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Remarks on Gauss's formula associated with the Gamma function *.

Abstract

The purpose of this paper is to prove the famous Gauss's formula for the Gamma function by means of simple and elementary calculas. In [1] and [2], Theorem 2.1. is proved by using the logarithmic convexity of the Gamma function. By using Lemma 1.1., however, we can prove it more concisely than in [1] or [2].

Keywords and Phrases: Gamma function, Gauss's formula.

1. Preliminaries and Notations

By [??], [[??].sup.+] and [[??].sup.+] we mean the set of all real numbers, the set of all positive real numbers and the set of all positive integers, respectively.

Now, the Gamma function [GAMMA](s) is defined by

[GAMMA](s) = [[integral].sup.[infinity].sub.0][e.sup.-x][x.sup.s-1]dx (1.1)

for s [member of] [[??].sup.+]. In order to prove Theorem 2.1 which is the above-mentioned Gauss's formula for the Gamma function, we recall several lemmas as follows (with proofs for completeness sake only).

First, we recall the following well-known result which is a key lemma for proving Theorem 2.1.

Lemma 1.1. The following inequalities:

[e.sup.x/[1+x]] [less than or equal to] 1 + x [less than or equal to] [e.sup.x] (1.2)

hold true for x > -1.

Proof (Only for the sake of completeness). As is well known, we have

1 + x [less than or equal to] [e.sup.x] (1.3)

for all x [member of] [??]. By substituting -[x/[1 + x]] for x in (1.3), we obtain

[e.sup.x/[1+x]] [less than or equal to] 1 + x (1.4)

for x > -1. Therefore, (1.2) follows from (1.3) and (1.4). []

Lemma 1.2. For K, n [member of] [[??].sup.+] such that K < n, the following inequality:

-x - [[K.sup.2]/[n - K]] [less than or equal to] -[nx/[n - x]] (1.5)

holds true for 0 [less than or equal to] x [less than or equal to] K.

Proof. Since the inequalities 0 [less than or equal to] x [less than or equal to] K < n holds, we have

nx/[n - x] = [(n - x)x + [x.sup.2]]/[n - x] = x + [[x.sup.2]/[n - x]] [less than or equal to] x + [[K.sup.2]/[n - K]]

and hence we obtain (1.5). []

Lemma 1.3. For n [member of] [[??].sup.+] and s [member of] [[??].sup.+], the following integral formula holds true:

[[integral].sup.n.sub.0][(1 - [x/n]).sup.n][x.sup.s-1]dx = n![n.sup.s]/s(s + 1) ... (s + n)

Proof. Again, only for the sake of completeness, by means of integration by parts, for any k [member of] [[??].sup.+] such that 1 [less than or equal to] k [less than or equal to] n, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Hence we complete the proof of Lemma 1.3. []

2. Proof of Gauss's Formula

By means of Lemmas 1.1., 1.2. and 1.3. given in Section 1, we can prove Gauss's formula for the Gamma function. We now recall a well-known (rather classical) form of the Gamma function as Theorem 2.1 below.

Theorem 2.1. (Gauss's formula). For n [member of] [[??].sup.+] and s [member of] [[??].sup.+], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Proof. By replacing -[x/n] for x in (1.2) and then by raising each side of (1.2) to n-th power, we obtain

[e.sup.-nx/[n-x]] [less than or equal to] [(1 - [x/n]).sup.n] [less than or equal to] [e.sup.-x] (2.1)

for 0 < x < n. For any given n [member of] [[??].sup.+] we take a number K [member of] [[??].sup.+] such that K < n. By multiplying each side of (2.1) by [x.sup.s-1] and then by integrating each side of (2.1), in view of Lemma 1.2., we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In the above inequalities, we let n [right arrow] [infinity] then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and furthermore by letting K [right arrow] [infinity] and by Lemma 1.3., we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Consequently, the proof of Theorem 2.1. is complete. []

Acknowledgements

The author would like to heartily thank Professor H.M.Srivastava (of the University of Victoria, Canada) for his invaluable advice and constant encouragement.

Received May 9, 2003, Accepted October 31, 2003.

References

[1] T. Takagi, Outline of Analysis (in Japanese), Iwanamisyoten Publisher, Tokyo, 1980.

[2] E. Artin, Einfuhrung in die Theorie der Gammafunction, Hamburg, 1932.

Shozo Niizeki ([dagger])

Department of Mathematics, Faculty of Science, Kochi University

Kochi 780-8520, Japan

* Mathematics Subject Classification. Primary 33B15.

([dagger]) E-mail:niizeki@math.Kochi-u.ac.jp
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