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


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


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


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


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


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


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.


[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.

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Author:Niizeki, Shozo
Publication:Tamsui Oxford Journal of Mathematical Sciences
Geographic Code:9JAPA
Date:Nov 1, 2005
Next Article:Solutions of the (n-1,1)-type multi-point boundary value problems for higher-order differential equations *.

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