You may find the next link interesting: http://dlmf.nist.gov/5

I think that you need the Mittag-Leffler functions - an extensions for exp.

Journal of Computational and Applied Mathematics 118 (1–2) (2000), 283–299

On Mittag-Leffler-type functions in fractional evolution processes, Francesco Mainardi, Rudolf Gorenflo, http://dx.doi.org/10.1016/S0377-0427(00)00294-6

Octav Olteanu, University Politehnica of Bucharest

For a.v.(x-1)

There are a few. I don't know for what you want to use it. However, look at my papers presented with texts here at the RG:

https://www.researchgate.net/publication/253871866_On_the_asymptotic_representation_of_the_Euler_gamma_function_by_Ramanujan?ev=prf_pub

https://www.researchgate.net/publication/246166981_Fast_evaluation_of_transcendental_functions?ev=prf_pubhttps://www.researchgate.net/publication/246166981_Fast_evaluation_of_transcendental_functions?ev=prf_pub

If you need such a series for effective calculation with high precision, you need to use the second paper. You can also see the approximation and the corresponding algorithm here:

http://www.ccas.ru/personal/karatsuba/algen.htm

in the chapter II. THE FEE METHOD

in the paragraphs:

6. Fast evaluation of the Euler gamma function for a rational value of the argument

and

7. Fast evaluation of the Euler gamma function for an algebraic argument

Good luck!

Article On the asymptotic representation of the Euler gamma function...

Article Fast evaluation of transcendental functions

You may find the next link interesting: http://www.math.leidenuniv.nl/~evertse/ant13-8.pdf. Your sincerely: Anna Tomova

Thanks alots

 Hi, You can see The Gamma function in terms of generalized Laquerre polynomials on the hyperlinks: https://en.wikipedia.org/wiki/Gamma_function#In_terms_of_generalized_Laguerre_polynomials and

https://en.wikipedia.org/wiki/Laguerre_polynomials#Generalized_Laguerre_polynomials . Best regards.

Here you can see some some expansions for the Euler Gamma function and the Riemann Zeta function

http://booksc.org/book/13200165/73fd9c

Thanks a lot.