The classical Euler gamma function defined by

Was first introduced by the Swiss mathematician Leonhard Euler (1707-1783) in his goal to generalize the factorial to non-integer values.

Today the Stirling’s formula

is one of the most well-known formulas for approximation of the factorial function by being widely applied in number theory, combinatorics, statistical physics, probability theory and other branches of science.

The Stirling’s formula for  has a generalization to the gamma function, namely that for  a positive real number,

Up to now, many researchers made great efforts in the area of establishing more accurate approximations for the gamma function, and had lots of inspiring results.

The Stirling’s series for the gamma function is presented (see [1, p.257, Eq. (7.1.40)]) by

Where  denotes the Bernoulli numbers defined by the generating formula

Then the first few terms of Bn are as follows:

Then the Burnside’s formula [3]

Is more precise than (1.2) and it’s also well-known.

Like the preceding,

The Burnside asymptotic series for the gamma function is presented (see [21, Corollary 1]) by

Also, the Gosper’s formula [14]

And the Ramanujan’s formula [27]

are better than (1.2) and (1.5).

The more exact formulas than former are the Windschitl’s formula [34]

the Nemes’ formula [25]

and the Chen’s formula [7]

More asymptotic expansion developed by some closed approximation formulas for the gamma function can be found in [2], [4-6], [8-11], [13], [15-17], [20], [24], [26], [28-32] and the references cited therein.

In our study, we focus the continued fraction approximations.

Recently, some authors have focused on continued fractions in order to obtain new asymptotic formulas. For example, Mortici [22] found Stieltjes’ continued fraction

On the other hand, Lu [18] provided a continued fraction approximation based on the Burnside’s formula (1.5) as follows:

Until now many continued fraction approximations for the gamma function were given, but it’s very uncomfortable to determine the parameters of the continued fractions because of computational difficulties. (see [33, Remark 1])

So, in this paper, using Euler connection, we provide a method for construction of continued fraction based on a given power series and determine all parameters of the continued fractions simply. Then we establish two complete continued fraction approximations for the gamma function as applications of our method.

For full article, Please go through this link: https://www.pubtexto.com/pdf/?complete-approximations-for-the-gamma-function-using-the-continued-fraction