DOCSLIB.ORG

On a Rapidly Converging Series for the Riemann Zeta Function

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On a Rapidly Converging Series for the Riemann Zeta Function On A Rapidly Converging Series For The Riemann Zeta Function Alois Pichler Department of Statistics and Operations Research, University of Vienna, Austria, Universitätsstraße 5, 1010 Vienna Abstract To evaluate Riemann’s zeta function is important for many investigations related to the area of number theory, and to have quickly converging series at hand in particular. We investigate a class of summation formulae and find, as a special case, a new proof of a rapidly converging series for the Riemann zeta function. The series converges in the entire complex plane, its rate of convergence being significantly faster than comparable representations, and so is a useful basis for evaluation algorithms. The evaluation of corresponding coefficients is not problematic, and precise convergence rates are elaborated in detail. The globally converging series obtained allow to reduce Riemann’s hypothesis to similar properties on polynomials. And interestingly, Laguerre’s polynomials form a kind of leitmotif through all sections. Keywords: Riemann Zeta function, Kummer function, Laguerre polynomials, Fourier transform, Riemann hypothesis. 1. Introduction and Definitions 1.1. Confluent Hypergeometric Functions Confluent hypergeometric functions are special hypergeometric functions, sometimes called also Kummer’s function of first and second kind. They are linear independent solutions of Kum- mer’s differential equation z y00 (z) + (b − z) y0 (z) − a y (z) = 0: The first solution, Kummer’s function of the first kind, is usually given as a globally converging power series − 1 a+i 1 i X i z M (a; b; z) := (1.1) b+i−1 i! arXiv:1201.6538v1 [math.NT] 31 Jan 2012 i=0 i (Kummer’s function of the first kind), and Kummer’s function of the second kind is often given by Γ (1 − b) Γ (b − 1) U (a; b; z) := M (a; b; z) − z1−b M (a − b + 1; 2 − b; z) (1.2) Γ (1 − b + a) Γ (a) Email address: [email protected] (Alois Pichler) Preprint submitted to nowhere April 3, 2019 1 INTRODUCTION AND DEFINITIONS 1.2 Laguerre’s Polynomials Kummer’s transformation states that M (a; b; z) = ez M (b − a; b; −z), which subsequently leads to the identity U (a; b; z) = z1−b U (1 + a − b; 2 − b; z). Another solution of this differential equation – which turns out to be identical to U and thus is simply another representation of U – is obtained as an integral representation by the Laplace transform 1 1 U (a; b; z) = e−ztta−1 (1 + t)b−a−1 dt Γ (a) ˆ0 1−b 1 a−b z −zt t = e a dt: (1.3) Γ (1 + a − b) ˆ0 (1 + t) Some particular and frequently used confluent hypergeometric functions are the upper and lower incomplete Gamma function, which have the representations 1 Γ (s; z) := ts−1e−tdt ˆz = e−zU (1 − s; 1 − s; z) = e−zzsU (1; 1 + s; z) (1.4) and z X (−1)k zs+k γ (s; z) := ts−1e−tdt = ˆ k! s + k 0 k=0 zs zs = M (s; s + 1; −z) = e−z M (1; s + 1; z) : (1.5) s s 1.2. Laguerre’s Polynomials For a a negative integer the defining series for M reduces to a polynomial, which turns out to be closely related to Laguerre’s polynomials: The explicit representation is ! i ! i + α X i + α z j L(α) (z) = M (−i; α + 1; z) = (−1) j : (1.6) i i i − j j! j=0 In view of (1.2) Laguerre’s polynomial may be given by Kummer’s second function as well, (α) (−1)i that is Li (z) = i! U (−i; α + 1; z). This somehow suggests that Laguerre’s polynomial are somewhat in between of both solutions M and U of Kummer’s differential equation. This is central in our investigations and reflected in the results of the next sections. In addition to that it is well-know that Laguerre’s polynomials are orthogonal with respect to the weight-function zαe−z; more precisely we find that 8 1 >0 i , j zαe−zL(α) (z) L(α) (z) dz = < (1.7) ˆ i j > i+α 0 : i Γ (α + 1) i = j; Laguerre’s polynomials thus are orthogonal with respect to the inner product 1 zαe−z hgj f i := g (z) f (z) dz: ˆ0 Γ (α + 1) As a result of the classical theory on Hilbert spaces we may expand a function f in a series p with respect to this orthogonal basis, provided that k f k2 := h f j f i < 1. The function has the 2 2 FOURIER SERIES P (α) (α) (α) expansion f (z) = i=0 fi Li (z) (and is therefore in the closed hull of all Li ), its coefficients taking the explicit form 1 L(α) (z) zαe−z f α = i f (z) dz (1.8) i i+α ˆ0 Γ (α + 1) i (cf. (1.7)). Conversely, the norm can be recovered from the function’s coefficients, as k f k2 = 2 P i+α α 2 i=0 i fi . An elementary example of such a representation in explicit terms is 1 X t i e−tz = L(α) (z) ; (1.9) α+1 i 1 + t (1 + t) i=0 as can be verified straight forward by evaluating the respective integrals for the coefficients (1.8). 1 Notably, this series converges point-wise if t > − 2 (k f k2 < 1), even if α ≤ −1. To give a reference of these classical ingredients aggregated in this section above we would like to refer to the standard work [AS64]. 2. Fourier Series Kummer’s functions allow some explicit representation as series of Laguerre polynomials. Theorem 1 (Expansion of Kummer’s functions in terms of Laguerre polynomials). Suppose that 5 Re (b − a) > 0 and Re (α − 2b) > − 2 , then b−1 −a a X i M (a; b; z) = (−1)i L(β−i) (z) b−β−1 i β−b a i=0 i and −a (1 + α − b)! X i U (a; b; z) = L(α) (z) ; (2.1) (1 + α − b + a)! i b−a−α−2 i=0 i moreover, X L(α) (z) Γ (s; z) = zse−z k (2.2) k+1+α−s k=0 (k + 1) k+1 α 1 for Re s − 2 < 4 . zi i Pi −β (β− j) Proof. As for the proof notice first that i! = (−1) j=0 i− j L j (z), which is a kind of con- verse of (1.6) and verified straight forward by comparing the coefficients of respective powers of z. Substituting this into (1.1), interchanging the order of summation and employing Gauss’ 3 2 FOURIER SERIES hypergeometric theorem (cf. [Koe98]) gives 1 a+i−1 i ! X i X −β M (a; b; z) = (−1)i L(β− j) (z) b+i−1 i − j j i=0 i j=0 1 1 a+i−1 ! X X i −β = L(β− j) (z) (−1)i j b+i−1 i − j j=0 i= j i X Γ (b) Γ (b − a − β) Γ (a + j) = L(β− j) (z)(−1) j j Γ (a) Γ (b − a) Γ (b − β + j) j=0 b−1 a+ j−1 X a j = L(β− j) (z)(−1) j j b−β−1 b−β−1+ j j=0 a j b−1 −a a X j = (−1) j L(β− j) (z) ; b−β−1 j β−b a j=0 j which is the desired assertion. As regards convergence it follows from the elementary recursion ! z − β − 1 z L(β− j) (z) = − 1 + L(β− j+1) (z) − L(β− j+2) (z) (2.3) j j j−1 j j−2 L(β− j)(z) j (β− j) −β−1 j z 1 that (−1) L j (z) = O j , a more thorough analysis even discloses that β = e 1 + O j . ( j) −a ( ) a−1 − j (β− j) j O −β−1 j O 1 Hence, ( 1) L j (z) β−b = j jb−β−1 = jb−a+1 , and the series converges – irrespective ( j ) of β – if Re (b − a) > 0. To verify the second statement notice that 1 L(α) (z) zαe−z Uα = i U (a; b; z) dz i i+α ˆ0 Γ (α + 1) i (α) 1 L (z) zαe−z 1 1 = i e−ztta−1 (1 + t)b−a−1 dtdz i+α ˆ0 Γ (α + 1) Γ (a) ˆ0 i 1 ta−1 1 L(α) (z) zαe−z = (1 + t)b−a−1 i e−ztdzdt i+α ˆ0 Γ (a) ˆ0 Γ (α + 1) i 1 1 ti ta−1 ( t)b−a−1 t; = 1 + α+i+1 d Γ (a) ˆ0 (1 + t) which is a consequence of (1.3) and (1.9). The latter integral is a beta function, we thus continue and find the coefficient 4 2 FOURIER SERIES (α) (α) α−1−z (α) α−1 (α) Algorithm 1 to evaluate Li (z), based on Li (z) = 2 + i Li−1 (z) − 1 + i Li−2 (z). L1 : = 0; Laguerre:= 1 For j := 1 t o i L0:= L1 ; L1:= L a g u e r r e ; L a g u e r r e := ( ( 2 ∗ j + alpha − 1− z ) ∗ L1− ( j + alpha − 1)∗ L0 ) / j ; Next j Return Laguerre (β−i) Algorithm 2 to evaluate Li (z), based on (2.3) L1 : = 0; Laguerre:= 1 For j := 1 t o i L0:= L1 ; L1:= L a g u e r r e ; L a g u e r r e := ( ( b e t a+ 1− j − z ) ∗ L1− z ∗ L0 ) / j ; Next j Return Laguerre 1 α 1 a−1+i b−a−α−i−2 Ui = t (1 + t) dt Γ (a) ˆ0 1 Γ (a + i) Γ (α + 2 − b) = Γ (a) Γ (i + a + α + 2 − b) a+i−1 (α + 1 − b)! i = (i + a + α + 1 − b)! i+a+α+2−b i which is the respective coefficient for U.
Load more

Recommended publications
  • Operational Techniques for Laguerre and Legendre Polynomials
    INTERNATIONAL JOURNAL of PURE MATHEMATICS Volume 2, 2015 Operational techniques for Laguerre and Legendre polynomials Clemente Cesarano, Dario Assante, A. R. El Dhaba Abstract— By starting from the concepts and the related n , (2) formalism of the Monomiality Principle, we exploit methods of δ xn =∏() xm − δ operational nature to describe different families of Laguerre m=0 polynomials, ordinary and generalized, and to introduce the Legendre polynomials through a special class of Laguerre polynomials whose associated multiplication and derivative operators read: themselves. Many of the identities presented, involving families of d different polynomials were derived by using the structure of the ^ δ operators who satisfy the rules of a Weyl group. M= xe dx In this paper, we first present the Laguerre and Legendre d δ −1 (3) polynomials, and their generalizations, from an operational point of ^ e dx P = . view, we discuss some operational identities and further we derive δ some interesting relations involving an exotic class of orthogonal polynomials in the description of Legendre polynomials. It is worth noting that, when δ = 0 then: Keywords— Laguerre polynomials, Legendre polynomials, Weyl group, Monomiality Principle, Generating Functions. ^ Mx= (4) ^ d I. INTRODUCTION P = . In this section we will present the concepts and the related aspects of dx the Monomiality Principle [1,2] to explore different approaches for More in general, a given polynomial Laguerre and Legendre polynomials [3]. The associated operational calculus introduced by the Monomiality Principle allows us to reformulate the theory of the generalized Laguerre and Legendre pxn (), nx∈∈, polynomials from a unified point of view. In fact, these are indeed shown to be particular cases of more general polynomials and can be can be considered a quasi-monomial if two operators also used to derive classes of isospectral problems, an interesting example is given by the generalized Bessel functions [4].
    [Show full text]
  • ' Unit 16 Her-Mite and Laguerre Polynomials
    ' UNIT 16 HER-MITE AND LAGUERRE POLYNOMIALS Structure Introduction Objectives Hermite's Differential Equation and ~eAitePolynomials Generating Function and Recurrence Relations for Hermite Polynomials Orthogonality Relations for Hermite Polynomials One Dimensional Harmonic Oscillator Laguerre's Differentiation Equation and Laguerre Polynomials Generating Polynomials and Recurrence Relations for Laguerre Polynomials Orthogonality Relations for Laguerre Polynomials Summary Terminal Questions Solutions and Answers 5 16.1 INTRODUCTION You have studied the properties of the Legendre polynomials, and the Bessel functions in Units 14 and 15, respectively. In this unit, you will learn the properties of Hermite polynomials and Laguerre polynomials. These polynomials are useful in studying the behaviour of a harmonic oscillator and the hydrogen atom using quantum mechanics. In fact, these polynomials enable us to describe the microscopic world of molecules, atoms, and sub- atomic particles in a mathematical language. A study of the properties of these polynomials and a knowledge of their zeros is also helpful in developing numerical methods. Like Legendre and Bessel polynomials, the Hermite and Laguerre polynomials are solutions of linear second order ordinary differential equations (ODE) known as the Hermite and the Laguerre differential equations. You will come across these ODES and the two polynomials in Sec. 16.2 and 16.6, respectively. The generating function and the recurrence relations for Hermite polynomials are given in Sec. 16.3. You have already seen the importance of the generating functions for Legendre and Bessel polynomials in the last units. You will now learn how the generating function for Hermite polynomials is helpful in obtaining many of their properties. The recurrence relations connecting Hermite polynomials of different orders and their derivatives are derived from the generating function in the same section.
    [Show full text]
  • Orthogonal Functions: the Legendre, Laguerre, and Hermite Polynomials
    ORTHOGONAL FUNCTIONS: THE LEGENDRE, LAGUERRE, AND HERMITE POLYNOMIALS THOMAS COVERSON, SAVARNIK DIXIT, ALYSHA HARBOUR, AND TYLER OTTO Abstract. The Legendre, Laguerre, and Hermite equations are all homogeneous second order Sturm-Liouville equations. Using the Sturm-Liouville Theory we will be able to show that polynomial solutions to these equations are orthogonal. In a more general context, finding that these solutions are orthogonal allows us to write a function as a Fourier series with respect to these solutions. 1. Introduction The Legendre, Laguerre, and Hermite equations have many real world practical uses which we will not discuss here. We will only focus on the methods of solution and use in a mathematical sense. In solving these equations explicit solutions cannot be found. That is solutions in in terms of elementary functions cannot be found. In many cases it is easier to find a numerical or series solution. There is a generalized Fourier series theory which allows one to write a function f(x) as a linear combination of an orthogonal system of functions φ1(x),φ2(x),...,φn(x),... on [a; b]. The series produced is called the Fourier series with respect to the orthogonal system. While the R b a f(x)φn(x)dx coefficients ,which can be determined by the formula cn = R b 2 , a φn(x)dx are called the Fourier coefficients with respect to the orthogonal system. We are concerned only with showing that the Legendre, Laguerre, and Hermite polynomial solutions are orthogonal and can thus be used to form a Fourier series. In order to proceed we must define an inner product and define what it means for a linear operator to be self- adjoint.
    [Show full text]
  • On Sobolev Orthogonality for the Generalized Laguerre Polynomials* Teresa E
    Journal of Approximation Theory AT2973 journal of approximation theory 86, 278285 (1996) article no. 0069 On Sobolev Orthogonality for the Generalized Laguerre Polynomials* Teresa E. Perez and Miguel A. Pin~ar- Departamento de Matematica Aplicada, Universidad de Granada, Granada, Spain Communicated by Walter Van Assche Received December 27, 1994; accepted in revised form September 4, 1995 (:) The orthogonality of the generalized Laguerre polynomials, [Ln (x)]n0,isa well known fact when the parameter : is a real number but not a negative integer. In fact, for &1<:, they are orthogonal on the interval [0, +) with respect to the weight function \(x)=x:e&x, and for :<&1, but not an integer, they are orthogonal with respect to a non-positive definite linear functional. In this work we will show that, for every value of the real parameter :, the generalized Laguerre polynomials are orthogonal with respect to a non-diagonal Sobolev inner product, that is, an inner product involving derivatives. 1996 Academic Press, Inc. 1. INTRODUCTION Classical Laguerre polynomials are the main subject of a very extensive (:) literature. If we denote by [Ln (x)]n0 the sequence of monic Laguerre polynomials, their crucial property is the following orthogonality condition + (:) (:) (:) (:) (:) : &x (Ln , Lm )L = Ln (x) L m (x) x e dx=kn $n, m , (1.1) |0 where m, n # N, kn{0 and the parameter : satisfies the condition &1<: in order to assure the convergence of the integrals. For the monic classical Laguerre polynomials, an explicit representation is well known (see Szego [5, p. 102]): n j (:) n (&1) n+: j Ln (x)=(&1) n! : x , n0, (1.2) j j! \n&j+ =0 a where ( b) is the generalized binomial coefficient a 1(a+1) = .
    [Show full text]
  • Some Identities for the Generalized Laguerre Polynomials
    Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (2016), 3388{3396 Research Article Some identities for the generalized Laguerre polynomials Wen-Kai Shaoa, Yuan Heb,∗, Jing Panb aDepartment of Mathematical Teaching and Research, Yibin Vocational & Technical College, 644003 Yibin, Sichuan, P. R. China. bFaculty of Science, Kunming University of Science and Technology, 650500 Kunming, Yunnan, P. R. China. Communicated by R. Saadati Abstract In this paper, we perform a further investigation for the generalized Laguerre polynomials. By applying the generating function methods and Pad´eapproximation techniques, we establish some new identities for the generalized Laguerre polynomials, and give some illustrative special cases as well as immediate consequences of the main results. c 2016 All rights reserved. Keywords: Generalized Laguerre polynomials, Pad´eapproximation, combinatorial identities. 2010 MSC: 11B83, 42C05, 41A21, 05A19. 1. Introduction (α) The generalized Laguerre polynomials Ln (x) associated with non-negative integer n and real number α > −1 are widely used in many problems of mathematical physics and quantum mechanics, for example, in the integration of Helmholtz's equation in paraboloidal coordinates, in the theory of the propagation of electromagnetic oscillations along long lines, etc., as well as in physics in connection with the solution of the second-order linear differential equation: xy00 + (α + 1 − x)y0 + ny = 0: (1.1) These polynomials satisfy some recurrence relations. One very useful, when extracting properties of the wave functions of the hydrogen atom, is the following three-term recurrence relation (see, e.g., [10, 19]): 2n + 1 + α − x n + α L(α) (x) = L(α)(x) − L(α) (x)(n ≥ 1); (1.2) n+1 n + 1 n n + 1 n−1 ∗Corresponding author Email addresses: wksh [email protected] (Wen-Kai Shao), [email protected],[email protected] (Yuan He), [email protected] (Jing Pan) Received 2016-02-28 W.-K.
    [Show full text]
  • Arxiv:1705.01190V1 [Math.CA]
    Noname manuscript No. (will be inserted by the editor) Uniform asymptotic expansions for Laguerre polynomials and related confluent hypergeometric functions T. M. Dunster · A. Gil · J. Segura Received: date / Accepted: date Abstract Uniform asymptotic expansions involving exponential and Airy (α) functions are obtained for Laguerre polynomials Ln (x), as well as comple- mentary confluent hypergeometric functions. The expansions are valid for n large and α small or large, uniformly for unbounded real and complex values of (α) x. The new expansions extend the range of computability of Ln (x) compared to previous expansions, in particular with respect to higher terms and large values of α. Numerical evidence of their accuracy for real and complex values of x is provided. Keywords Asymptotic expansions Laguerre polynomials Confluent hypergeometric functions Turning point· theory, WKB methods· Numerical computation · · Mathematics Subject Classification (2000) MSC 34E05 33C45 33C15 34E20 33F05 · · · · The authors acknowledge support from Ministerio de Econom´ıa y Competitividad, project MTM2015-67142-P (MINECO/FEDER, UE). T. M. Dunster Department of Mathematics and Statistics San Diego State University. 5500 Campanile Drive San Diego, CA, USA. E-mail: [email protected] A. Gil Departamento de Matem´atica Aplicada y CC. de la Computaci´on. ETSI Caminos. Universidad de Cantabria. 39005-Santander, Spain. arXiv:1705.01190v1 [math.CA] 2 May 2017 E-mail: [email protected] J. Segura Departamento de Matem´aticas, Estad´ıstica y Computaci´on. Universidad de Cantabria. 39005-Santander, Spain. E-mail: [email protected] 2 T. M. Dunster et al. 1 Introduction In this paper we shall obtain computable asymptotic expansions for Laguerre (α) polynomials Ln (x), for the case n large and α small or large.
    [Show full text]
  • On a Different Kind of D-Orthogonal Polynomials That Generalize the Laguerre Polynomials 1 Introduction
    Mathematica Aeterna, Vol. 2, 2012, no. 6, 561 - 572 On a different kind of d-orthogonal polynomials that generalize the Laguerre polynomials Serhan Varma Ankara University, Faculty of Science, Department of Mathematics TR-06100 Tando˘gan Ankara, Turkey [email protected] Fatma Ta¸sdelen Ankara University, Faculty of Science, Department of Mathematics TR-06100 Tando˘gan Ankara, Turkey [email protected] Abstract The purpose of this work is to give an another generalization of the Laguerre polynomials in the context of d-orthogonality by a generat- ing function of a certain form. We derive the d-dimensional functional vector for which the d-orthogonality holds. Some properties of the ob- tained polynomials are determined: expilicit representation, relation with a known polynomial, a recurrence relation of order-(d + 1) and a differential equation of order-(d + 1). Mathematics Subject Classification: 33C45, 42C05. Keywords: d-Orthogonal polynomials, Generating functions, d-Dimensional functionals, Recurrence relations, Laguerre polynomials. 1 Introduction Lately, there has been an accelerating interest in extensions of the notion of orthogonal polynomials. One of them is the multiple orthogonality [1 − 4]. This notion has many applications in various fields of mathematics as approx- imation theory, spectral theory of operators and special functions theory. A subclass of multiple orthogonal polynomials known as d-orthogonal polynomi- als, introduced by Van Iseghem [23] and Maroni [20]. Let P be the linear space of polynomials with complex coefficients and ′ let P be its algebraic dual. A polynomial sequence {Pn}n≥0 in P is called a polynomial set (PS for short) if and only if deg Pn = n for all non-negative 562 S.
    [Show full text]
  • Certain Sequences and Its Integral Representations in Terms Of
    Global Journal of Science Frontier Research: F Mathematics and Decision Sciences Volume 15 Issue 5 Version 1.0 Year 2015 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 2249-4626 & Print ISSN: 0975-5896 Certain Sequences and its Integral Representations in Terms of Laguerre Polynomials By Baghdadi Aloui Abstract- In this paper, we introduce a connection formula between the monomial basis and the shifted Laguerre basis. As an application, some integral representations in terms of Laguerre polynomials for certain sequences are obtained. Keywords: Laguerre polynomials, special functions, integral formulas. GJSFR-F Classification : MSC 2010: Primary 33C45; Secondary 42C05 Certain SequencesanditsIntegralRepresentationsinTermsofLaguerrePolynomials Strictly as per the compliance and regulations of : © 2015. Baghdadi Aloui. This is a research/review paper, distributed under the terms of the Creative Commons Attribution- Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. Notes Certain Sequences and its Integral Representations in Terms of Laguerre 2015 r ea Polynomials Y 1 Baghdadi Aloui Abstract- In this paper, we introduce a connection formula between the monomial basis and the shifted Laguerre basis. V As an application, some integral representations in terms of Laguerre polynomials for certain sequences are obtained. V Keywords: Laguerre polynomials, special functions, integral formulas. ue ersion I s s I. Introduction and Main Results I By using some special functions and some particular integrals, we recall some integral rep- XV resentations for certain integer (or real) sequences. a) Some special functions ) The Gamma function is defined by the definite integral F ) Z +1 Γ(z) = xz−1e−x dx; <e(z) > 0: 0 Research Volume We can see directly, that Γ(1) = 1, and using integration by parts, that Γ(z + 1) = zΓ(z).
    [Show full text]
  • Some Identities of Laguerre Polynomials Arising from Differential Equations Taekyun Kim1,2,Daesankim3, Kyung-Won Hwang4* Andjongjinseo5
    Kim et al. Advances in Difference Equations (2016)2016:159 DOI 10.1186/s13662-016-0896-1 R E S E A R C H Open Access Some identities of Laguerre polynomials arising from differential equations Taekyun Kim1,2,DaeSanKim3, Kyung-Won Hwang4* andJongJinSeo5 *Correspondence: [email protected] Abstract 4Department of Mathematics, Dong-A University, Busan, 49315, In this paper, we derive a family of ordinary differential equations from the generating Republic of Korea function of the Laguerre polynomials. Then these differential equations are used in Full list of author information is order to obtain some properties and new identities for those polynomials. available at the end of the article MSC: 05A19; 33C45; 11B37; 35G35 Keywords: Laguerre polynomials; differential equations 1 Introduction The Laguerre polynomials, Ln(x)(n ≥ ), are defined by the generating function xt ∞ e– –t = L (x)tn (see [, ]). () –t n n= Indeed, the Laguerre polynomial Ln(x) is a solution of the second order linear differential equation xy +(–x)y + ny = (see[–]). () From (), we can get the following equation: ∞ xt ∞ e– –t (–)mxmtm L (x)tn = = ( – t)–m– n –t m! n= m= ∞ ∞ (–)mxmtm m + l = tl m! l m= l= ∞ n (–)m n xm = m tn.() m! n= m= Thus by (), we get immediately the following equation: n (–)m n xm L (x)= m (n ≥ ) see [, –] .() n m! m= © 2016 Kim et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
    [Show full text]
  • Arxiv:1509.06465V1 [Math.CV] 22 Sep 2015 (1.3) Uin Fscn-Re Ieeta Qain,Woesadr F (1.1) Standard Whose Equations, Differential Second-Order of Lutions 1.1
    SOME NEW PROPERTIES OF CONFLUENT HYPERGEOMETRIC FUNCTIONS XU-DAN LUO AND WEI-CHUAN LIN Abstract. The confluent hypergeometric functions (the Kummer functions) ∞ (α)n n defined by 1F1(α; γ; z) := z (γ =6 0, −1, −2, ··· ), which are of Pn=0 n!(γ)n many properties and great applications in statistics, mathematical physics, engineering and so on, have been given. In this paper, we investigate some new properties of 1F1(α; γ; z) from the perspective of value distribution the- ory. Specifically, two different growth orders are obtained for α ∈ Z≤0 and α 6∈ Z≤0, which are corresponding to the reduced case and non-degenerated case of 1F1(α; γ; z). Moreover, we get an asymptotic estimation of character- 1F1′ (α;γ;z) istic function T (r, 1F1(α; γ; z)) and a more precise result of m r, , 1F1(α;γ;z) compared with the Logarithmic Derivative Lemma. Besides, the distribution of zeros of the confluent hypergeometric functions is discussed. Finally, we show how a confluent hypergeometric function and an entire function are uniquely determined by their c-values. 1. Introduction 1.1. Background. It is a well-known fact that some special functions are the so- lutions of second-order differential equations, whose standard form is d2y dy (1.1) + p(z) + q(z)y(z)=0, dz2 dz where p(z) and q(z) are given complex-valued functions. The singularities of (1.1) may be those of solutions, and solutions are analytic at the ordinary points of (1.1). Definition 1.1. If z = z0 is a singularity of (1.1), then z = z0 is a regular singu- larity of (1.1) if and only if 2 (1.2) (z z0) p(z), (z z0) q(z) arXiv:1509.06465v1 [math.CV] 22 Sep 2015 − · − · are analytic in z : z z < R , where R is a positive real number.
    [Show full text]
  • Effective Laguerre Asymptotics
    Effective Laguerre asymptotics D. Borwein,∗ Jonathan M. Borwein† and Richard E. Crandall‡ May 28, 2008 Abstract It is known that the generalized Laguerre polynomials can enjoy subexponential growth for large primary index. In particular, for certain fixed parameter pairs (a, z) one has the large-n asymptotic behavior √ (−a) −a/2−1/4 2 nz Ln (−z) ∼ C(a, z)n e . We introduce a computationally motivated contour integral that allows efficient numerical Laguerre evaluations, yet also leads to the complete asymptotic series over the full parameter domain of subex- ponential behavior. We present a fast algorithm for symbolic generation of the rather formidable expansion coefficients. Along the way we address the difficult problem of establishing effective (i.e. rigorous and explicit) error bounds on the general expansion. A primary tool for these developments is an “exp-arc” method giving a natural bridge between converging series and effective asymptotics. 1 The challenge of “effectiveness” For primary integer indices n = 0, 1, 2,... we define the Laguerre polynomial thus: n X n − azk L(−a)(−z) := . (1) n n − k k! k=0 Our use of negated parameters −a, −z is intentional, for convenience in our analysis and in connection with related research, as we later explain. We shall work on the difficult problem of establishing asymptotics with effective error bounds for the two-parameter domain D := {(a, z) ∈ C × C : z 6∈ (−∞, 0]}. That is, a is any complex number, while z is a complex number not on the negative-closed cut (−∞, 0]. Herein, we say a function f(n) is of subexponential growth if log f(n) ∼ CnD as n → ∞ for positive constants C, D with D < 1.
    [Show full text]
  • UNIT 3 LEGENDRE, HERMITE and Laguerre Polynomials LAGUERRE POLYNOMIALS
    Legendre, Hermite and UNIT 3 LEGENDRE, HERMITE AND Laguerre Polynomials LAGUERRE POLYNOMIALS Structure Page No. 3.1 Introduction 83 Objectives 3.2 Legendre Polynomials 84 Rodrigue’s Formula Generating Function Recurrence Relations Orthogonality Property Legendre Function of the Second Kind 3.3 Hermite and Laguerre Polynomials 100 3.4 Applications to Physical Situations 106 3.5 Summary 111 3.6 Solutions/Answers 112 3.1 INTRODUCTION In Unit 2, we discussed methods of finding series solutions of second order, linear, homogeneous differential equations with variable coefficients. In this unit and the unit to follow, we shall apply these methods to a few special second order equations occurring frequently in applied mathematics, engineering and physics. These equations are Legendre’s, Bessel’s, Hermite’s and Laguerre’s equations. The solution to these equations, that occur in applications, are referred to as special functions . They are called ‘special’ as they are different from the standard functions like sine, cosine, exponential, logarithmic etc. In this unit, we shall concentrate on Legendre, Hermite and Laguerre polynomials which are polynomial solutions to Legendre’s, Hermite’s and Laguerre’s differential equations. Legendre polynomials first arose in the problem of expressing the Newtonian potential of a conservative force field in an infinite series involving the distance variable of two points and their included central angle. Other similar problems dealing with either gravitational potential or electrostatic potentials and steady-state heat conduction problems in spherical solids, also lead to Legendre polynomials. Other polynomials which commonly occur in applications are the Hermite and Laguerre. They play an important role in quantum mechanics, and in probability theory.
    [Show full text]