The American Mathematical Monthly A Generalization of Wallis Product --Manuscript Draft-- Manuscript Number: 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] Additional Information: Question Response 5-Character 2000 AMS/Math Reviews 00-xx Math. Classification Codes for your manuscript. 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Then some corollaries can be obtained by this formula. 9 10 Wallis’ product (see [1]) is a famous formula, which has been named in honor of 11 English mathematician John Wallis (1616-1703). There are some generalization of this 12 formula, for example see [2]. This note generalizes Wallis product by a simple proof. 13 The generalized formula is as follows: 14 15 Y1 mk(mk + m) −(2m+1) 2 16 Q = 2πm m−1 (m)! m−1 ; m ≥ 2: m−1 2 m−1 17 k=1 l=1 (mk + l) 18 19 Lemma 1. Let m ≥ 2 be an integer number, then 20 2nm m+1 2m Yn − − − 21 mk(mk + m) 2 m m 1 (n + 1) m 1 n! m 1 Q = m! m−1 : ; 8n 2 N: 22 m−1 2 2 (1) m−1 m−1 23 k=1 l=1 (mk + l) (m(n + 1))! 24 25 Proof. The following identity can be proved by induction on n. 26 Yn 27 1 (n + 1)m! Q − = ; 28 m 1(mk + l) (m(n + 1))! 29 k=1 l=0 30 Therefore, 31 32 Yn mk(mk + m) Yn 1 33 Q = m2nn!2(n + 1) Q m−1 2 m−1 2 m−1 m−1 34 k=1 l=1 (mk + l) k=1 l=1 (mk + l) 35 Yn 1 1 1 36 = m2n(1+ m−1 )n!2(1+ m−1 )(n + 1) Q m−1 2 37 m−1 k=1 (mk + l) 38 l=0 2nm m+1 2m − − − 39 2 m m 1 (n + 1) m 1 n! m 1 m−1 40 = m! : 2 : − 41 (m(n + 1))! m 1 42 43 44 Theorem 1. The following formula is true for m = 2; 3; 4;::: 45 46 Y1 mk(mk + m) −(2m+1) 2 Q = 2πm m−1 (m)! m−1 : 47 m−1 2 (2) m−1 48 k=1 l=1 (mk + l) 49 50 Proof. The Stirling’s formula yields 51 p 1 − 52 2πnn+ 2 e n lim = 1; (3) 53 n!1 n! 54 55 January 2014] A GENERALIZATION OF WALLIS’ PRODUCT 1 56 57 58 59 60 61 62 63 64 65 Mathematical Assoc. of America American Mathematical Monthly 121:1 October 26, 2014 6:06 p.m. A-Generalization-of-Wallis-Product.tex page 2 and 1 p 1 − 2 2π(m(n + 1))m(n+1)+ 2 e m(n+1) lim = 1: (4) 3 n!1 (m(n + 1))! 4 5 Then (3) and (4) imply that 6 (p ) 2m 7 ! 2nm m+1 1 − m−1 2nm m+1 2m m−1 m−1 n+ n 8 m m−1 (n + 1) m−1 n! m−1 m (n + 1) 2πn 2 e lim = lim ( ) 9 n!1 2 n!1 p 2 (m(n + 1))! m−1 1 − m−1 10 2π(m(n + 1))m(n+1)+ 2 e m(n+1) 11 0 1 ( ) −m(2n+1) 12 − 2m −(2m+1) n + 1 m 1 13 = lim @2πe m−1 m m−1 A : 14 n!1 n 15 16 The following relation is obvious, 17 ( ) −m(2n+1) 18 − n + 1 m 1 −2m 19 lim = e m−1 : 20 n!1 n 21 22 Hence, 23 ! 2nm m+1 2m 24 m m−1 (n + 1) m−1 n! m−1 −(2m+1) lim = 2πm m−1 : (5) 25 !1 2 n (m(n + 1))! m−1 26 27 Proof of the theorem will be completed by Lemma and (5). 28 29 Note that (2) is Wallis product for m = 2, 30 1 31 Y 2k(2k + 2) π 32 = : (6) (2k + 1)2 4 33 k=1 34 35 It shows that (2) is a generalization of Wallis formula. Also as a result of (2), we can 36 prove 37 Y1 38 4k(4k + 4) 3π = p : (7) 39 (4k + 1)(4k + 3) 8 2 40 k=1 41 The theorem implies the following equality 42 43 Y1 4k(4k + 4) −9 2 44 3 3 2 = 2π:4 :4! ; (8) 3 45 k=1 ((4k + 1)(4k + 2)(4k + 3)) 46 47 when m = 4. Wallis product (6) immediately yields 48 ( ) Y1 2 1 49 (4k + 2) 3 4 3 50 1 = : (9) 3 π 51 k=1 (4k(4k + 4)) 52 53 Finally, equality (7) will be obtained by (8) and (9). 54 55 2 ⃝c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121 56 57 58 59 60 61 62 63 64 65 Mathematical Assoc. of America American Mathematical Monthly 121:1 October 26, 2014 6:06 p.m. A-Generalization-of-Wallis-Product.tex page 3 REFERENCES 1 1. M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and 2 Mathematical Tables, 9th printing, Dover, New York, 1972. 3 2. T. J. Osler, The general Vieta-Wallis product for π, The Mathematical Gazette, 89 No. 516. (2005) 371– 4 377. 5 6 7 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. 8 Department of Mathematics, University of Qom, Qom, Iran. P. O. Box. 37185-3766. 9 [email protected] & [email protected] 10 11 12 HAMID NADERI YEGANEH is a Bachelor student of mathematics in University of Qom. He won gold 13 medal at the 38th Iranian Mathematical Society’s Competition (2014). [email protected] 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 January 2014] A GENERALIZATION OF WALLIS’ PRODUCT 3 56 57 58 59 60 61 62 63 64 65 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 publica- tion in the American Mathematical Monthly. This manuscript has not been published and is not under consideration for publication else- where. Best Regards, Mahdi Ahmadinia Assistant Professor of Department of Mathematics University of Qom Qom, Iran. 1.