Gamma and Beta Integrals 1 Gamma Function
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Introduction to Analytic Number Theory the Riemann Zeta Function and Its Functional Equation (And a Review of the Gamma Function and Poisson Summation)
Math 229: Introduction to Analytic Number Theory The Riemann zeta function and its functional equation (and a review of the Gamma function and Poisson summation) Recall Euler’s identity: ∞ ∞ X Y X Y 1 [ζ(s) :=] n−s = p−cps = . (1) 1 − p−s n=1 p prime cp=0 p prime We showed that this holds as an identity between absolutely convergent sums and products for real s > 1. Riemann’s insight was to consider (1) as an identity between functions of a complex variable s. We follow the curious but nearly universal convention of writing the real and imaginary parts of s as σ and t, so s = σ + it. We already observed that for all real n > 0 we have |n−s| = n−σ, because n−s = exp(−s log n) = n−σe−it log n and e−it log n has absolute value 1; and that both sides of (1) converge absolutely in the half-plane σ > 1, and are equal there either by analytic continuation from the real ray t = 0 or by the same proof we used for the real case. Riemann showed that the function ζ(s) extends from that half-plane to a meromorphic function on all of C (the “Riemann zeta function”), analytic except for a simple pole at s = 1. The continuation to σ > 0 is readily obtained from our formula ∞ ∞ 1 X Z n+1 X Z n+1 ζ(s) − = n−s − x−s dx = (n−s − x−s) dx, s − 1 n=1 n n=1 n since for x ∈ [n, n + 1] (n ≥ 1) and σ > 0 we have Z x −s −s −1−s −1−σ |n − x | = s y dy ≤ |s|n n so the formula for ζ(s) − (1/(s − 1)) is a sum of analytic functions converging absolutely in compact subsets of {σ + it : σ > 0} and thus gives an analytic function there. -
Probabilistic Proofs of Some Beta-Function Identities
1 2 Journal of Integer Sequences, Vol. 22 (2019), 3 Article 19.6.6 47 6 23 11 Probabilistic Proofs of Some Beta-Function Identities Palaniappan Vellaisamy Department of Mathematics Indian Institute of Technology Bombay Powai, Mumbai-400076 India [email protected] Aklilu Zeleke Lyman Briggs College & Department of Statistics and Probability Michigan State University East Lansing, MI 48825 USA [email protected] Abstract Using a probabilistic approach, we derive some interesting identities involving beta functions. These results generalize certain well-known combinatorial identities involv- ing binomial coefficients and gamma functions. 1 Introduction There are several interesting combinatorial identities involving binomial coefficients, gamma functions, and hypergeometric functions (see, for example, Riordan [9], Bagdasaryan [1], 1 Vellaisamy [15], and the references therein). One of these is the following famous identity that involves the convolution of the central binomial coefficients: n 2k 2n − 2k =4n. (1) k n − k k X=0 In recent years, researchers have provided several proofs of (1). A proof that uses generating functions can be found in Stanley [12]. Combinatorial proofs can also be found, for example, in Sved [13], De Angelis [3], and Miki´c[6]. A related and an interesting identity for the alternating convolution of central binomial coefficients is n n n 2k 2n − 2k 2 n , if n is even; (−1)k = 2 (2) k n − k k=0 (0, if n is odd. X Nagy [7], Spivey [11], and Miki´c[6] discussed the combinatorial proofs of the above identity. Recently, there has been considerable interest in finding simple probabilistic proofs for com- binatorial identities (see Vellaisamy and Zeleke [14] and the references therein). -
Leibniz's Rule and Fubini's Theorem Associated with Hahn Difference
View metadata, citation and similar papers at core.ac.uk brought to you by CORE I S S N 2provided 3 4 7 -by1921 KHALSA PUBLICATIONS Volume 12 Number 06 J o u r n a l of Advances in Mathematics Leibniz’s rule and Fubini’s theorem associated with Hahn difference operators å Alaa E. Hamza † , S. D. Makharesh † † Jeddah University, Department of Mathematics, Saudi, Arabia † Cairo University, Department of Mathematics, Giza, Egypt E-mail: [email protected] å † Cairo University, Department of Mathematics, Giza, Egypt E-mail: [email protected] ABSTRACT In 1945 , Wolfgang Hahn introduced his difference operator Dq, , which is defined by f (qt ) f (t) D f (t) = , t , q, (qt ) t 0 where = with 0 < q <1, > 0. In this paper, we establish Leibniz’s rule and Fubini’s theorem associated with 0 1 q this Hahn difference operator. Keywords. q, -difference operator; q, –Integral; q, –Leibniz Rule; q, –Fubini’s Theorem. 1Introduction The Hahn difference operator is defined by f (qt ) f (t) D f (t) = , t , (1) q, (qt ) t 0 where q(0,1) and > 0 are fixed, see [2]. This operator unifies and generalizes two well known difference operators. The first is the Jackson q -difference operator which is defined by f (qt) f (t) D f (t) = , t 0, (2) q qt t see [3, 4, 5, 6]. The second difference operator which Hahn’s operator generalizes is the forward difference operator f (t ) f (t) f (t) = , t R, (3) (t ) t where is a fixed positive number, see [9, 10, 13, 14]. -
INTEGRALS of POWERS of LOGGAMMA 1. Introduction The
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 139, Number 2, February 2011, Pages 535–545 S 0002-9939(2010)10589-0 Article electronically published on August 18, 2010 INTEGRALS OF POWERS OF LOGGAMMA TEWODROS AMDEBERHAN, MARK W. COFFEY, OLIVIER ESPINOSA, CHRISTOPH KOUTSCHAN, DANTE V. MANNA, AND VICTOR H. MOLL (Communicated by Ken Ono) Abstract. Properties of the integral of powers of log Γ(x) from 0 to 1 are con- sidered. Analytic evaluations for the first two powers are presented. Empirical evidence for the cubic case is discussed. 1. Introduction The evaluation of definite integrals is a subject full of interconnections of many parts of mathematics. Since the beginning of Integral Calculus, scientists have developed a large variety of techniques to produce magnificent formulae. A partic- ularly beautiful formula due to J. L. Raabe [12] is 1 Γ(x + t) (1.1) log √ dx = t log t − t, for t ≥ 0, 0 2π which includes the special case 1 √ (1.2) L1 := log Γ(x) dx =log 2π. 0 Here Γ(x)isthegamma function defined by the integral representation ∞ (1.3) Γ(x)= ux−1e−udu, 0 for Re x>0. Raabe’s formula can be obtained from the Hurwitz zeta function ∞ 1 (1.4) ζ(s, q)= (n + q)s n=0 via the integral formula 1 t1−s (1.5) ζ(s, q + t) dq = − 0 s 1 coupled with Lerch’s formula ∂ Γ(q) (1.6) ζ(s, q) =log √ . ∂s s=0 2π An interesting extension of these formulas to the p-adic gamma function has recently appeared in [3]. -
2020 Mathsoc Integration Bee Qualifiers Solutions
2020 MathSoc Integration Bee Qualifiers Solutions 1. Standard integral, but the bounds are tricky: Z 507 1 x dx = 5072 − 5032 503 2 1 = (507 + 503) (507 − 503) 2 1 = · 1010 · 4 2 = 2020: 2. Rewrite 3ln x as xln 3 using log laws, so now we have a standard integral: Z Z x1+ln 3 3ln x dx = xln 3 dx = + C: 1 + ln 3 3. This is simply integrating ex=2: Z p Z p ex dx = ex=2 dx = 2 ex + C: p 4. Substitute u = ex − 1: Z p Z u2 Z 1 ex − 1 dx = 2 du = 2 1 − du u2 + 1 1 + u2 = 2 u − tan−1 u + C p p = 2 ex − 1 − tan−1 ex − 1 + C: 5. Substitute x = eu then the integral becomes Z eu cos u du: Apply integration by parts twice: Z Z eu cos u du = eu cos u + eu sin u du Z = eu cos u + eu sin u − eu cos u du: Rearrange to obtain Z 1 eu cos u du = eu (cos u + sin u) ; 2 then Z 1 cos (ln x) dx = x (cos (ln x) + sin (ln x)) + C: 2 6. Substitute u = e2x, then Z e2x 1 Z 1 1 1 p dx = p du = sin−1 u + C = sin−1 e2x + C: 1 − e4x 2 1 − u2 2 2 Z 7π=4 4x cos x 7. This is an odd function from "−a to a" so 2 dx = 0. −7π=4 x − sin jxj + cos jxj c UNSW Mathematics Society 2020 π 8. -
Standard Distributions
Appendix A Standard Distributions A.1 Standard Univariate Discrete Distributions I. Binomial Distribution B(n, p) has the probability mass function n f(x)= px (x =0, 1,...,n). x The mean is np and variance np(1 − p). II. The Negative Binomial Distribution NB(r, p) arises as the distribution of X ≡{number of failures until the r-th success} in independent trials each with probability of success p. Thus its probability mass function is r + x − 1 r x fr(x)= p (1 − p) (x =0, 1, ··· , ···). x Let Xi denote the number of failures between the (i − 1)-th and i-th successes (i =2, 3,...,r), and let X1 be the number of failures before the first success. Then X1,X2,...,Xr and r independent random variables each having the distribution NB(1,p) with probability mass function x f1(x)=p(1 − p) (x =0, 1, 2,...). Also, X = X1 + X2 + ···+ Xr. Hence ∞ ∞ x x−1 E(X)=rEX1 = r p x(1 − p) = rp(1 − p) x(1 − p) x=0 x=1 ∞ ∞ d d = rp(1 − p) − (1 − p)x = −rp(1 − p) (1 − p)x x=1 dp dp x=1 © Springer-Verlag New York 2016 343 R. Bhattacharya et al., A Course in Mathematical Statistics and Large Sample Theory, Springer Texts in Statistics, DOI 10.1007/978-1-4939-4032-5 344 A Standard Distributions ∞ d d 1 = −rp(1 − p) (1 − p)x = −rp(1 − p) dp dp 1 − (1 − p) x=0 =p r(1 − p) = . (A.1) p Also, one may calculate var(X) using (Exercise A.1) var(X)=rvar(X1). -
Elementary Calculus
Elementary Calculus 2 v0 2g 2 0 v0 g Michael Corral Elementary Calculus Michael Corral Schoolcraft College About the author: Michael Corral is an Adjunct Faculty member of the Department of Mathematics at School- craft College. He received a B.A. in Mathematics from the University of California, Berkeley, and received an M.A. in Mathematics and an M.S. in Industrial & Operations Engineering from the University of Michigan. This text was typeset in LATEXwith the KOMA-Script bundle, using the GNU Emacs text editor on a Fedora Linux system. The graphics were created using TikZ and Gnuplot. Copyright © 2020 Michael Corral. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. Preface This book covers calculus of a single variable. It is suitable for a year-long (or two-semester) course, normally known as Calculus I and II in the United States. The prerequisites are high school or college algebra, geometry and trigonometry. The book is designed for students in engineering, physics, mathematics, chemistry and other sciences. One reason for writing this text was because I had already written its sequel, Vector Cal- culus. More importantly, I was dissatisfied with the current crop of calculus textbooks, which I feel are bloated and keep moving further away from the subject’s roots in physics. In addi- tion, many of the intuitive approaches and techniques from the early days of calculus—which I think often yield more insights for students—seem to have been lost. -
Sums of Powers and the Bernoulli Numbers Laura Elizabeth S
Eastern Illinois University The Keep Masters Theses Student Theses & Publications 1996 Sums of Powers and the Bernoulli Numbers Laura Elizabeth S. Coen Eastern Illinois University This research is a product of the graduate program in Mathematics and Computer Science at Eastern Illinois University. Find out more about the program. Recommended Citation Coen, Laura Elizabeth S., "Sums of Powers and the Bernoulli Numbers" (1996). Masters Theses. 1896. https://thekeep.eiu.edu/theses/1896 This is brought to you for free and open access by the Student Theses & Publications at The Keep. It has been accepted for inclusion in Masters Theses by an authorized administrator of The Keep. For more information, please contact [email protected]. THESIS REPRODUCTION CERTIFICATE TO: Graduate Degree Candidates (who have written formal theses) SUBJECT: Permission to Reproduce Theses The University Library is rece1v1ng a number of requests from other institutions asking permission to reproduce dissertations for inclusion in their library holdings. Although no copyright laws are involved, we feel that professional courtesy demands that permission be obtained from the author before we allow theses to be copied. PLEASE SIGN ONE OF THE FOLLOWING STATEMENTS: Booth Library of Eastern Illinois University has my permission to lend my thesis to a reputable college or university for the purpose of copying it for inclusion in that institution's library or research holdings. u Author uate I respectfully request Booth Library of Eastern Illinois University not allow my thesis -
Extended K-Gamma and K-Beta Functions of Matrix Arguments
mathematics Article Extended k-Gamma and k-Beta Functions of Matrix Arguments Ghazi S. Khammash 1, Praveen Agarwal 2,3,4 and Junesang Choi 5,* 1 Department of Mathematics, Al-Aqsa University, Gaza Strip 79779, Palestine; [email protected] 2 Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India; [email protected] 3 Institute of Mathematics and Mathematical Modeling, Almaty 050010, Kazakhstan 4 Harish-Chandra Research Institute Chhatnag Road, Jhunsi, Prayagraj (Allahabad) 211019, India 5 Department of Mathematics, Dongguk University, Gyeongju 38066, Korea * Correspondence: [email protected] Received: 17 August 2020; Accepted: 30 September 2020; Published: 6 October 2020 Abstract: Various k-special functions such as k-gamma function, k-beta function and k-hypergeometric functions have been introduced and investigated. Recently, the k-gamma function of a matrix argument and k-beta function of matrix arguments have been presented and studied. In this paper, we aim to introduce an extended k-gamma function of a matrix argument and an extended k-beta function of matrix arguments and investigate some of their properties such as functional relations, inequality, integral formula, and integral representations. Also an application of the extended k-beta function of matrix arguments to statistics is considered. Keywords: k-gamma function; k-beta function; gamma function of a matrix argument; beta function of matrix arguments; extended gamma function of a matrix argument; extended beta function of matrix arguments; k-gamma function of a matrix argument; k-beta function of matrix arguments; extended k-gamma function of a matrix argument; extended k-beta function of matrix arguments 1. -
TYPE BETA FUNCTIONS of SECOND KIND 135 Operators
Adv. Oper. Theory 1 (2016), no. 1, 134–146 http://doi.org/10.22034/aot.1609.1011 ISSN: 2538-225X (electronic) http://aot-math.org (p, q)-TYPE BETA FUNCTIONS OF SECOND KIND ALI ARAL 1∗ and VIJAY GUPTA2 Communicated by A. Kaminska Abstract. In the present article, we propose the (p, q)-variant of beta func- tion of second kind and establish a relation between the generalized beta and gamma functions using some identities of the post-quantum calculus. As an ap- plication, we also propose the (p, q)-Baskakov–Durrmeyer operators, estimate moments and establish some direct results. 1. Introduction The quantum calculus (q-calculus) in the field of approximation theory was discussed widely in the last two decades. Several generalizations to the q variants were recently presented in the book [3]. Further there is possibility of extension of q-calculus to post-quantum calculus, namely the (p, q)-calculus. Actually such extension of quantum calculus can not be obtained directly by substitution of q by q/p in q-calculus. But there is a link between q-calculus and (p, q)-calculus. The q calculus may be obtained by substituting p = 1 in (p, q)-calculus. We mentioned some previous results in this direction. Recently, Gupta [8] introduced (p, q) gen- uine Bernstein–Durrmeyer operators and established some direct results. (p, q) generalization of Sz´asz–Mirakyan operators was defined in [1]. Also authors inves- tigated a Durrmeyer type modifications of the Bernstein operators in [9]. We can also mention other papers as Bernstein operators [10], Bernstein–Stancu oper- ators [11]. -
Notes on Riemann's Zeta Function
NOTES ON RIEMANN’S ZETA FUNCTION DRAGAN MILICIˇ C´ 1. Gamma function 1.1. Definition of the Gamma function. The integral ∞ Γ(z)= tz−1e−tdt Z0 is well-defined and defines a holomorphic function in the right half-plane {z ∈ C | Re z > 0}. This function is Euler’s Gamma function. First, by integration by parts ∞ ∞ ∞ Γ(z +1)= tze−tdt = −tze−t + z tz−1e−t dt = zΓ(z) Z0 0 Z0 for any z in the right half-plane. In particular, for any positive integer n, we have Γ(n) = (n − 1)Γ(n − 1)=(n − 1)!Γ(1). On the other hand, ∞ ∞ Γ(1) = e−tdt = −e−t = 1; Z0 0 and we have the following result. 1.1.1. Lemma. Γ(n) = (n − 1)! for any n ∈ Z. Therefore, we can view the Gamma function as a extension of the factorial. 1.2. Meromorphic continuation. Now we want to show that Γ extends to a meromorphic function in C. We start with a technical lemma. Z ∞ 1.2.1. Lemma. Let cn, n ∈ +, be complex numbers such such that n=0 |cn| converges. Let P S = {−n | n ∈ Z+ and cn 6=0}. Then ∞ c f(z)= n z + n n=0 X converges absolutely for z ∈ C − S and uniformly on bounded subsets of C − S. The function f is a meromorphic function on C with simple poles at the points in S and Res(f, −n)= cn for any −n ∈ S. 1 2 D. MILICIˇ C´ Proof. Clearly, if |z| < R, we have |z + n| ≥ |n − R| for all n ≥ R. -
Lp-Solution to the Random Linear Delay Differential Equation with a Stochastic Forcing Term
mathematics Article Lp-solution to the Random Linear Delay Differential Equation with a Stochastic Forcing Term Juan Carlos Cortés * and Marc Jornet Instituto Universitario de Matemática Multidisciplinar, Building 8G, access C, 2nd floor, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain; [email protected] * Correspondence: [email protected] Received: 25 May 2020; Accepted: 18 June 2020; Published: 20 June 2020 Abstract: This paper aims at extending a previous contribution dealing with the random autonomous-homogeneous linear differential equation with discrete delay t > 0, by adding a random forcing term f (t) that varies with time: x0(t) = ax(t) + bx(t − t) + f (t), t ≥ 0, with initial condition x(t) = g(t), −t ≤ t ≤ 0. The coefficients a and b are assumed to be random variables, while the forcing term f (t) and the initial condition g(t) are stochastic processes on their respective time domains. The equation is regarded in the Lebesgue space Lp of random variables with finite p-th moment. The deterministic solution constructed with the method of steps and the method of variation of constants, which involves the delayed exponential function, is proved to be an Lp-solution, under certain assumptions on the random data. This proof requires the extension of the deterministic Leibniz’s integral rule for differentiation to the random scenario. Finally, we also prove that, when the delay t tends to 0, the random delay equation tends in Lp to a random equation with no delay. Numerical experiments illustrate how our methodology permits determining the main statistics of the solution process, thereby allowing for uncertainty quantification.