric functions considered in your recent preprint \cite{BBP14}

are expressed in terms of the incomplete beta function because all of them

are some $_2F_1$'s with $c=a+1$.

The same is immediately seen from the fifth equation at p.1, where a change

of the integration variable $t^p\to t$ gives

$$\arcsin_p(x)=p^{-1}B_{x^p}(1/p,1-1/p).$$

Best regards,

Mykola

PS The same is attached as pdf.

Sorry, something went wrong with the citation.

Here is the pdf once again.

Best,

M

Hello,

here is the corrected final part of "Mykola_Shpot_2F1.pdf".

Best,

Mykola

News are here:

https://www.researchgate.net/publication/283354759_Feynman_integral_in_mathbb_R1oplusmathbb_Rm_and_complex_expansion_of__2F_1

Link for 50 downloads of the journal article:

http://www.tandfonline.com/eprint/T7j8YFwGmNSnujn9SvcA/full

Some corrections to the ITSF paper can be found in

R. B. Paris and M. A. Shpot,

A Feynman integral in Lifshitz-point and Lorentz-violating theories in R^D⊕ R^m ,

Mathematical Methods in the Applied Sciences 2018:

http://onlinelibrary.wiley.com/doi/10.1002/mma.4763/epdf?r3_referer=wol&tracking_action=preview_click&show_checkout=1&purchase_referrer=onlinelibrary.wiley.com&purchase_site_license=LICENSE_DENIED

I see that this discussion is over 4 years old, but the RG computer just brought it to my attention. Would you be kind enough to provide a link to a publicly accessible version of your ITSF article for those who don't wish to pay to examine the fruits of your labour. Thank you.

Here are the links.

The ArXviv preprint of the ITSF paper:

https://arxiv.org/abs/1510.08876

Some corrections concerning the misprinted formula (2.7) of the ITSF paper (Eq. (17) in ArXviv1510.08876)

and the convergence of the Horn function H_4 are given in a recent MMAS paper with Richard Paris.

The ArXviv link is

https://arxiv.org/abs/1707.03018

I refer you to the excellent Book: Involving a lot of expressions of the Gaussian hyper-geometric series 2F1

A.P. Prudnikov ,Yury Brychkov ,O.I. Marichev ,Integrals and Series. Volume 3: More Special Functions,Publisher: Gordon and Breach (New York-London),1989.

Thank you, Professor Bin-Saad.

Mykola