A method which generates sharp estimations for big factorials.

1. INTRODUCTION

The famous Stirling's formula and its different generalizations have a wide class of applications in science as statistical physiscs or probability theory. In consequence, it has been deeply studied by a large number of authors due to its practical importance. The Stirling's formula:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.1)

in the sense that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

is a good approximation for big factorials. In fact, the formula (1.1) was discovered by the French mathematician Abraham de Moivre (1667-1754) in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

while the Scottish mathematician James Stirling (1692-1770) discovered the constant [square root of 2 [pi]] in the previous formula. For details, [1,3] can be consulted.

2. MAIN RESULT

The next step in this direction is to define more and more accurate approximations for n!. Such a result is the following formula introduced by W. Burnside in [2], then rediscovered by Y. Weissman in [4]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.1)

The Burnside's formula has great superiority over the Stirling's formula, in fact, the formula (2.1) is one of the most performant ever known. Moreover, the following inequalities hold true:

[[sigma]n <n! < [[beta].sub.n] (2.2)

and the advantage of the formula (2.1) is described by the relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For proofs and other details, see [1,3].

We develop here the idea to define new approximations of the form

n! [approximately equal to] M([[sigma].sub.n],[[beta].sub.n]),

where M([[sigma].sub.n], [[beta].sub.n]) is a certain mean of [[sigma].sub.n] and [[beta].sub.n]. If we think to the fact that products are involved in the expressions of [[sigma].sub.n], [[beta].sub.n], then it seems natural to consider geometric means.

On the other side, remark that it is enough to have the ordering

[[sigma].sub.n] <n! < M([[sigma].sub.n], [[beta].sub.n]) < [[beta].sub.n]),

to obtain a better approximation than the Burnside's formula. We tried with the means [square root of [[sigma].sub.n], [[beta].sub.n] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], but we arrived to the situation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

After some numerical computations, we deduced that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

so one approximation we are talking about is the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (2.3)

Some comparative values of this approximation with Stirling's formula and Burnside's formula are given in the following table:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Now we are concentrate to simplify the expression of the approximations Tn and moreover, to give a new stronger estimation. More precisely, we are looking for an estimation [in which remains greater than n!, but less than [[tau].sub.n]. This can be possible for example, by replacing the polynomial from (2.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

by a smaller one, denoted by Q, such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (2.4)

After some simple computations, we can deduce that (2.4) is fulled as soon as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

We are also careful to obtain a new estimation which is greater than n!, in order to be sure that it is stronger than all previous estimations. We arrive at the conclusion that if we use the polynomial

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

then add the free term 1/12, the approximation (2.3) becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which is most accurate, as we can see from the next table.

REFERENCES

[1] M. Abramowitz and I. Stegun: Handbook of Mathematical Functions, Dover Publications, New York, 1972.

[2] W. Burnside: A rapidly convergent series for log N!, Messenger Math., 46(1917), 157-159.

[3] J. O'Connor and E. F. Robertson: James Stirling, MacTutor History of Mathematics Archive.

[4] Y. Weissman: An improved analytical approximation to n!, Amer. J. Phys., 51(9)(1983).

CRISTINEL MORTICI

Valahia University of Targoviste

Department of Mathematics

Bd. Unirii 18, 130082, Targoviste, Romania

```n [[sigma].sub.n] n!

5 118.02 120
7 4980.4 5040
10 3.5987 x [10.sup.6] 3628800
15 1.3004 x [10.sup.12] 1307674368000
17 3.5395 x [10.sup.14] 355687428096000
20 2.4228 x [10.sup.18] 2432902008176640000

n [[tau].sub.n] [[beta].sub.n]

5 120.18 120.91
7 5046.0 5068.0
10 3.632 x [10.sup.6] 3.6422 x [10.sup.6]
15 1.3085 x [10.sup.12] 1.3112 x [10.sup.12]
17 3.5589 x [10.sup.14] 3.5654 x [10.sup.14]
20 2.4341 x [10.sup.18] 2.4379 x [10.sup.18]

n n! [[mu].sub.n]

5 120 120.08
7 5040 5042.7
10 3628800 3.6303 x [10.sup.6]
15 1307674368000 1.3081 x [10.sup.12]
17 355687428096000 3.5578 x [10.sup.14]
20 2432902008176640000 2.4335 x [10.sup.18]

n [[tau].sub.n]

5 120.18
7 5046.0
10 3.632 x [10.sup.6]
15 1.3085 x [10.sup.12]
17 3.5589 x [10.sup.14]
20 2.4341 x [10.sup.18]
```
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