DOCSLIB.ORG

Infinite Product Representations for Gamma Function and Binomial

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Infinite Product Representations for Gamma Function and Binomial Innite Product Representations for Gamma Func- tion and Binomial Coecient By Edigles Guedes November 16, 2016 In the beginning was the Word, and the Word was with God, and the Word was God. - John 1:1. Abstract. In this paper, I demonstrate one new innite product for binomial coecient and news Euler's and Weierstrass's innite product for Gamma function among other things. 1. Introduction In 1729, Leonhard Euler (1707-1783) gave the innite product expansion for gamma function [1, p. 33] z 1 1 1 z 1 ¡(z) = 1 + 1 + ¡ ; (1) z j j j=1 Y which is valid in C, except for z 0; 1; 2; ::: . In 1854, Karl Weierstrass (18215f-18¡97) ¡gave tghe innte product expansion for gamma function [1, p. 34-35] 1 z ¡(z) = ze z 1 + e z/j; (2) j ¡ j=1 which is valid for all C. Y The binomial coecient may be dened by the nite product [2] n ` (`)n ` + r 1 = = ¡ ; (3) n n! r r=1 Y which is valid for ` C 0; 1; 2; ::: and n N. In [3, p. 3, form2ula (¡1.f1.1¡.2)],¡the Pgochhamm2er's symbol is dened by ¡(` + n) (`) = : (4) n ¡(`) In [4], I gave the innite product representation for the Pochhammer's symbol n 1 1 n 1 (`) = 1 + 1 + ¡ ; (5) n j j + ` 1 j=1 Y ¡ which is valid for ` C 0; 1; 2; ::: and n N. 2 ¡ f ¡ ¡ g 2 The beta function may be dened by [5] (a 1)!(b 1)! B(a; b) ¡ ¡ ; (6) (a + b 1)! when a; b are positive integers. ¡ In this paper, I prove the innite product representation for binomial coecient given by 1 ` (`) 1 n n ¡ = n = 1 + 1 + ; n n! j j + ` 1 j=1 Y ¡ and the gamma function is given by news innite product representations 1 1 z+n z n 1 z ¡(z + n) = 1 + 1 + 1 + ¡ j j j + z 1 j=1 Y ¡ 1 2 Infinite Product Representations for Gamma Function and Binomial Coefficient and 1 1 z n = e (z+n) 1 + 1 + e (z+n)/j; z ¡(z + n) j j + z 1 ¡ j=1 Y ¡ z 1 1 1 z n n ¡ (z n)¡(z) = 1 + 1 + ¡ 1 + ¡ j j j + z n 1 j=1 and Y ¡ ¡ 1 1 z z n n z/j = e 1 + ¡ 1 + e¡ ; (z n)¡(z) j j + z n 1 j=1 ¡ Y ¡ ¡ among other things. 2. Preliminary Lemma 1. If a; b R and b =/ 0, then 2 a 1 (a + j 1)(b + j) = ¡ : b (a + j)(b + j 1) j=1 Y ¡ Proof. I well-know the identity a b!(a 1)! a!(b 1)! ¡ = ¡ ; (7) b [b + (a 1) + 1]! [a + (b 1) + 1]! ¡ ¡ using the denition for beta function (6), I obtain a B(a + 1; b) = : (8) b B(a; b + 1) On the other hand, the beta function have the following innite product representation [6, p. 899] 1 j(a + b + j) (a + b + 1)B(a + 1; b + 1) = ; (9) (a + j)(b + j) j=1 Y valid for a;b =/ 1; 2;::: Setting a a 1 and b b 1, respectively, in both members of (9), I nd ¡ ¡ ! ¡ ! ¡ 1 1 j(a + b + j 1) B(a; b + 1) = ¡ ; (10) a + b (a + j 1)(b + j) j=1 and Y ¡ 1 1 j(a + b + j 1) B(a + 1; b) = ¡ : (11) a + b (a + j)(b + j 1) j=1 Y ¡ Substituting (10) and (11) into the right hand side of (9), I get a a + b 1 j(a + b + j 1)(a + j 1)(b + j) = ¡ ¡ : b a + b (a + j)(b + j 1)j(a + b + j 1) j=1 Y ¡ ¡ Eliminate the same terms in the numerator and denominator of the above equation, and encounter a 1 (a + j 1)(b + j) = ¡ ; b (a + j)(b + j 1) j=1 Y ¡ which is the desired result. 3. Binomial Coefficient: the Infinite Product 3.1. Innite Product Representation for Binomial Coecient. Theorem 2. If ` C 1; 2; ::: and n N, then 2 ¡ f¡ ¡ g 2 1 ` (`) 1 n n ¡ = n = 1 + 1 + ; n n! j j + ` 1 j=1 Y ¡ 3 ` where denotes the binomial coecient, (`) denotes the Pochhammer's symbol and n! n n denotes the factorial. Proof. Setting a = ` + r 1 and b = r in both members of the Lemma 1, I obtain ¡ ` + r 1 1 (` + r + j 2)(r + j) ¡ = ¡ : (12) r (` + r + j 1)(r + j 1) j=1 Y ¡ ¡ Substituting (12) into the right hand side of (3), I nd n ` (`)n 1 (` + r + j 2)(r + j) = = ¡ n n! (` + r + j 1)(r + j 1) r=1 j=1 ¡ ¡ n Y Y 1 (` + r + j 2)(r + j) = ¡ (` + r + j 1)(r + j 1) j=1 r=1 ¡ ¡ Y Y 1 1 (j + n)(j + ` 1) 1 n n ¡ = ¡ = 1 + 1 + ; j(j + ` + n 1) j j + ` 1 j=1 j=1 Y ¡ Y ¡ which is the desired result. 4. News Euler's and Weierstrass's Infinite Product Representation for Gamma Function 4.1. New Euler's Innite Product Representation for Gamma Function. Theorem 3. If z C 0; 1; 2; ::: and n N, then 2 ¡ f ¡ ¡ g 2 1 1 z+n z n 1 z¡(z + n) = 1 + 1 + 1 + ¡ ; j j j + z 1 j=1 Y ¡ where ¡(z) denotes the gamma function. Proof. From denition (4), I give ¡(` + n) (`) = ¡(` + n) = (`) ¡(`): (13) n ¡(`) ) n Setting the right hand side of (1) and (5) into the right hand side of (13), I get n 1 1 n 1 1 1 1 ` ` 1 ¡(` + n) = 1 + 1 + ¡ 1 + 1 + ¡ j j + ` 1 ` j j j=1 ¡ j=1 Y n Y 1 1 n 1 1 1 ` ` 1 `¡(` + n) = 1 + 1 + ¡ 1 + 1 + ¡ ) j j + ` 1 j j j=1 ¡ j=1 Y n Y 1 1 n 1 1 ` ` 1 = 1 + 1 + ¡ 1 + 1 + ¡ j j + ` 1 j j j=1 Y ¡ 1 1 `+n ` n 1 = 1 + 1 + 1 + ¡ : j j j + ` 1 j=1 Y ¡ Changing ` by z in previous equation, I obtain the desired result. 4.2. New Weierstrass's Innite Product Representation for Gamma Function. Theorem 4. If z C and n N, then 2 2 1 1 z n = e (z+n) 1 + 1 + e (z+n)/j; z ¡(z + n) j j + z 1 ¡ j=1 Y ¡ 4 Infinite Product Representations for Gamma Function and Binomial Coefficient where ¡(z) denotes the gamma function, ex denotes the exponential function and denotes the Euler-Mascheroni constant. Proof. The inverse of the Theorem 3, give me 1 1 1 (z+n) z n = 1 + ¡ 1 + 1 + z¡(z + n) j j j + z 1 j=1 ¡ Y m 1 z+n 1 (z+n) z n = lim 1 + lim 1 + ¡ 1 + 1 + m m m j j j + z 1 !1 !1j=1 Y ¡ m 1 z+n 1 (z+n) z n = lim 1 + 1 + ¡ 1 + 1 + m 2 m j j j + z 1 3 !1 j=1 Y ¡ 4 m 1 m 5 ¡ 1 (z+n) z n = lim 1 + ¡ 1 + 1 + m 2 j j j + z 1 3 !1 j=1 j=1 Y Y ¡ 4 m 5 (z+n) z n = lim m¡ 1 + 1 + m 2 j j + z 1 3 !1 j=1 Y ¡ 4 m 5 1 1 1 1 z n = lim exp 1 1 + + ::: + ln m (z + n) 1 + 1 + m 2 ¡ 2 ¡ 2 m ¡ m ¡ j j + z 1 3 !1 j=1 Y ¡ 4 1 1 (z + n) (z + n) (z + n) 5 = lim exp 1 + + ::: + ln m (z + n) exp ¡ + ¡ + ::: + ¡ m 2 m ¡ 1 2 m !1 m z n 1 + 1 + j j + z 1 j=1 Y ¡ m 1 1 z n (z+n)/j = lim exp 1 + + ::: + ln m (z + n) 1 + 1 + e¡ m 2 2 m ¡ j j + z 1 3 !1 j=1 Y ¡ 4 m 5 1 1 z n (z+n)/j = lim exp 1 + + ::: + ln m (z + n) lim 1 + 1 + e¡ m 2 m ¡ m 0 j j + z 1 1 !1 !1 j=1 Y ¡ 1 z @n A =e (z+n) 1 + 1 + e (z+n)/j; j j + z 1 ¡ j=1 Y ¡ which is the desired result. Example 5. Set n = 1 in Theorem 3 and Theorem 4, and encounter 1 1 z+1 z 1 1 z2 ¡(z) = 1 + 1 + 1 + ¡ j j j + z 1 j=1 and Y ¡ 1 1 z 1 = e (z+1) 1 + 1 + e (z+1)/j: z2 ¡(z) j j + z 1 ¡ j=1 Y ¡ 4.3. New Euler's Innite Product Representation for Gamma Function. Theorem 6. If z C 0; 1; 2; ::: , n N and Re(z) > n, then 2 ¡ f ¡ ¡ g 2 z 1 1 1 z n n ¡ (z n)¡(z) = 1 + 1 + ¡ 1 + ; ¡ j j j + z n 1 j=1 Y ¡ ¡ where ¡(z) denotes the gamma function. Proof. In [3, p. 240, I.29], I found the formula ¡(z) ¡(z n) = ¡(z) = ¡(z n) (z n) : (14) ¡ (z n) ) ¡ ¡ n ¡ n 5 Setting the right hand side of (1) and (5) into the right hand side of (14), I get z n 1 n 1 1 1 1 ¡ z n ¡ 1 1 n ¡ ¡(z) = 1 + 1 + ¡ 1 + 1 + z n j j j j + z n 1 ¡ j=1 j=1 ¡ ¡ Y z n 1 Y n 1 1 1 1 ¡ z n ¡ 1 n ¡ = 1 + 1 + ¡ 1 + 1 + z n j j j j + z n 1 ¡ j=1 ¡ ¡ Y z 1 1 1 1 z n n ¡ = 1 + 1 + ¡ 1 + z n j j j + z n 1 ¡ j=1 ¡ ¡ Y z 1 1 1 z n n ¡ (z n)¡(z) = 1 + 1 + ¡ 1 + ; ) ¡ j j j + z n 1 j=1 Y ¡ ¡ which is the desired result.
Load more

Recommended publications
  • Infinite Product of a Number
    International Journal of Pure and Applied Mathematical Sciences. ISSN 0972-9828 Volume 10, Number 2 (2017), pp. 107-110 © Research India Publications http://www.ripublication.com Infinite Product of a Number Rushal Sohal Student The Lexicon International School, Wagholi, Pune, India. Abstract It seems that multiplying a positive number infinite times would give something extremely huge (infinity), but is it so? In this paper, we will show that any positive integer (>1) raised to the power infinity (i.e. infinite product of n ∞ ∞ =∏푖=1 푛=푛×푛×푛...= 푛 , 푛 > 1) is not infinity; we will use simple mathematical equations and also use some known sums to prove that the infinite product of a positive integer (>1) is not equal to infinity, ∞i.e. 푛∞ ≠ . Keywords: positive integer(퐼+); infinity*(∞); Riemann zeta function 휁(푠). 1. INTRODUCTION So, what is the infinite product of a positive integer (>1), i.e. what is 푛∞? Is it ∞ or something else? Till now, not much research has been done on this topic, and by convention we assume 푛∞ to be equal to ∞. In this paper, we will make simple equations and also use some known sums and prove that 푛∞ = 0 and that, it is not equal to ∞. We are going to approach the problem by observing patterns, assigning and re- assigning variables, and finally solving the equations. But first, we need to prove that 푛∞ ≠ ∞. 2. METHODS The methods for solving the problem are simple but a bit counter-intuitive. In this paper, we will make simple mathematical equations and firstly prove that 푛∞ ≠ ∞.
    [Show full text]
  • Regularized Integral Representations of the Reciprocal Function
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 25 December 2018 doi:10.20944/preprints201812.0310.v1 Peer-reviewed version available at Fractal Fract 2019, 3, 1; doi:10.3390/fractalfract3010001 Article Regularized Integral Representations of the Reciprocal G Function Dimiter Prodanov Correspondence: Environment, Health and Safety, IMEC vzw, Kapeldreef 75, 3001 Leuven, Belgium; [email protected]; [email protected] Version December 25, 2018 submitted to Preprints 1 Abstract: This paper establishes a real integral representation of the reciprocal G function in terms 2 of a regularized hypersingular integral. The equivalence with the usual complex representation is 3 demonstrated. A regularized complex representation along the usual Hankel path is derived. 4 Keywords: gamma function; reciprocal gamma function; integral equation 5 MSC: 33B15; 65D20, 30D10 6 1. Introduction 7 Applications of the Gamma function are ubiquitous in fractional calculus and the special function 8 theory. It has numerous interesting properties summarized in [1]. It is indispensable in the theory of 9 Laplace transforms. The history of the Gamma function is surveyed in [2]. In a previous note I have 10 investigated an approach to regularize derivatives at points, where the ordinary limit diverges [5]. 11 This paper exploits the same approach for the purposes of numerical computation of singular integrals, 12 such as the Euler G integrals for negative arguments. The paper also builds on an observations in [4]. 13 The present paper proves a real singular integral representation of the reciprocal G function. The 14 algorithm is implemented in the Computer Algebra System Maxima for reference and demonstration 15 purposes.
    [Show full text]
  • Functions of a Complex Variable II Math 562, Spring 2020
    Functions of a Complex Variable II Math 562, Spring 2020 Jens Lorenz April 27, 2020 Department of Mathematics and Statistics, UNM, Albuquerque, NM 87131 Contents 1 Notations 5 2 The Schwarz Lemma and Aut(D) 6 2.1 Schwarz Lemma . .6 2.2 Biholomorphic Maps and Automorphisms . .7 2.3 The Automorphism Group of D ...................7 2.4 The Schwarz{Pick Lemma . .9 2.5 Remarks . 11 3 Linear Fractional Transformations and the Riemann Sphere 12 3.1 The Riemann Sphere . 12 3.2 Linear Fractional Transformations . 14 3.3 The Automorphisms of the Riemann Sphere . 16 3.4 A Remarkable Geometric Property of Linear Fractional Trans- formations . 17 3.4.1 Analytical Description of Circles . 18 3.4.2 Circles under Reciprocation . 18 3.4.3 Straight Lines under Reciprocation . 20 3.5 Example: The Cayley Transform in the Complex Plane . 21 3.6 The Cayley Transform of a Matrix . 22 4 The Riemann Mapping Theorem 24 4.1 Statement and Overview of Proof . 24 4.2 The Theorem of Arzela{Ascoli . 26 4.3 Montel's Theorem . 29 4.4 Auxiliary Results on Logarithms and Square Roots . 32 4.5 Construction of a Map in ..................... 33 4.6 A Bound of f 0(P ) for all Ff .................. 35 4.7 Application ofj thej Theorems2 of F Montel and Hurwitz . 36 4.8 Proof That f is Onto . 37 1 4.9 Examples of Biholomorphic Mappings . 39 5 An Introduction to Schwarz–Christoffel Formulas 41 5.1 General Power Functions . 41 5.2 An Example of a Schwarz–Christoffel Map .
    [Show full text]
  • Numerous Proofs of Ζ(2) = 6
    π2 Numerous Proofs of ζ(2) = 6 Brendan W. Sullivan April 15, 2013 Abstract In this talk, we will investigate how the late, great Leonhard Euler P1 2 2 originally proved the identity ζ(2) = n=1 1=n = π =6 way back in 1735. This will briefly lead us astray into the bewildering forest of com- plex analysis where we will point to some important theorems and lemmas whose proofs are, alas, too far off the beaten path. On our journey out of said forest, we will visit the temple of the Riemann zeta function and marvel at its significance in number theory and its relation to the prob- lem at hand, and we will bow to the uber-famously-unsolved Riemann hypothesis. From there, we will travel far and wide through the kingdom of analysis, whizzing through a number N of proofs of the same original fact in this talk's title, where N is not to exceed 5 but is no less than 3. Nothing beyond a familiarity with standard calculus and the notion of imaginary numbers will be presumed. Note: These were notes I typed up for myself to give this seminar talk. I only got through a portion of the material written down here in the actual presentation, so I figured I'd just share my notes and let you read through them. Many of these proofs were discovered in a survey article by Robin Chapman (linked below). I chose particular ones to work through based on the intended audience; I also added a section about justifying the sin(x) \factoring" as an infinite product (a fact upon which two of Euler's proofs depend) and one about the Riemann Zeta function and its use in number theory.
    [Show full text]
  • On Infinite Products
    :?!t--:v.- -, View metadata, citation and similar papers at core.ac.ukbrought to you by CORE provided by K-State Research Exchange ON INFINITE PRODUCTS by ROSE KORDONOWY SHAW B. S. , Dickinson State College, 1961^ A MASTER'S REPORT submitted in partial fulfillment of the requirements for the degree MASTER OF SCIENCE Department of Mathematics KANSAS STATE UNIVERSITY" Manhattan, Kansas 1966 Approved by; :o7 U) '^ JScGV ii '3^ TABLE OP CONTENTS INTRODUCTION 1 BASIC DEFINITIONS 1 CONVERGENCE OF INFINITE PRODUCTS i; RELATING CONVERGENCE OF INFINITE PRODUCTS TO CONVERGENCE OF INFINITE SERIES 9 ABSOLUTE CONVERGENCE 21 THE ASSOCIATE LOGARITHMIC SERIES 2i|. UNIFORM CONVERGENCE 28 DEVELOPMENT OF TRIGONOMETRIC FUNCTIONS AS INFINITE PRODUCTS 3i| ACKNOWLEDGMENT l^^ BIBLIOGRAPHY l^]^ INTRODUCTION Many functions can be represented in many different ways. They can be represented by Fourier series, orthogonal poly- nomials, infinite series, and infinite products. Infinite products will be considered as the background for the develop- ment of expansion of functions. The infinite product repre- sentations of sin X and cos x will be considered. The theory of infinite products is very closely related to that of infinite series; so closely, in fact, that the test for convergence of such products will actually be reduced to the test for the convergence of certain infinite series. The object of this paper is to begin with the basic defini- tion of an infinite product and to take the study of infinite products to the point where a basic understanding of infinite products and how they can be handled is reached. BASIC DEFINITIONS An infinite product is of the form .
    [Show full text]
  • Infinite Product Group
    Brigham Young University BYU ScholarsArchive Theses and Dissertations 2007-07-13 Infinite Product Group Keith G. Penrod Brigham Young University - Provo Follow this and additional works at: https://scholarsarchive.byu.edu/etd Part of the Mathematics Commons BYU ScholarsArchive Citation Penrod, Keith G., "Infinite Product Group" (2007). Theses and Dissertations. 976. https://scholarsarchive.byu.edu/etd/976 This Thesis is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact [email protected], [email protected]. INFINITE PRODUCT GROUPS by Keith Penrod A thesis submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics Brigham Young University August 2007 Copyright c 2007 Keith Penrod All Rights Reserved BRIGHAM YOUNG UNIVERSITY GRADUATE COMMITTEE APPROVAL of a thesis submitted by Keith Penrod This thesis has been read by each member of the following graduate committee and by majority vote has been found to be satisfactory. Date Gregory Conner, Chair Date James Cannon Date Eric Swenson BRIGHAM YOUNG UNIVERSITY As chair of the candidate’s graduate committee, I have read the thesis of Keith Penrod in its final form and have found that (1) its format, citations, and bibliographical style are consistent and acceptable and fulfill university and department style requirements; (2) its illustrative materials including figures, tables, and charts are in place; and (3) the final manuscript is satisfactory to the graduate committee and is ready for submission to the university library.
    [Show full text]
  • Chapter 1 Euler's Product Formula
    Chapter 1 Euler’s Product Formula 1.1 The Product Formula The whole of analytic number theory rests on one marvellous formula due to Leonhard Euler (1707-1783): −1 X n−s = Y 1 − p−s . n∈N, n>0 primes p Informally, we can understand the formula as follows. By the Funda- mental Theorem of Arithmetic, each n ≥ 1 is uniquely expressible in the form n = 2e2 3e3 5e5 ··· , where e2, e3, e5,... are non-negative integers (all but a finite number being 0). Raising this to the power −s, n−s = 2−e2s3−e3s5−e5s ··· . Adding for n = 1, 2, 3,... , X n−s = 1 + 2−s + 2−2s + ··· 1 + 3−s + 3−2s + ··· 1 + 5−s + 5−2s + ··· ··· , each term on the left arising from just one product on the right. But for each prime p, −1 1 + p−s + p−2s + ··· = 1 − p−s , and the result follows. Euler’s Product Formula equates the Dirichlet series P n−s on the left with the infinite product on the right. 1–1 1.2. INFINITE PRODUCTS 1–2 To make the formula precise, we must develop the theory of infinite prod- ucts, which we do in the next Section. To understand the implications of the formula, we must develop the the- ory of Dirichlet series, which we do in the next Chapter. 1.2 Infinite products 1.2.1 Definition Infinite products are less familiar than infinite series, but are no more com- plicated. Both are examples of limits of sequences. Definition 1.1. The infinite product Y cn n∈N is said to converge to ` 6= 0 if the partial products Y Pn = cm → ` as n → ∞.
    [Show full text]
  • The Weierstrass Factorization Theorem for Slice Regular Functions Over the Quaternions
    Ann Glob Anal Geom DOI 10.1007/s10455-011-9266-0 ORIGINAL PAPER The Weierstrass factorization theorem for slice regular functions over the quaternions Graziano Gentili · Irene Vignozzi Received: 8 November 2010 / Accepted: 5 April 2011 © Springer Science+Business Media B.V. 2011 Abstract The class of slice regular functions of a quaternionic variable has been recently introduced and is intensively studied, as a quaternionic analogue of the class of holomorphic functions. Unlike other classes of quaternionic functions, this one contains natural quatern- ionic polynomials and power series. Its study has already produced a rather rich theory having steady foundations and interesting applications. The main purpose of this article is to prove a Weierstrass factorization theorem for slice regular functions. This result holds in a formula- tion that reflects the peculiarities of the quaternionic setting and the structure of the zero set of such functions. Some preliminary material that we need to prove has its own independent interest, like the study of a quaternionic logarithm and the convergence of infinite products of quaternionic functions. Keywords Functions of a quaternionic variable · Weierstrass factorization theorem · Zeros of hyperholomorphic functions Mathematics Subject Classification (2000) 30G35 · 30C15 · 30B10 1 Introduction The well-known theory of Fueter regular functions [9,10,25] is the first and most celebrated candidate among a few others, in the search for a quaternionic analogue of the theory of holo- morphic functions of one complex variable. This well-developed theory [3,19–21] inspired also the setting of Clifford algebras (see, e.g. [2]). Recently, Gentili and Struppa proposed a different approach, which led to a new notion of holomorphicity (called slice regular- ity) for quaternion-valued functions of a quaternionic variable [15,16].
    [Show full text]
  • Gamma and Beta Functions
    Appendix A Gamma and Beta Functions A.1 A Useful Formula The following formula is valid: Z 2 √ n e−|x| dx = π . Rn This is an immediate consequence of the corresponding one-dimensional identity Z +∞ 2 √ e−x dx = π , −∞ which is usually proved from its two-dimensional version by switching to polar coordinates: Z +∞ Z +∞ 2 2 Z ∞ 2 I2 = e−x e−y dydx = 2π re−r dr = π . −∞ −∞ 0 A.2 Definitions of Γ (z) and B(z,w) For a complex number z with Rez > 0 define Z ∞ Γ (z) = tz−1e−t dt. 0 Γ (z) is called the gamma function. It follows from its definition that Γ (z) is analytic on the right half-plane Rez > 0. Two fundamental properties of the gamma function are that Γ (z + 1) = zΓ (z) and Γ (n) = (n − 1)!, where z is a complex number with positive real part and n ∈ Z+. Indeed, integration by parts yields Z ∞ tze−t ∞ 1 Z ∞ 1 Γ (z) = tz−1e−t dt = + tze−t dt = Γ (z + 1). 0 z 0 z 0 z Since Γ (1) = 1, the property Γ (n) = (n − 1)! for n ∈ Z+ follows by induction. Another important fact is that 417 418 A Gamma and Beta Functions √ 1 Γ 2 = π . This follows easily from the identity Z ∞ 1 Z ∞ 2 √ 1 − 2 −t −u Γ 2 = t e dt = 2 e du = π . 0 0 Next we define the beta function. Fix z and w complex numbers with positive real parts. We define Z 1 Z 1 B(z,w) = tz−1(1 −t)w−1 dt = tw−1(1 −t)z−1 dt.
    [Show full text]
  • On Wallis-Type Products and Polya's Urn Schemes
    On Wallis-type products and P´olya's urn schemes Iddo Ben-Ari Diana Hay Alexander Roitershtein Abstract A famous \curious identity" of Wallis gives a representation of the constant π in terms of a simply structured infinite product of fractions. Sondow and Yi [Amer. Math. Monthly 117 (2010), 912-917] identified a general scheme for evaluating Wallis-type infinite products. The main purpose of this paper is to discuss an interpretation of the scheme by means of P´olya urn models. 1 Introduction This paper is motivated by the work of Sondow and Yi [20], where several exam- ples of Wallis-type products (see Definition 2.6 below) have been constructed. The method used in [20] yields \cyclically structured" converging infinite prod- ucts of fractions and evaluates their limit by means of the gamma function (see Section 2 below). Only in a limited range of cases is an expression of the limit in terms of powers of π and algebraic numbers known. Section 3 contains an in- structive survey of generic examples. In Section 4 we discuss a relation between the Wallis-type products and the P´olya urn scheme. Our main observation is contained in the statement of Theorem 4.4. Throughout the paper, Γ( · ) is the gamma function and N; Q; R; C are, respectively, natural, rational, real, and complex numbers. 2 Wallis-type infinite products Our starting point is a representation of the constant π discovered by Wallis [3, p. 68]: π 2 2 4 4 6 6 8 8 10 10 12 12 = ··· : (1) 2 1 3 3 5 5 7 7 9 9 11 11 13 A standard proof of the identity (1) relies on the evaluation of Wallis's integral R π=2 2n+1 0 sin t dt [2].
    [Show full text]
  • Regularized Integral Representations of the Reciprocal Gamma Function
    fractal and fractional Article Regularized Integral Representations of the Reciprocal Gamma Function Dimiter Prodanov 1,2 1 Environment, Health and Safety, IMEC, 3001 Leuven, Belgium; [email protected] or [email protected] 2 Neuroscience Research Flanders, 3001 Leuven, Belgium Received: 16 November 2018; Accepted: 8 January 2019; Published: 12 January 2019 Abstract: This paper establishes a real integral representation of the reciprocal Gamma function in terms of a regularized hypersingular integral along the real line. A regularized complex representation along the Hankel path is derived. The equivalence with the Heine’s complex representation is demonstrated. For both real and complex integrals, the regularized representation can be expressed in terms of the two-parameter Mittag-Leffler function. Reference numerical implementations in the Computer Algebra System Maxima are provided. Keywords: Gamma function; reciprocal Gamma function; Mittag-Leffler function; integral equation MSC: 33B15; 65D20; 30D10 1. Introduction Applications of the Gamma function in fractional calculus and the special function theory are ubiquitous. For example, the function is indispensable in the theory of Laplace transforms. The history of the Gamma function is surveyed in [1], in which the author further states that “of the so-called ’higher mathematical functions’, the Gamma function is undoubtedly the most fundamental”. A classical reference on the Gamma function is given by Artin [2]. The Gamma function has numerous remarkable properties that are surveyed in [3]. For example, one of its classical applications is the formula for the volume of an n-dimensional ball. The reciprocal Gamma function is a normalization constant in all of the classical fractional derivative operators: the Riemann–Liouville, Caputo, and Grünwald–Letnikov.
    [Show full text]
  • Lectures on the Riemann Zeta–Function
    Lectures on The Riemann Zeta–Function By K. Chandrasekharan Tata Institute of Fundamental Research, Bombay 1953 Lectures on the Riemann Zeta-Function By K. Chandrasekharan Tata Institute of Fundamental Research 1953 The aim of these lectures is to provide an intorduc- tion to the theory of the Riemann Zeta-function for stu- dents who might later want to do research on the subject. The Prime Number Theorem, Hardy’s theorem on the Zeros of ζ(s), and Hamburger’s theorem are the princi- pal results proved here. The exposition is self-contained, and required a preliminary knowledge of only the ele- ments of function theory. Contents 1 The Maximum Principle 1 2 The Phragmen-Lindelof principle 9 3 Schwarz’s Lemma 17 4 Entire Functions 25 5 Entire Functions (Contd.) 35 6 The Gamma Function 45 1 Elementary Properties . 45 2 Analytic continuation of Γ(z)............... 49 3 TheProductFormula ................... 51 7 The Gamma Function: Contd 55 4 The Bohr-Mollerup-Artin Theorem . 55 5 Gauss’s Multiplication Formula . 58 6 Stirling’s Formula . 59 8 The Zeta Function of Riemann 63 1 Elementary Properties of ζ(s)............... 63 9 The Zeta Function of Riemann (Contd.) 69 2 Elementary theory of Dirichlet Series . 69 v vi Contents 10 The Zeta Function of Riemann (Contd) 75 2 (Contd). Elementary theory of Dirichlet series . 75 11 The Zeta Function of Riemann (Contd) 87 3 Analytic continuation of ζ(s). First method . 87 4 Functional Equation (First method) . 90 5 Functional Equation (Second Method) . 92 12 The Zeta Function of Riemann (Contd) 97 6 Some estimates for ζ(s).................
    [Show full text]