The American Mathematical Monthly
A Generalization of Wallis Product
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Full Title:
A Generalization of Wallis Product
Article Type:
Note
Keywords:
Wallis Product
Corresponding Author:
Mahdi Ahmadinia, Ph.D.
University of Qom
Qom, Qom IRAN (ISLAMIC REPUBLIC OF)
Corresponding Author Secondary
Information:
Corresponding Author's Institution:
University of Qom
Corresponding Author's Secondary
Institution:
First Author:
Mahdi Ahmadinia, Ph.D.
First Author Secondary Information:
Order of Authors:
Mahdi Ahmadinia, Ph.D.
Hamid Naderi Yeganeh, Undergraduate Student
Order of Authors Secondary Information:
Abstract:
This note presents a generalization of Wallis' product. We prove this generalization by
Stirling's formula. Then some corollaries can be obtained by this formula.
Corresponding Author E-Mail:
[email protected]
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Mathematical Assoc. of America
American Mathematical Monthly 121:1
October 26, 2014 6:06 p.m.
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A-Generalization-of-Wallis-Product.tex
A Generalization of Wallis Product
Mahdi Ahmadinia and Hamid Naderi Yeganeh
Abstract. This note presents a generalization of Wallis’ product. We prove this generalization
by Stirling’s formula. Then some corollaries can be obtained by this formula.
Wallis’ product (see [1]) is a famous formula, which has been named in honor of
English mathematician John Wallis (1616-1703). There are some generalization of this
formula, for example see [2]. This note generalizes Wallis product by a simple proof.
The generalized formula is as follows:
∞
∏
−(2m+1)
mk(mk + m)
2
= 2πm m−1 (m)! m−1 ,
∏m−1
2
m−1
k=1
l=1 (mk + l)
m ≥ 2.
Lemma 1. Let m ≥ 2 be an integer number, then
n
∏
∏m−1
k=1
l=1
m+1
2nm
mk(mk + m)
2
= m!
2
m−1
.
2m
m m−1 (n + 1) m−1 n! m−1
2
(mk + l) m−1
∀n ∈ N.
,
(m(n + 1))! m−1
(1)
Proof. The following identity can be proved by induction on n.
n
∏
k=1
∏m−1
l=0
1
(n + 1)m!
=
,
(m(n
+ 1))!
(mk + l)
Therefore,
n
∏
k=1
mk(mk + m)
∏m−1
l=1
(mk + l)
2
m−1
= m2n n!2 (n + 1)
n
∏
∏m−1
k=1
1
l=1
1
2
(mk + l) m−1
1
= m2n(1+ m−1 ) n!2(1+ m−1 ) (n + 1)
n
∏
k=1
m+1
2nm
= m!
2
m−1
.
∏m−1
l=0
1
2
(mk + l) m−1
2m
m m−1 (n + 1) m−1 n! m−1
2
.
(m(n + 1))! m−1
Theorem 1. The following formula is true for m = 2, 3, 4, . . .
∞
∏
−(2m+1)
mk(mk + m)
2
= 2πm m−1 (m)! m−1 .
∏m−1
2
m−1
k=1
l=1 (mk + l)
Proof. The Stirling’s formula yields
√
lim
n→∞
January 2014]
(2)
1
2πnn+ 2 e−n
= 1,
n!
A GENERALIZATION OF WALLIS’ PRODUCT
(3)
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page 1
Mathematical Assoc. of America
American Mathematical Monthly 121:1
October 26, 2014
6:06 p.m.
A-Generalization-of-Wallis-Product.tex
and
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√
lim
n→∞
1
2π(m(n + 1))m(n+1)+ 2 e−m(n+1)
= 1.
(m(n + 1))!
(4)
Then (3) and (4) imply that
(
lim
m
n→∞
2nm
m−1
(n + 1)
m+1
m−1
(m(n + 1))!
n!
2m
m−1
2m
(√
) m−1
m+1
2nm
1
m m−1 (n + 1) m−1
2πnn+ 2 e−n
= lim (
2
) m−1
n→∞ √
1
2π(m(n + 1))m(n+1)+ 2 e−m(n+1)


(
) −m(2n+1)
m−1
−(2m+1)
n+1
2m
.
= lim 2πe m−1 m m−1
n→∞
n
)
2
m−1
The following relation is obvious,
(
lim
n→∞
n+1
n
) −m(2n+1)
m−1
−2m
= e m−1 .
Hence,
(
lim
n→∞
2nm
m+1
2m
m m−1 (n + 1) m−1 n! m−1
(m(n + 1))!
)
= 2πm
2
m−1
−(2m+1)
m−1
.
(5)
Proof of the theorem will be completed by Lemma and (5).
Note that (2) is Wallis product for m = 2,
∞
∏
2k(2k + 2)
π
= .
2
(2k + 1)
4
k=1
(6)
It shows that (2) is a generalization of Wallis formula. Also as a result of (2), we can
prove
∞
∏
4k(4k + 4)
3π
= √ .
(4k + 1)(4k + 3)
8 2
k=1
(7)
The theorem implies the following equality
∞
∏
k=1
4k(4k + 4)
((4k + 1)(4k + 2)(4k + 3))
2
3
= 2π.4
−9
3
2
.4! 3 ,
(8)
when m = 4. Wallis product (6) immediately yields
∞
∏
(4k + 2) 3
k=1
(4k(4k + 4)) 3
2
1
( ) 13
4
=
.
π
(9)
Finally, equality (7) will be obtained by (8) and (9).
2
c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121
⃝
page 2
Mathematical Assoc. of America
American Mathematical Monthly 121:1
October 26, 2014
6:06 p.m.
A-Generalization-of-Wallis-Product.tex
REFERENCES
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1. M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing, Dover, New York, 1972.
2. T. J. Osler, The general Vieta-Wallis product for π , The Mathematical Gazette, 89 No. 516. (2005) 371–
377.
MAHDI AHMADINIA received his PhD in mathematics from Kerman University under the guidance of
professor Mehdi Radjabalipour and was supported by Mahani Mathematical Research Center.
Department of Mathematics, University of Qom, Qom, Iran. P. O. Box. 37185-3766.
[email protected] & [email protected]
HAMID NADERI YEGANEH is a Bachelor student of mathematics in University of Qom. He won gold
medal at the 38th Iranian Mathematical Society’s Competition (2014).
[email protected]
January 2014]
A GENERALIZATION OF WALLIS’ PRODUCT
3
page 3
Cover Letter
Mahdi Ahmadinia
Department of Mathematics,
University of Qom,
Qom, Iran.
P. O. Box. 37185-3766.
Scott T. Chapman
Professor and Scholar in Residence
Sam Houston State University
Department of Mathematics and Statistics
Box 2206
Huntsville, Texas 77341-2206.
Oct 26, 2014
Dear Professor Scott T. Chapman
I am pleased to submit an original research article entitled: ”A Generalization of Wallis
Product” by M. Ahmadinia and H. Naderi Yeganeh for consideration for possible publication in the American Mathematical Monthly.
This manuscript has not been published and is not under consideration for publication elsewhere.
Best Regards,
Mahdi Ahmadinia
Assistant Professor of Department of Mathematics
University of Qom
Qom, Iran.
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